tag:blogger.com,1999:blog-59817063655006040462024-02-07T01:01:05.307-08:00A Medley of MathematicsJessehttp://www.blogger.com/profile/13804151393945151752noreply@blogger.comBlogger47125tag:blogger.com,1999:blog-5981706365500604046.post-71188161244939120072011-06-07T08:32:00.000-07:002011-06-07T09:14:07.201-07:00Mind-reading card trickAlice draws 5 cards from a deck, keeps one secret, and shows the other 4 to Bob in a particular order that allows him to guess the secret card. To anyone observing who doesn't know the trick, it looks like Bob is reading Alice's mind.<div><br /></div><div>This trick works because of two facts:</div><div><ul><li>When Alice draws a 5-card hand from a deck of cards, she must get at least two cards of the same suit.</li><li>Think of a deck of cards as going from 1 (ace) to 13 (king) and put these numbers around a clock. The farthest apart any two cards can be is 6 steps, so long as you start on the correct card (it takes 7 steps to get from 13 to 7 but only 6 steps to get from 7 to 13).</li></ul><div>Alice draws her hand and chooses two cards in the same suit. One of these is the "base" card and the other is the secret card. It must be possible to get from the base card to the secret card in 6 or fewer steps.</div><div><br /></div><div>Alice shows Bob the base card. Now Bob knows the suit of the secret card, and he knows he has to count upward some amount from the base card to find the number of the secret card.</div></div><div><br /></div><div>To tell Bob how much to count, Alice hands Bob her three remaining cards in the appropriate order:</div><div>low medium high - count up 1 step</div><div>low high medium - count up 2</div><div>medium low high - count up 3</div><div>medium high low - count up 4</div><div>high low medium - count up 5</div><div>high medium low - count up 6</div><div><br /></div><div>If Alice has multiple cards with the same number, she orders those cards by suit. From lowest to highest, the suits go Clubs, Diamonds, Hearts, Spades.</div><div><br /></div><div>Here are a couple of examples.</div><div><br /></div><div><b>Example 1</b></div><div><br /></div><div>Alice draws: A clubs, 4 diamonds, 5 hearts, 6 spades, 7 spades. </div><div><br /></div><div>Since spades is the only suit in which she has 2 cards, she has to use spades for the base card and the secret card. Since it takes only 1 step to go from 6 to 7, she shows Bob the 6 of spades as the base card.</div><div><br /></div><div>Her remaining cards are A clubs (low), 4 diamonds (medium), 5 hearts (high). She shows Bob these cards in this order.</div><div><br /></div><div>Bob knows he must step up 1 from the 6 of spades, so the secret card must be the 7 of spades.</div><div><br /></div><div><b>Example 2</b></div><div><br /></div><div>Alice draws: K hearts, 4 hearts, 2 clubs, 2 diamonds, J spades.</div><div><br /></div><div>Since hearts is the only suit in which she has 2 cards, the secret card and the base card must be hearts. The base card has to be the king of hearts, since we can get from K to 4 in 5 steps but it takes 9 steps to get from 4 to K.</div><div><br /></div><div>The remaining cards are 2 clubs (low), 2 diamonds (medium), J spades (high). She needs to tell Bob to step up 5, so she shows him the cards in the order</div><div style="text-align: center;">J spades (high), 2 clubs (low), 2 diamonds (medium)</div><div style="text-align: left;"><br /></div><div style="text-align: left;">Bob knows to step up 5 from the king of hearts, so the secret card must be the 4 of hearts.</div>Jessehttp://www.blogger.com/profile/13804151393945151752noreply@blogger.com0tag:blogger.com,1999:blog-5981706365500604046.post-83640886841305262252011-06-06T09:57:00.001-07:002011-06-06T10:36:49.218-07:00Cryptology - the final foray<div style="text-align: left;"><span class="Apple-style-span" style="font-size:medium;">This was my favorite day. During the previous five weeks of the EB I felt that I was having trouble finding the optimal working arrangement for this group - they didn't want to work alone, or in pairs, or in small groups, or all together with me at the chalkboard. What to do?</span></div><div><span class="Apple-style-span" style="font-size:medium;"><br /></span></div><div><span class="Apple-style-span" style="font-size:medium;">On week six I handed them these rules:</span></div><div><ul><li><span class="Apple-style-span" style="font-size:medium;">If there are prizes, they must be divided evenly.</span></li><li><span class="Apple-style-span" style="font-size:medium;">Cooperate! Help each other! Also, make sure everyone gets to help.</span></li><li><span class="Apple-style-span" style="font-size:medium;">Leave the room as you found it.</span></li></ul></div><div><span class="Apple-style-span" style="font-size:medium;">Then I handed them a box labeled "Spy Tools" containing alphabet clocks, enigma machines, sheets with the alphabet for solving substitution ciphers, and some odd-looking index cards with weird holes cut out.</span></div><div><br /></div><div><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhBZO-eT-IF2aQxI4DDHGdOSppwcA8u-wdkctWGQqTouWVP9ONqJJ_SsbQWX8ViO4MLVrLUn7MLBcoIzBUbzLU-f2oPfj0mEZ0mYrD2FmWSzxICvVL1tBwwJY9D5BZPvO-gdZoFwxcHSfc/s1600/2start.JPG" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhBZO-eT-IF2aQxI4DDHGdOSppwcA8u-wdkctWGQqTouWVP9ONqJJ_SsbQWX8ViO4MLVrLUn7MLBcoIzBUbzLU-f2oPfj0mEZ0mYrD2FmWSzxICvVL1tBwwJY9D5BZPvO-gdZoFwxcHSfc/s320/2start.JPG" border="0" alt="" id="BLOGGER_PHOTO_ID_5615153081359249378" style="display: block; margin-top: 0px; margin-right: auto; margin-bottom: 10px; margin-left: auto; text-align: center; cursor: pointer; width: 320px; height: 214px; " /></a></div><div><br /></div><div style="text-align: left;"><span class="Apple-style-span" style="font-size:medium;">Then I handed them the first clue and stood back. </span></div><div style="text-align: left;"><span class="Apple-style-span" style="font-size:medium;"><br /></span></div><p style="text-align: center;margin-bottom: 0in; "><span style="font-family:Courier, monospace;"><b><span class="Apple-style-span" style="font-size: x-large;">1. Easiest first! </span></b></span><span class="Apple-style-span" style="font-family:Courier, monospace;"><b><span class="Apple-style-span" style="font-size: x-large;">XQGHU WKH WUDVK FDQ</span></b></span></p> <div><br /></div><div><span class="Apple-style-span" style="font-size:medium;">They correctly figured out that "easiest first" referred to the Caesar cipher, the easiest code we had learned. They found the next clue UNDER THE TRASH CAN:</span></div><div><br /></div><div><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjeBxQq7iW61poVnTnivyEnV4MNTpccGNj8SfKbscL8_r8W0rNE7vjPbVvcOoZsTr_0UDT17Jmd81L-Tcz5MYNbLpIpRCoK71L06ZpGMgeAe9P7rX5bZtaUJ7IyoeCnR5kI8RU5xOYQme8/s1600/3trash.JPG" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjeBxQq7iW61poVnTnivyEnV4MNTpccGNj8SfKbscL8_r8W0rNE7vjPbVvcOoZsTr_0UDT17Jmd81L-Tcz5MYNbLpIpRCoK71L06ZpGMgeAe9P7rX5bZtaUJ7IyoeCnR5kI8RU5xOYQme8/s320/3trash.JPG" border="0" alt="" id="BLOGGER_PHOTO_ID_5615153091871703682" style="display: block; margin-top: 0px; margin-right: auto; margin-bottom: 10px; margin-left: auto; text-align: center; cursor: pointer; width: 320px; height: 214px; " /></a></div><div><br /></div><div><p style="text-align: center;margin-bottom: 0in; "><span style="font-family:Courier, monospace;"><b><span class="Apple-style-span" style="font-size:x-large;"><span class="Apple-style-span" style="font-size: x-large;">2</span><span class="Apple-style-span" style="font-family:'courier new';"><span class="Apple-style-span" style="font-size: x-large;">. The mystery starts with AAA. </span></span></span></b></span><b><span class="Apple-style-span" style="font-family:'courier new';"><span class="Apple-style-span" style="font-size: x-large;">AIHFACJQMBJOLYW</span></span></b></p> </div><div><br /></div><div><span class="Apple-style-span" style="font-size:medium;">"Starts with AAA" indicated that they needed to use an Enigma machine with all its rotors initially set to A.</span></div><div><span class="Apple-style-span" style="font-size:medium;"><br /></span></div><div><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhXvkStWWS7kAZCBB9RHGx1FQKEBjG6rkahnaerLkISAbMg2s_c5u2wQxigcwwq9LvDSs6jUthwF5xU-DUw1hNJkcoZi42MDpC-jFLGepXBe9ST7cUZXQ7IaKmrWYI0pV_UA_BEYtBhdoo/s1600/4enigma.JPG" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhXvkStWWS7kAZCBB9RHGx1FQKEBjG6rkahnaerLkISAbMg2s_c5u2wQxigcwwq9LvDSs6jUthwF5xU-DUw1hNJkcoZi42MDpC-jFLGepXBe9ST7cUZXQ7IaKmrWYI0pV_UA_BEYtBhdoo/s320/4enigma.JPG" border="0" alt="" id="BLOGGER_PHOTO_ID_5615153098365096306" style="text-align: left;display: block; margin-top: 0px; margin-right: auto; margin-bottom: 10px; margin-left: auto; cursor: pointer; width: 320px; height: 214px; " /></a></div><div><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhnGhzepcpP-BsvRhRonnZCW60huBBKm_REkkDldGTaUEYnHhU5HUC6GblDN9zW9YorTcyVxWZ-JImu-qkDOTHRrp_NP1MpioDoc6KvCzaIej0CUDRvjL9nprMiJ1p4AmIi22oVhc5UwCE/s1600/5enigma.JPG" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhnGhzepcpP-BsvRhRonnZCW60huBBKm_REkkDldGTaUEYnHhU5HUC6GblDN9zW9YorTcyVxWZ-JImu-qkDOTHRrp_NP1MpioDoc6KvCzaIej0CUDRvjL9nprMiJ1p4AmIi22oVhc5UwCE/s320/5enigma.JPG" border="0" alt="" id="BLOGGER_PHOTO_ID_5615153102523866258" style="text-align: left;display: block; margin-top: 0px; margin-right: auto; margin-bottom: 10px; margin-left: auto; cursor: pointer; width: 214px; height: 320px; " /></a></div><div><span class="Apple-style-span" style="font-size:medium;"><br /></span></div><div><span class="Apple-style-span" style="font-size:medium;">The next clue was found on the BOTTOMBOOKSHELF.</span></div> <div><br /></div><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjCO7WDEL2DLRZIb8aLBvhu4hHLTQTtd9m0sy2VW8PQcyixd2ThTpcZuBLD-WlXH02YpfkAeLI1P2O_BDmKf2BlMBKruDigo7_Tz90GcJ6_oq3Es67sb1LETtklCsqp9RWoUvyb8E2OOko/s1600/6bookshelf.JPG" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;width: 320px; height: 214px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjCO7WDEL2DLRZIb8aLBvhu4hHLTQTtd9m0sy2VW8PQcyixd2ThTpcZuBLD-WlXH02YpfkAeLI1P2O_BDmKf2BlMBKruDigo7_Tz90GcJ6_oq3Es67sb1LETtklCsqp9RWoUvyb8E2OOko/s320/6bookshelf.JPG" border="0" alt="" id="BLOGGER_PHOTO_ID_5615153112886943570" /></a><span class="Apple-style-span" style=" ;font-family:Courier, monospace;font-size:27px;"><div style="text-align: center;"><span class="Apple-style-span" style="font-size:x-large;"><b><br /></b></span></div><div style="text-align: center;"><b><span class="Apple-style-span" style="font-family:'courier new';"><span class="Apple-style-span" style="font-size: x-large;">3. VMVVPNPMFGQVGGQCETDI - Jesse</span></span></b></div><div style="text-align: left;"><span class="Apple-style-span" style="font-family:georgia;"><span class="Apple-style-span" style="font-size: medium;"><br /></span></span></div><div style="text-align: left;"><span class="Apple-style-span" style="font-family:georgia;"><span class="Apple-style-span" style="font-size: medium;">This one didn't even hint at which method was needed, but since they hadn't used the alphabet clock yet they decided to try that, with my name as the key. It worked. </span></span></div><div style="text-align: left;"><span class="Apple-style-span" style="font-family:Georgia, serif;"><span class="Apple-style-span" style="font-family:georgia;"><span class="Apple-style-span" style="font-size: medium;"><br /></span></span></span></div><div style="text-align: left;"><span class="Apple-style-span" style=" ;font-family:Georgia, serif;font-size:medium;"><span class="Apple-style-span" style=" ;font-size:16px;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiHMUDVJ7FzDJwsD_LAjX-5VdB_clGaIYMOfYZ4nl0xtpXN8Zzz1_jZq1oV2QbSIsYGWghJx1VGi9Pk7G8FnS5UtFCjF8-9j9PxugOkkNr8ehOPiAT0HnAc0iWsDoqxkqgZOM74lng2NMI/s1600/10decoding.JPG" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiHMUDVJ7FzDJwsD_LAjX-5VdB_clGaIYMOfYZ4nl0xtpXN8Zzz1_jZq1oV2QbSIsYGWghJx1VGi9Pk7G8FnS5UtFCjF8-9j9PxugOkkNr8ehOPiAT0HnAc0iWsDoqxkqgZOM74lng2NMI/s320/10decoding.JPG" border="0" alt="" id="BLOGGER_PHOTO_ID_5615156525955197826" style="display: block; margin-top: 0px; margin-right: auto; margin-bottom: 10px; margin-left: auto; text-align: center; cursor: pointer; width: 320px; height: 214px; " /></a></span></span></div><div style="text-align: left;"><span class="Apple-style-span" style=" ;font-family:Georgia, serif;font-size:medium;"><br /></span></div><div style="text-align: left;"><span class="Apple-style-span" style="font-family:Georgia, serif;"><span class="Apple-style-span" style="font-size: medium;"> The next clue was found under the MIDDLELUNCHROOMTABLE.</span></span></div><div style="text-align: left;"><span class="Apple-style-span" style="font-family:Georgia, serif;"><span class="Apple-style-span" style="font-size: medium;"><br /></span></span></div><div style="text-align: left;"><span class="Apple-style-span" style="font-family: Georgia, serif; "><b><span class="Apple-style-span" style="font-size: x-large;"><span class="Apple-style-span" style="font-family:'courier new';">4. Congratulations! Sorry there aren't any prizes, but at least you're not getting painted red, boxed up, and given as an inter-office Christmas gift.</span></span></b></span></div><div style="text-align: left;"><span class="Apple-style-span" style="font-family: Georgia, serif; "><span class="Apple-style-span" style="font-family: georgia; font-size: medium; "><br /></span></span></div><div style="text-align: left;"><span class="Apple-style-span" style="font-family: Georgia, serif; "><span class="Apple-style-span" style="font-family: georgia; font-size: medium; "></span><span class="Apple-style-span" style="font-family: georgia; font-size: medium; ">This is very unhelpful! However, when they held one of those funny-looking index cards over it, the cut-out spaces revealed this message:</span></span></div><div style="text-align: left;"><span class="Apple-style-span" style="font-family: Georgia, serif; "><span class="Apple-style-span" style="font-family: georgia; font-size: medium; "><br /></span></span></div><div style="text-align: left;"><b><span class="Apple-style-span" style="font-family:'courier new';"><span class="Apple-style-span" style="font-size: x-large;">4. Congratulations! Sorry there aren't any </span></span><span class="Apple-style-span" style="color:#FF0000;"><span class="Apple-style-span" style="font-family:'courier new';"><span class="Apple-style-span" style="font-size: x-large;">prizes</span></span></span><span class="Apple-style-span" style="font-family:'courier new';"><span class="Apple-style-span" style="font-size: x-large;">, but at least you're not getting pa</span></span><span class="Apple-style-span" style="color:#FF0000;"><span class="Apple-style-span" style="font-family:'courier new';"><span class="Apple-style-span" style="font-size: x-large;">in</span></span></span><span class="Apple-style-span" style="font-family:'courier new';"><span class="Apple-style-span" style="font-size: x-large;">ted </span></span><span class="Apple-style-span" style="color:#FF0000;"><span class="Apple-style-span" style="font-family:'courier new';"><span class="Apple-style-span" style="font-size: x-large;">red</span></span></span><span class="Apple-style-span" style="font-family:'courier new';"><span class="Apple-style-span" style="font-size: x-large;">, </span></span><span class="Apple-style-span" style="color:#FF0000;"><span class="Apple-style-span" style="font-family:'courier new';"><span class="Apple-style-span" style="font-size: x-large;">b</span></span></span><span class="Apple-style-span" style="font-family:'courier new';"><span class="Apple-style-span" style="font-size: x-large;">oxed up, </span></span><span class="Apple-style-span" style="color:#FF0000;"><span class="Apple-style-span" style="font-family:'courier new';"><span class="Apple-style-span" style="font-size: x-large;">a</span></span></span><span class="Apple-style-span" style="font-family:'courier new';"><span class="Apple-style-span" style="font-size: x-large;">nd </span></span><span class="Apple-style-span" style="color:#FF0000;"><span class="Apple-style-span" style="font-family:'courier new';"><span class="Apple-style-span" style="font-size: x-large;">g</span></span></span><span class="Apple-style-span" style="font-family:'courier new';"><span class="Apple-style-span" style="font-size: x-large;">iven as an </span></span><span class="Apple-style-span" style="color:#FF0000;"><span class="Apple-style-span" style="font-family:'courier new';"><span class="Apple-style-span" style="font-size: x-large;">in</span></span></span><span class="Apple-style-span" style="font-family:'courier new';"><span class="Apple-style-span" style="font-size: x-large;">ter-</span></span><span class="Apple-style-span" style="color:#FF0000;"><span class="Apple-style-span" style="font-family:'courier new';"><span class="Apple-style-span" style="font-size: x-large;">office</span></span></span><span class="Apple-style-span" style="font-family:'courier new';"><span class="Apple-style-span" style="font-size: x-large;"> Christmas gift.</span></span></b></div><div style="text-align: left;"><span class="Apple-style-span" style=" ;font-family:Georgia, serif;font-size:medium;"><span style="font-family:Courier, monospace;"><div><span class="Apple-style-span" style="font-family:georgia;"><span class="Apple-style-span" style="font-size: medium;"><br /></span></span></div><div><span class="Apple-style-span" style="font-family: georgia; font-size: medium; ">They found the PRIZES IN RED BAG IN OFFICE as advertised. One bottle of bubbles, one grow-beast dinosaur, and two pencil erasers for everyone. Max took charge of the prize distribution in a very effective way, declaring that dinosaurs would be picked from youngest to oldest. It was subsequently decided that pencil erasers would be chosen first from youngest to oldest, then from oldest to youngest.</span></div><div><span class="Apple-style-span" style="font-family:georgia;"><span class="Apple-style-span" style="font-size: medium;"><br /></span></span></div><div><span class="Apple-style-span" style="font-family:Courier, monospace;"><span class="Apple-style-span" style=" ; serif;font-size:16px;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjHQGMi4sVfCf8MWmlaos5zFqBow02jmUeEecfyXhasC5Lu96i54dtopmZTsR8CYM-wWsLel8RSuk90XlKghtOV0rjU7CBFaufQPEACTdmoudlIOtqMLWCMTy_-kXiV7Eycn6mKqLhJBd0/s1600/11prizedistribution.JPG" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjHQGMi4sVfCf8MWmlaos5zFqBow02jmUeEecfyXhasC5Lu96i54dtopmZTsR8CYM-wWsLel8RSuk90XlKghtOV0rjU7CBFaufQPEACTdmoudlIOtqMLWCMTy_-kXiV7Eycn6mKqLhJBd0/s320/11prizedistribution.JPG" border="0" alt="" id="BLOGGER_PHOTO_ID_5615156538146602610" style="display: block; margin-top: 0px; margin-right: auto; margin-bottom: 10px; margin-left: auto; text-align: center; cursor: pointer; width: 320px; height: 214px; " /></a></span></span></div></span></span><span style="font-family:Courier, monospace;"></span><p></p> </div></span>Jessehttp://www.blogger.com/profile/13804151393945151752noreply@blogger.com1tag:blogger.com,1999:blog-5981706365500604046.post-47003066041703841722011-06-06T09:32:00.001-07:002011-06-06T09:56:57.538-07:00Catching up on the blog!<div><b>In which we build all the Platonic solids and do one real mathematical proof:</b></div><div><br /></div><div>With the big group, we spent a lot of time playing with zome tools to build the <a href="http://en.wikipedia.org/wiki/Platonic_solid">Platonic solids</a>. Platonic solids are <i>convex regular polyhedra</i>. Here's what that mouthful means:</div><div><ul><li>A poly<i>hedron</i> is a 3-D object, as opposed to a poly<i>gon</i>, which is a 2-D object. </li><li><i>Regular</i> in this case means "all the same." A regular polygon is a polygon where all the sides and angles are the same length, such as an equilateral triangle or a square. A regular polyhedron is one for which all the sides are the same regular polygon. </li><li>A polygon or polyhedron is <i>convex</i> if, when we pick two points inside it and connect them by the shortest straight line, the line is entirely contained inside the polygon or polyhedron.</li></ul></div><div>There are three different Platonic solids whose faces are triangles (tetrahedron, octahedron, and icosahedron), one whose faces are squares (cube), and one whose faces are pentagons (dodecahedron). We tried to build one whose faces were hexagons but we ended up with a soccer ball instead - some faces were hexagons, some faces were pentagons.</div><div><br /></div><div>We talked about what a<i> proof</i> is. To a mathematician, a proof is an argument that convinces a reasonable listener of some claim. I made the claim that we had in fact built all the Platonic solids there were to be built, then went through the <a href="http://en.wikipedia.org/wiki/Platonic_solid#Geometric_proof">geometric proof</a> of this claim. Most of the reasonable listeners were convinced.</div><div><br /></div><div><b>In which we do calculus:</b></div><div><br /></div><div>With the small group, I had gotten some requests to do calculus, so we did. </div><div><br /></div><div>We talked a little about limits: the sequence 1, 1/2, 1/3, 1/4, ... has a limit of 0. The sequence 1, 1, 1,... has a limit of 1.</div><div><br /></div><div>We found the area of a circle by filling the circle with concentric rings (using yarn or playdoh), then unrolling the rings into a triangle and finding the area of the triangle, which is the same as the area of the circle. This is explained beautifully at the website <a href="http://betterexplained.com/articles/a-gentle-introduction-to-learning-calculus/">betterexplained.com</a>.</div><div><br /></div><div>We had a refresher of what "slope" means. I like the phrase </div><div style="text-align: center;">"slope equals rise over run"</div><div> since I think it's easier to remember "rise over run" than it is to remember whether x or y goes on top. Then we tried to figure out what "slope" should mean if the line isn't straight. We found the slope of the function y=x^2 pretty much as you would do it in a calculus class, by calculating the slope of the line between the points </div><div style="text-align: center;">(x,x^2) and (x+delta x, (x+delta x)^2)</div><div style="text-align: left;">and talking about what happens as delta x gets smaller and smaller.</div><div style="text-align: left;"><br /></div><div style="text-align: left;"><b>In which we do origami:</b></div><div style="text-align: left;"><b><br /></b></div><div style="text-align: left;">When the schedules were crazy with field trips and such, I caved and let them do origami again. Rabbits and turtles and boxes, oh my. The small group did some modular origami and we talked about what "modular" means.</div>Jessehttp://www.blogger.com/profile/13804151393945151752noreply@blogger.com0tag:blogger.com,1999:blog-5981706365500604046.post-9664903195649003332011-05-13T10:03:00.000-07:002011-05-13T10:07:49.137-07:00Creating Hydrogen and Oxygen from Water<span class="Apple-style-span" style="font-family: arial, sans-serif; font-size: 13px; border-collapse: collapse; "><div>Renata blogged about the <a href="http://renatas-class.blogspot.com/2011/04/food-science-experiments.html">electrolysis demonstration</a> Zach did for her class. Here are his directions for a version of that experiment that the kids can do at home.</div><div><br /></div>Simple Electrolysis Experiment<br /><br />This is a smaller scale experiment than what we did in class. This can be safely done at home, and will not involve the collection of the gasses.<br /><br />Here are the items you will need.<br /><br />1: Glass container (a jam jar works well here).<br />2: Two pencils<br />3: A piece of cardboard slightly bigger than the glass.<br />4: Two pieces of thin electrical wire about 8-10 inches in length.<br />5: Electrical tape.<br />6: Epsom Salt (this is NOT table salt. Ask your parents!)<br />7: A 9 volt battery.<br /><br />Experiment steps:<br /><br />1: Remove the erasers from the pencils and sharpen both ends.<br /><br />2: Attach one wire (about 8-10 inches long) to each pencil by wrapping the exposed wire around one end of the pencil, and using the electrical tape to secure it.<br /><br />3: Fill the glass container about 3/4 of the way with water.<br /><br />4: Mix several tablespoons of the Epsom salt in with the water. You want to saturate the solution, so keep mixing until no more will dissolve. Do this about 1 tablespoon at a time.<br /><br />5: Put the cardboard piece on top of the glass, and poke two holes about 2 inches apart. Push the end of each pencil that does not have the wire attached through the holes, and into the water.<br /><br />6: Finally, attach the other end of the wire to the 9 volt battery with the electrical tape to help hold them in place.<br /><br />At this point you will see bubbles forming at the tips of the pencils in the water. If you look closely, one will be forming bubbles more quickly than the other. This is the Hydrogen. The one with the fewer<br />bubbles is the Oxygen.<br /><br />That's it! You have successfully split hydrogen and oxygen from water.</span>Jessehttp://www.blogger.com/profile/13804151393945151752noreply@blogger.com0tag:blogger.com,1999:blog-5981706365500604046.post-10963240083397359012011-05-11T10:54:00.000-07:002011-05-16T09:35:23.546-07:00Cryptology - Enigma Machine<div style="text-align: left;">Today we learned that the Germans used a device called the Engima machine to encrypt their messages during WWII. They thought it was completely secure, but the Allies broke the code!</div><div><br /></div><div>We watched <a href="http://www.youtube.com/watch?v=DnBsndE1IkA">this video</a> about how the Engima machine works. The general idea is that when you encode a letter, it goes through three different substitution ciphers, bounces off a "reflector," and then goes back through those same three substitution ciphers again. Each substitution cipher is on a rotor. Each time you encrypt a new letter, at least one rotor moves, so you end up using a different set of substitution ciphers for each letter!</div><div><br /></div><div><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjvIBt22WzMzgFHZUFbhF1RIfsIMalwDIdHTA-Wj8ECTT5nlIVxs2malBLaiHkES3AhgMCplsMuxxjC7yRzNcLI77GhTUDo1dyEMry4Bt_8mZKvjj_pT0JdN56JxHYcRettqQrJB5GJP-I/s1600/enigmaboard.jpg" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjvIBt22WzMzgFHZUFbhF1RIfsIMalwDIdHTA-Wj8ECTT5nlIVxs2malBLaiHkES3AhgMCplsMuxxjC7yRzNcLI77GhTUDo1dyEMry4Bt_8mZKvjj_pT0JdN56JxHYcRettqQrJB5GJP-I/s320/enigmaboard.jpg" border="0" alt="" id="BLOGGER_PHOTO_ID_5607352417348686434" style="display: block; margin-top: 0px; margin-right: auto; margin-bottom: 10px; margin-left: auto; text-align: center; cursor: pointer; width: 320px; height: 214px; " /></a></div><div><br /></div><div>We <a href="http://dave-reed.com/DIYenigma/DIYenigma.pdf">made our own Enigma machines</a> out of paper and tape. Here's the Enigma machine construction in progress:</div><div><br /></div><div><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEigFm8t_kAQOaQq6qCgDfafwprSZNpLJ_sI5bTZrKd1Eq-Aq0JxeuTNSOjPhb36JJicu7ZnwLfAeMyS75kOzV1vBjfoo1LSSe6qhyN5t0kYn9d-tFIV1pdZl0MbwQ24dUG5aBg7Kzm0-A4/s1600/enigmaboys.jpg" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEigFm8t_kAQOaQq6qCgDfafwprSZNpLJ_sI5bTZrKd1Eq-Aq0JxeuTNSOjPhb36JJicu7ZnwLfAeMyS75kOzV1vBjfoo1LSSe6qhyN5t0kYn9d-tFIV1pdZl0MbwQ24dUG5aBg7Kzm0-A4/s320/enigmaboys.jpg" border="0" alt="" id="BLOGGER_PHOTO_ID_5607352421486502306" style="display: block; margin-top: 0px; margin-right: auto; margin-bottom: 10px; margin-left: auto; text-align: center; cursor: pointer; width: 320px; height: 214px; " /></a></div><div><br /></div><div>Using the engima machine to decode a message:</div><div><br /></div><div><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhQh6vySf_B9DgqwL3ik0P7Vbyx4v54CUfcaasS8LU1-HIKOMjvMqGvCiBzAohzAR7XA0icNT9KfdNoSEXo_ihNlGNpCxiKL-uUBRkDquCh3PSQDEPZWMwjem1zRpV1yoYopBem9f4ISX8/s1600/enigmalydia.jpg" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhQh6vySf_B9DgqwL3ik0P7Vbyx4v54CUfcaasS8LU1-HIKOMjvMqGvCiBzAohzAR7XA0icNT9KfdNoSEXo_ihNlGNpCxiKL-uUBRkDquCh3PSQDEPZWMwjem1zRpV1yoYopBem9f4ISX8/s320/enigmalydia.jpg" border="0" alt="" id="BLOGGER_PHOTO_ID_5607352426707029874" style="display: block; margin-top: 0px; margin-right: auto; margin-bottom: 10px; margin-left: auto; text-align: center; cursor: pointer; width: 320px; height: 214px; " /></a></div><div><br /></div><div>There's a nice article <a href="http://plus.maths.org/content/os/issue34/features/ellis/index">here</a> about how the Engima machine works and about some of the things that helped the Allies figure out what was going on.</div>Jessehttp://www.blogger.com/profile/13804151393945151752noreply@blogger.com0tag:blogger.com,1999:blog-5981706365500604046.post-70384930667180272982011-05-11T09:46:00.001-07:002011-05-11T11:14:32.582-07:00Cryptology - The Alphabet Clock<div style="text-align: left;"><blockquote></blockquote><span class="Apple-style-span" style="font-family:arial;"><span class="Apple-style-span" style="font-size:medium;">In classes 3 and 4 we learned about modular arithmetic and the one-time pad.</span></span></div><div><span class="Apple-style-span" style="font-family:arial;"><span class="Apple-style-span" style="font-size:medium;"><br /></span></span></div><div><i><span class="Apple-style-span" style="font-family:arial;"><span class="Apple-style-span" style="font-size:medium;">Modular arithmetic</span></span></i><span class="Apple-style-span" style="font-family:arial;"><span class="Apple-style-span" style="font-size:medium;"> is the sort of arithmetic you do on a clock. On a normal clock, if you start at 7 and add 11 hours you end up at 6. Mathematicians would write it like this:</span></span></div><div style="text-align: center;"><span class="Apple-style-span" style="font-family:arial;"><span class="Apple-style-span" style="font-size:medium;">7+11 = 6 mod 12</span></span></div><div><span class="Apple-style-span" style="font-family:arial;"><span class="Apple-style-span" style="font-size:medium;">(actually, we use three horizontal lines instead of two for the equal sign, but I don't know how to make that symbol on the blog). The "mod 12'' part means that we're on a clock with 12 numbers, starting with 0 at the top (instead of 12) and continuing around to 11.</span></span></div><div><span class="Apple-style-span" style="font-family:arial;"><span class="Apple-style-span" style="font-size:medium;"><br /></span></span></div><div><span class="Apple-style-span" style="font-family:arial;"><span class="Apple-style-span" style="font-size:medium;">For cryptology, we use a clock with the 26 letters on it. These correspond to the numbers from 0 to 25.</span></span></div><div><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_kV_uAZxjbgUUQxF_2a3ifOFhRAHV8AbhIg5yI5zDIUiDmBOvKSrC1o7UzAcPLCF_K1nGhafHWwfWR0fPc19XVnSgmIHzeM3UblJgEwVlsrmnorH59rlbIYLptvQLJuhGqc4bGnTlcMc/s1600/alphabetclock.jpg" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_kV_uAZxjbgUUQxF_2a3ifOFhRAHV8AbhIg5yI5zDIUiDmBOvKSrC1o7UzAcPLCF_K1nGhafHWwfWR0fPc19XVnSgmIHzeM3UblJgEwVlsrmnorH59rlbIYLptvQLJuhGqc4bGnTlcMc/s320/alphabetclock.jpg" border="0" alt="" id="BLOGGER_PHOTO_ID_5605504523356842690" style="display: block; margin-top: 0px; margin-right: auto; margin-bottom: 10px; margin-left: auto; text-align: center; cursor: pointer; width: 320px; height: 320px; " /></a></div><div><span class="Apple-style-span" style="font-family:arial;"><span class="Apple-style-span" style="font-size:medium;">On the alphabet clock,</span></span></div><div style="text-align: center;"><span class="Apple-style-span" style="font-family:arial;"><span class="Apple-style-span" style="font-size:medium;">0 and 26 both mean A</span></span></div><div style="text-align: center;"><span class="Apple-style-span" style="font-family:arial;"><span class="Apple-style-span" style="font-size:medium;">1 and 27 both mean B</span></span></div><div style="text-align: center;"><span class="Apple-style-span" style="font-family:arial;"><span class="Apple-style-span" style="font-size:medium;">25 and -1 both mean Z</span></span></div><div><span class="Apple-style-span" style="font-family:arial;"><span class="Apple-style-span" style="font-size:medium;">and so on.</span></span></div><div><span class="Apple-style-span" style="font-family:arial;"><span class="Apple-style-span" style="font-size:medium;"><br /></span></span></div><div><span class="Apple-style-span" style="font-family:arial;"><span class="Apple-style-span" style="font-size:medium;">To use this clock to encrypt a message, we need some starting plaintext and a key</span></span><i><span class="Apple-style-span" style="font-family:arial;"><span class="Apple-style-span" style="font-size:medium;">.</span></span></i><span class="Apple-style-span" style="font-family:arial;"><span class="Apple-style-span" style="font-size:medium;"> A </span></span><i><span class="Apple-style-span" style="font-family:arial;"><span class="Apple-style-span" style="font-size:medium;">key</span></span></i><span class="Apple-style-span" style="font-family:arial;"><span class="Apple-style-span" style="font-size:medium;"> is a secret bit of information that the person sending the message and the person receiving the message both have to know in order for this to work</span></span><i><span class="Apple-style-span" style="font-family:arial;"><span class="Apple-style-span" style="font-size:medium;">.</span></span></i></div><div><i><span class="Apple-style-span" style="font-family:arial;"><span class="Apple-style-span" style="font-size:medium;"><br /></span></span></i></div><div><span class="Apple-style-span" style="font-family:arial;"><span class="Apple-style-span" style="font-size:medium;">Let's use CHICKEN as the plaintext and MOOFAZA as the key.</span></span></div><div><span class="Apple-style-span" style="font-family:arial;"><span class="Apple-style-span" style="font-size:medium;"><br /></span></span></div><div><span class="Apple-style-span" style="font-family:arial;"><span class="Apple-style-span" style="font-size:medium;">We translate the plaintext and key into numbers, using the clock. We add the first number of plaintext and the first number of key, then the second number of plaintext and the second number of key, and so on.</span></span></div><blockquote><div style="text-align: center;"><blockquote><pre><span class="Apple-style-span" style="font-family:arial;"><pre><span class="Apple-style-span" style="font-size: small;">plaintext CHICKEN 2 7 8 2 10 4 13<br />key MOOFAZA 12 14 14 5 0 25 0<br />sum 14 21 22 7 10 29 13</span></pre></span><p style="text-align: center;margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 16px/normal 'Courier New'; "><span class="Apple-style-span" style="font-size: small;"><span class="Apple-style-span" style="font-family:arial;"></span></span></p></pre><span class="Apple-style-span" style="font-size: small;"><span class="Apple-style-span" style="font-family:arial;"></span></span><p></p></blockquote></div></blockquote><div style="text-align: center;"><blockquote><p style="text-align: left;margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 16px/normal 'Courier New'; "></p></blockquote><p style="text-align: left;margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 16px/normal 'Courier New'; "></p><p></p><p></p><p style="text-align: left;margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 16px/normal Georgia; "><span class="Apple-style-span" style="font-family:arial;"><span class="Apple-style-span" style="font-size:medium;">Then we translate the numbers back into letters. On the alphabet clock, 29 and 3 are both D.</span></span></p><pre><span class="Apple-style-span" style="font-size:medium;">14 21 22 7 10 29 13<br />O V W H K D N</span></pre><p style="text-align: left;margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 16px/normal Georgia; "><span class="Apple-style-span" style="font-family:arial;"><span class="Apple-style-span" style="font-size:medium;">We send the ciphertext OVWHKDN.</span></span></p><p style="text-align: left;margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 16px/normal Georgia; "><span class="Apple-style-span" style="font-family:arial;"><span class="Apple-style-span" style="font-size:medium;"><br /></span></span></p><p style="text-align: left;margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 16px/normal Georgia; "><span class="Apple-style-span" style="font-family:arial;"><span class="Apple-style-span" style="font-size:medium;">To decrypt the message, we need to know the ciphertext and the key. Since we added the key to get the ciphertext, we have to subtract the key to get the plaintext. Translate the ciphertext and key into numbers, and take the difference between each pair of numbers:</span></span></p><pre><span class="Apple-style-span" style="font-size:medium;">ciphertext OVWHKDN 14 21 22 7 10 29 13<br />key MOOFAZA 12 14 14 5 0 25 0<br />difference 2 7 8 2 10 4 13</span></pre><p style="text-align: left;margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 16px/normal Georgia; "><span class="Apple-style-span" style="font-family:arial;"><span class="Apple-style-span" style="font-size:medium;">Finally we translate those numbers back into letters:</span></span></p><pre><span class="Apple-style-span" style="font-size:medium;">2 7 8 2 10 4 13<br />C H I C K E N</span></pre><p style="text-align: left;margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 16px/normal Georgia; "><span class="Apple-style-span" style="font-family:arial;"><span class="Apple-style-span" style="font-size:medium;">If the key is totally random and as long as the message, we have what's called a </span></span><i><span class="Apple-style-span" style="font-family:arial;"><span class="Apple-style-span" style="font-size:medium;">one-time pad</span></span></i><span class="Apple-style-span" style="font-family:arial;"><span class="Apple-style-span" style="font-size:medium;">. In one sense, this is the best cryptography there is: if you don't know the key, you can't figure out the message. I don't care how good your computer is - it can tell all the possible messages, but it can't tell which was the real one.</span></span></p><p style="text-align: left;margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 16px/normal Georgia; "><span class="Apple-style-span" style="font-family:arial;"><span class="Apple-style-span" style="font-size:medium;"><br /></span></span></p><p style="text-align: left;margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 16px/normal Georgia; "><span class="Apple-style-span" style="font-family:arial;"><span class="Apple-style-span" style="font-size:medium;">In another sense, this is horribly impractical. You have to get a gigantic list of completely random letters to your buddy, without anyone else seeing them, and you and your buddy have to always be at the same place in the gigantic list of letters. What a mess!</span></span></p><p></p></div>Jessehttp://www.blogger.com/profile/13804151393945151752noreply@blogger.com0tag:blogger.com,1999:blog-5981706365500604046.post-25830354407337704042011-04-20T11:35:00.000-07:002011-04-20T11:54:41.424-07:00EB: Cryptology<div style="text-align: left;">The first class we learned about the <a href="http://www.cs.trincoll.edu/~crypto/historical/caesar.html">Caesar Cipher and other shift ciphers</a>, as well as learning some useful cryptology words.</div><div><ul><li><i>Plaintext</i> - the meaningful English message</li><li><i>Ciphertext</i> - what you actually send; the secret coded message</li><li><i>Encrypt</i> - turn the plaintext into ciphertext</li><li><i>Decrypt</i> - turn the ciphertext back into plaintext</li></ul><div>The students paired off and sent their partners encrypted messages to decrypt.</div><div><br /></div><div>Today I gave the class several encrypted messages to break, of increasing degrees of difficulty. We worked them all out on the board. It was helpful that I forgot what the plaintext messages were!</div></div><div><br /></div><div>Here's the first one:</div><div><br /></div><div>MJQQT, HQFXX.</div><div>HTSLWFYZQFYNTSX TS GWJFPNSL YMNX HNUMJW!</div><div>- OJXXJ</div><div><br /></div><div>This one is pretty easy. It looks like a letter, and I wrote it, so "OJXXJ" stands for "JESSE." The first word, "MJQQT," is "HELLO." Since I used a shift cipher, the rest follows from there.</div><div><br /></div><div>For the next message, I didn't give them any helpful formatting.</div><div><br /></div><div>KXNDRSCYXOKVCYFOBIXSMO</div><div><br /></div><div>The most common English letter is "E," and the most common characters in this ciphertext are "X" and "O." I was still using a shift cipher, so there were really only two things to try. That didn't take long either.</div><div><br /></div><div>Then, since they kept breaking my shift ciphers, I changed to a cipher that randomly mixed up the letters. I did give them back the formatting, though - I didn't want it to be impossible (or take more than the hour-long class)!</div><div><br /></div><div>VCPT PT E DPYYRHRNV VMSR FY ZFDR</div><div><br /></div><div>The letter "E" in the ciphertext is a one-letter word, so it must be either "I" or "A" (assuming I'm using grammatical English!). The class went with "A". </div><div><br /></div><div>Then the two-letter words in the ciphertext, "PT" and "FY," must translate to two-letter English words that don't contain the letter "A." Also, the two-letter word "PT" is the second half of the four-letter word "VCPT." It was agreed that "THIS IS" was a reasonable guess at those first two words.</div><div><br /></div><div>From there, they got the answer!</div><div><br /></div><div>The picture shows, from left to right:</div><div><ul><li>The ciphertext-to-plaintext translation (we used this more for the first two puzzles)</li><li>The ciphertext, with plaintext underneath</li><li>A list of common 2-letter English words</li></ul><div><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgp6RZnhbOcjxydKMcttNAv9aUItMqmaOPW0aXgPyzLk45CzqeFa_8k0uL9-bsolYTgYwe_TQSA8-2PqPTsAgsXw_frfNRahWd7WoqXf7ABEKP1OnrnSnrKMcqEMDEEWXWtaS6WmeX9YoI/s1600/boardwork.jpg" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgp6RZnhbOcjxydKMcttNAv9aUItMqmaOPW0aXgPyzLk45CzqeFa_8k0uL9-bsolYTgYwe_TQSA8-2PqPTsAgsXw_frfNRahWd7WoqXf7ABEKP1OnrnSnrKMcqEMDEEWXWtaS6WmeX9YoI/s400/boardwork.jpg" border="0" alt="" id="BLOGGER_PHOTO_ID_5597740645288936706" style="display: block; margin-top: 0px; margin-right: auto; margin-bottom: 10px; margin-left: auto; text-align: center; cursor: pointer; width: 400px; height: 267px; " /></a></div></div><div><br /></div>Jessehttp://www.blogger.com/profile/13804151393945151752noreply@blogger.com0tag:blogger.com,1999:blog-5981706365500604046.post-81486732500481198922011-04-19T08:56:00.000-07:002011-04-19T09:01:55.712-07:00More Killer SudokuYesterday and today we worked on <a href="http://www.killersudokuonline.com/archives/2011/apage931vcn1913.html#kdaily">this puzzle</a>. Renata and I were impressed with how well everyone worked on it - almost every moment was full of raised hands and kids saying "I know something!" We finished before the end of class!<div><br /></div><div><br /><div><br /><div><br /></div></div></div>Jessehttp://www.blogger.com/profile/13804151393945151752noreply@blogger.com0tag:blogger.com,1999:blog-5981706365500604046.post-40696412132826830092011-04-14T08:49:00.000-07:002011-04-14T09:06:39.538-07:00Origami and Pascal's TriangleThe first quarter of April was spring break, and we haven't had time to do a whole lot since then. We've made some neat origami vases and <a href="http://www.origami-diagram.com/spanish-box">boxes</a>. The vase required folding the paper exactly in thirds, so we had to figure out how to do that (Google "origami thirds" and you'll find several different ways).<div><br /></div><div>We also did a little more stuff with Pascal's Triangle. If we left-justify all the numbers, we get a staircase instead of a pyramid:</div><div><br /></div><div>1</div><div>1 1</div><div>1 2 1</div><div>1 3 3 1</div><div>1 4 6 4 1</div><div>1 5 10 10 5 1</div><div><br /></div><div>If we sum the numbers in each row, we get doubles:</div><div><div><br /></div><div><span class="Apple-tab-span" style="white-space:pre"> </span>Sum</div><div>1 <span class="Apple-tab-span" style="white-space:pre"> </span>1</div><div>1 1 <span class="Apple-tab-span" style="white-space:pre"> </span>2</div><div>1 2 1 <span class="Apple-tab-span" style="white-space:pre"> </span>4 </div><div>1 3 3 1 <span class="Apple-tab-span" style="white-space:pre"> </span>8</div><div>1 4 6 4 1 <span class="Apple-tab-span" style="white-space:pre"> </span>16</div><div>1 5 10 10 5 1 <span class="Apple-tab-span" style="white-space:pre"> </span>32</div></div><div><br /></div><div>If we sum the numbers on each finite diagonal, we get the Fibonacci numbers. I think this is really cool.</div><div><br /></div><div>Sum</div><div><div><span class="Apple-style-span" style="color:#FF0000;">1<span class="Apple-tab-span" style="white-space:pre"> </span><span class="Apple-tab-span" style="white-space:pre"> </span><span class="Apple-tab-span" style="white-space:pre"> </span> 1</span></div><div><span class="Apple-style-span" style="color:#FF9900;">1<span class="Apple-tab-span" style="white-space:pre"> </span><span class="Apple-tab-span" style="white-space:pre"> </span><span class="Apple-tab-span" style="white-space:pre"> </span> 1</span> <span class="Apple-style-span" style="color:#33CCFF;">1</span></div><div><span class="Apple-style-span" style="color:#33CCFF;">2<span class="Apple-tab-span" style="white-space:pre"> </span><span class="Apple-tab-span" style="white-space:pre"> </span><span class="Apple-tab-span" style="white-space:pre"> </span> 1</span> <span class="Apple-style-span" style="color:#33FF33;">2 </span><span class="Apple-style-span" style="color:#FF0000;">1</span></div><div><span class="Apple-style-span" style="color:#33FF33;">3<span class="Apple-tab-span" style="white-space:pre"> </span><span class="Apple-tab-span" style="white-space:pre"> <span class="Apple-tab-span" style="white-space:pre"> </span><span class="Apple-tab-span" style="white-space:pre"> </span></span> 1</span> <span class="Apple-style-span" style="color:#FF0000;">3</span> <span class="Apple-style-span" style="color:#FF9900;">3</span> 1</div><div><span class="Apple-style-span" style="color:#FF0000;">5<span class="Apple-tab-span" style="white-space:pre"> </span><span class="Apple-tab-span" style="white-space:pre"> </span><span class="Apple-tab-span" style="white-space:pre"> </span> 1</span> <span class="Apple-style-span" style="color:#FF9900;">4</span> 6 4 1</div><div><span class="Apple-style-span" style="color:#FF9900;">8<span class="Apple-tab-span" style="white-space:pre"> </span><span class="Apple-tab-span" style="white-space:pre"> </span><span class="Apple-tab-span" style="white-space:pre"> </span> 1</span> 5 10 10 5 1</div></div><div><br /></div>Jessehttp://www.blogger.com/profile/13804151393945151752noreply@blogger.com0tag:blogger.com,1999:blog-5981706365500604046.post-13037266808940895042011-03-31T11:51:00.000-07:002011-03-31T12:37:08.467-07:00Pascal's Triangle and Tetrahedrons<div>Here's Pascal's Triangle as we left it yesterday. Comparing to the one I posted yesterday, it looks like we made a few mistakes in the bottom two rows. Oops! I think the infinity signs on the right are there to show that the triangle can keep going on and on forever.</div><div><br /></div><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhLd3yBJZuZkN0CpxH5Lc4LUwz5SMitk28e-C68CXKSlFgE3yTseMP6p6-5M8SllEhLqDqBlqAkkKW5K3PVPkFZohmgFTgZtywRaQ8Aw9jt0_efDTcUdhjZe2wwnWfLtBBXibzVwtogJPA/s1600/Triangle.JPG" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhLd3yBJZuZkN0CpxH5Lc4LUwz5SMitk28e-C68CXKSlFgE3yTseMP6p6-5M8SllEhLqDqBlqAkkKW5K3PVPkFZohmgFTgZtywRaQ8Aw9jt0_efDTcUdhjZe2wwnWfLtBBXibzVwtogJPA/s320/Triangle.JPG" border="0" alt="" id="BLOGGER_PHOTO_ID_5590325548855623346" style="display: block; margin-top: 0px; margin-right: auto; margin-bottom: 10px; margin-left: auto; text-align: center; cursor: pointer; width: 320px; height: 240px; " /></a><br /><div>Today we started with square numbers, so called because they're the numbers of dots you can arrange into squares of increasing sizes (see upper right corner of chalkboard). There are also "triangular numbers," which are the numbers of dots you can arrange into equilateral triangles of increasing sizes (see middle right chalkboard).<div><br /></div><div><div><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEijd9DMa64ww6SXrh0-XAv8mPPz3eTlebXrnIF1YJxuK25htW9NDSgL2vZWG8_tpGAtAGG1FbB0f8lx444vnZ35gkPeOKjXTOeHD0PLG_P5yXCKAjP8b38bR82ueLrrnLh_8FqmbS3gYCA/s1600/trianglenumbers.jpg" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"></a><div><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEijd9DMa64ww6SXrh0-XAv8mPPz3eTlebXrnIF1YJxuK25htW9NDSgL2vZWG8_tpGAtAGG1FbB0f8lx444vnZ35gkPeOKjXTOeHD0PLG_P5yXCKAjP8b38bR82ueLrrnLh_8FqmbS3gYCA/s1600/trianglenumbers.jpg" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img style="text-align: left;display: block; margin-top: 0px; margin-right: auto; margin-bottom: 10px; margin-left: auto; cursor: pointer; width: 240px; height: 320px; " src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEijd9DMa64ww6SXrh0-XAv8mPPz3eTlebXrnIF1YJxuK25htW9NDSgL2vZWG8_tpGAtAGG1FbB0f8lx444vnZ35gkPeOKjXTOeHD0PLG_P5yXCKAjP8b38bR82ueLrrnLh_8FqmbS3gYCA/s320/trianglenumbers.jpg" border="0" alt="" id="BLOGGER_PHOTO_ID_5590325553985703522" /></a><br /></div><div>The diagonal of Pascal's triangle that contains the numbers 1, 3, 6, 10, 15, etc. consists of the triangular numbers.</div><div><br /></div><div>Pascal's triangle also contains "tetrahedral numbers." To make sense of this we had to know what a tetrahedron is. First we tried to make one out of ourselves:</div><div><br /><div style="text-align: center;"><span class="Apple-style-span" style="color:#0000EE;"><u><span class="Apple-style-span" style="color: rgb(0, 0, 0); -webkit-text-decorations-in-effect: none; "><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjKj3_XQiLdoMWObCeoBG0HnwE6r8_bUBbp6XDRIviBQIwcONCq-RZ5Oa8bcTdH85fVKkcnUVylEd9nLpfqz0NFSnByTkhPDaheOitlymBB2rJJ8V5ol024o8-McRyDdxmRhFR-vWJUmXU/s1600/humantetra1.JPG" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjKj3_XQiLdoMWObCeoBG0HnwE6r8_bUBbp6XDRIviBQIwcONCq-RZ5Oa8bcTdH85fVKkcnUVylEd9nLpfqz0NFSnByTkhPDaheOitlymBB2rJJ8V5ol024o8-McRyDdxmRhFR-vWJUmXU/s320/humantetra1.JPG" border="0" alt="" id="BLOGGER_PHOTO_ID_5590324283523195490" style="display: block; margin-top: 0px; margin-right: auto; margin-bottom: 10px; margin-left: auto; text-align: center; cursor: pointer; width: 320px; height: 240px; " /></a></span></u></span></div><div><br /></div><div>Since our building pieces (the kids) were different heights, this didn't exactly work. I started to build a tetrahedron using pencils, and then the kids brought out the <a href="http://www.zometool.com/products">zometools</a>. I had never played with these before. Now I want buckets of zometools. So much fun!</div></div><div><br /></div><div>We all built tetrahedra, myself included.</div><div><br /></div><div><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj6Vr6allrecGLMymDsmZY08HNhjm5pufkyr8prgLoCOG963Dwvh7rXF7Si4jEHvSuYSPFkrGIwtXVDbg7beDwFafXa5bGUXOqu7jnSn1fWpSrlEMgQQBV4FSssNsFsV1k9ROVN3g6g49E/s1600/Atty.JPG" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj6Vr6allrecGLMymDsmZY08HNhjm5pufkyr8prgLoCOG963Dwvh7rXF7Si4jEHvSuYSPFkrGIwtXVDbg7beDwFafXa5bGUXOqu7jnSn1fWpSrlEMgQQBV4FSssNsFsV1k9ROVN3g6g49E/s320/Atty.JPG" border="0" alt="" id="BLOGGER_PHOTO_ID_5590324274562476706" style="display: block; margin-top: 0px; margin-right: auto; margin-bottom: 10px; margin-left: auto; text-align: center; cursor: pointer; width: 320px; height: 240px; " /></a></div><div><br /></div><div>Lydia built a tetrahedron tree.</div><div><br /><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjGoPyOS-U7jhK4vTr39pqtNh8Q5l8gNTh14INl50PN1iCsyVnbjSnpX0ChqILCkvVJ775E5K22k7q3QoHz1wuBVBRNwHTFV0Jvw22_VhC5SrVTHz6977d_LMZJAMeZdPZqZNIytuHQhAM/s1600/tetratree2.jpg" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;width: 240px; height: 320px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjGoPyOS-U7jhK4vTr39pqtNh8Q5l8gNTh14INl50PN1iCsyVnbjSnpX0ChqILCkvVJ775E5K22k7q3QoHz1wuBVBRNwHTFV0Jvw22_VhC5SrVTHz6977d_LMZJAMeZdPZqZNIytuHQhAM/s320/tetratree2.jpg" border="0" alt="" id="BLOGGER_PHOTO_ID_5590325545516088514" /></a><br /><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhPjYw-8p_VAJbizSOWwf3yupj3CI0IEHG84UCUGxvdduUfmjjT-Bip07T8moc-4MhwBXGna8yFd_ZHREKqR6LgbaLjlZzwRZb8KvbrGRT3inTq542R63dQeO_2Fi2jqPOSKD62rvIwRFE/s1600/tetratree.jpg" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;width: 240px; height: 320px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhPjYw-8p_VAJbizSOWwf3yupj3CI0IEHG84UCUGxvdduUfmjjT-Bip07T8moc-4MhwBXGna8yFd_ZHREKqR6LgbaLjlZzwRZb8KvbrGRT3inTq542R63dQeO_2Fi2jqPOSKD62rvIwRFE/s320/tetratree.jpg" border="0" alt="" id="BLOGGER_PHOTO_ID_5590324294194421794" /></a><br /></div><div>Saul made a very large tetrahedron and filled it with the little white zomespheres. This is where the "tetrahedral numbers" come in - by counting up the number of spheres it takes to make tetrahedra of increasing sizes.</div><div><br /><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjat95LSBfWJc065rpeKCmPDkZ7NGVppmikf2GTqOMESW6JXhvscuNXWTWG3REAfkM7nHSnVIuMZi9HQavsSb4__rXM8VafCF8b2i49HYCCZJlSiFVJhJ6sNkJP0IqZL-GFOGKd_z8Nfj8/s1600/Saulalmost165.JPG" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;width: 320px; height: 240px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjat95LSBfWJc065rpeKCmPDkZ7NGVppmikf2GTqOMESW6JXhvscuNXWTWG3REAfkM7nHSnVIuMZi9HQavsSb4__rXM8VafCF8b2i49HYCCZJlSiFVJhJ6sNkJP0IqZL-GFOGKd_z8Nfj8/s320/Saulalmost165.JPG" border="0" alt="" id="BLOGGER_PHOTO_ID_5590324290824317570" /></a><br /></div><div>He very kindly let Noah put on the last one:</div><div><br /><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgn9O0klyZZ_WflmeXA-Qla1O-I-eO-hcFVLnTVLEX19Zks1xgVl6DDxKOAmVg0g373EJ9hkPt7qYhH3VeOrwuWM7eh9DkShCzQ2fXj9LsU60_vf-DaGfsc3C-MKNukd7romtFpx0zDxsk/s1600/Saul165.JPG" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;width: 320px; height: 240px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgn9O0klyZZ_WflmeXA-Qla1O-I-eO-hcFVLnTVLEX19Zks1xgVl6DDxKOAmVg0g373EJ9hkPt7qYhH3VeOrwuWM7eh9DkShCzQ2fXj9LsU60_vf-DaGfsc3C-MKNukd7romtFpx0zDxsk/s320/Saul165.JPG" border="0" alt="" id="BLOGGER_PHOTO_ID_5590324287451111474" /></a><br /></div><div>The tetrahedral numbers are 1, 4, 10, ... which just happen to be the next diagonal in Pascal's Triangle. We have the <span class="Apple-style-span" style="color:#FF0000;">natural numbers</span>, the <span class="Apple-style-span" style="color:#FF9900;">triangular numbers</span>, and the <span class="Apple-style-span" style="color:#33CCFF;">tetrahedral numbers</span>:</div><div><br /></div><div><span class="Apple-style-span" style=" color: rgb(99, 32, 53); font-family:arial;font-size:small;"><p style="line-height: 1.4em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; height: 15px; overflow-x: visible; overflow-y: visible; text-align: center; font: normal normal normal 13px/normal 'Courier New'; ">1</p><p style="line-height: 1.4em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; height: 15px; overflow-x: visible; overflow-y: visible; text-align: center; font: normal normal normal 13px/normal 'Courier New'; ">1 <span style="color: rgb(232, 39, 25); ">1</span></p><p style="line-height: 1.4em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; height: 15px; overflow-x: visible; overflow-y: visible; text-align: center; font: normal normal normal 13px/normal 'Courier New'; ">1 <span style="color: rgb(232, 39, 25); ">2</span> <span style="color: rgb(237, 109, 35); ">1</span></p><p style="line-height: 1.4em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; height: 15px; overflow-x: visible; overflow-y: visible; text-align: center; font: normal normal normal 13px/normal 'Courier New'; ">1 <span style="color: rgb(232, 39, 25); ">3</span> <span style="color: rgb(237, 109, 35); ">3</span> <span style="color: rgb(87, 201, 250); ">1</span></p><p style="line-height: 1.4em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; height: 15px; overflow-x: visible; overflow-y: visible; text-align: center; font: normal normal normal 13px/normal 'Courier New'; ">1 <span style="color: rgb(232, 39, 25); ">4</span> <span style="color: rgb(237, 109, 35); ">6</span> <span style="color: rgb(87, 201, 250); ">4</span> 1</p><p style="line-height: 1.4em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; height: 15px; overflow-x: visible; overflow-y: visible; text-align: center; font: normal normal normal 13px/normal 'Courier New'; ">1 <span style="color: rgb(232, 39, 25); ">5</span> <span style="color: rgb(237, 109, 35); ">10</span> <span style="color: rgb(87, 201, 250); ">10</span> 5 1</p><p style="line-height: 1.4em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; height: 15px; overflow-x: visible; overflow-y: visible; text-align: center; font: normal normal normal 13px/normal 'Courier New'; ">1 <span style="color: rgb(232, 39, 25); ">6</span> <span style="color: rgb(237, 109, 35); ">15</span> <span style="color: rgb(87, 201, 250); ">20</span> 15 6 1</p><p style="line-height: 1.4em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; height: 15px; overflow-x: visible; overflow-y: visible; text-align: center; font: normal normal normal 13px/normal 'Courier New'; ">1 <span style="color: rgb(232, 39, 25); ">7</span> <span style="color: rgb(237, 109, 35); ">21</span> <span style="color: rgb(87, 201, 250); ">35</span> 35 21 7 1</p><p style="line-height: 1.4em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; height: 15px; overflow-x: visible; overflow-y: visible; text-align: center; font: normal normal normal 13px/normal 'Courier New'; ">1 <span style="color: rgb(232, 39, 25); ">8</span> <span style="color: rgb(237, 109, 35); ">28 </span><span style="color: rgb(87, 201, 250); ">56 </span>70 56 28 8 1</p><p style="line-height: 1.4em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; height: 15px; overflow-x: visible; overflow-y: visible; text-align: center; font: normal normal normal 13px/normal 'Courier New'; ">1 <span style="color: rgb(232, 39, 25); ">9 </span><span style="color: rgb(237, 109, 35); ">36</span> <span style="color: rgb(87, 201, 250); ">84</span> 126 126 84 36 9 1</p><div><p style="line-height: 1.4em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; height: 15px; overflow-x: visible; overflow-y: visible; text-align: center; font: normal normal normal 13px/normal 'Courier New'; ">1 <span class="Apple-style-span" style="color:#FF0000;">10</span> <span class="Apple-style-span" style="color:#FF9900;">45</span><span class="Apple-style-span" style="color:#33CCFF;"> 120</span> 210 252 210 120 45 10 1</p><p style="line-height: 1.4em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; height: 15px; overflow-x: visible; overflow-y: visible; text-align: center; font: normal normal normal 13px/normal 'Courier New'; ">1 <span class="Apple-style-span" style="color:#FF0000;">11 </span><span class="Apple-style-span" style="color:#FF9900;">55</span> <span class="Apple-style-span" style="color:#33CCFF;">165</span> 330 462 462 300 165 55 11</p></div></span><div style="text-align: center;"><span class="Apple-style-span" style="color:#0000EE;"><u><br /></u></span></div>When Saul counted up all the zomespheres he stacked inside his tetrahedron, he got 165, which is indeed one of the tetrahedral numbers. No zomespheres were lost in the counting of this tetrahedral number.</div><div><br /><div style="text-align: center;"><span class="Apple-style-span" style="color:#0000EE;"><u><br /></u></span></div></div></div></div></div>Jessehttp://www.blogger.com/profile/13804151393945151752noreply@blogger.com0tag:blogger.com,1999:blog-5981706365500604046.post-69647262922144375062011-03-30T07:38:00.000-07:002011-03-30T08:39:20.309-07:00Patterns in Pascal's Triangle<div><span class="Apple-style-span" style="font-family:'courier new';"><span class="Apple-style-span" style="font-family:'courier new';"><div><span class="Apple-style-span" style="font-family:'courier new';"><span class="Apple-style-span" style=" ;font-size:small;"><span class="Apple-style-span" style=" ;font-family:'Courier New';font-size:12px;"><span class="Apple-style-span" style="font-family:'courier new';"><div><span class="Apple-style-span" style="font-family:'courier new';"><span class="Apple-style-span" style="font-family:'courier new';"><p style="text-align: center; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 12px/normal 'Courier New'; "><span class="Apple-style-span" style="font-family:'courier new';"><span class="Apple-style-span" style="font-size:small;"><span class="Apple-style-span" style="font-family:arial;"><span class="Apple-style-span" style="font-size:medium;"></span></span></span></span></p><span class="Apple-style-span" style="font-family:'courier new';"><span class="Apple-style-span" style="font-size:small;"><span class="Apple-style-span" style="font-family:arial;"><p style="text-align: left;margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 16px/normal Arial; ">Today we got all the way to row 14. The very first row, the point of the triangle that only has one number, is row zero.</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px Georgia; min-height: 19.0px"><span class="Apple-style-span" style="font-family:'Courier New';font-size:100%;"><span class="Apple-style-span" style="font-size:12px;"><span class="Apple-style-span" style="font-family:Georgia;font-size:130%;"><span class="Apple-style-span" style="font-size:16px;"><br /></span></span></span></span></p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1 1</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1 2 1</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1 3 3 1</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1 4 6 4 1</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1 5 10 10 5 1</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1 6 15 20 15 6 1</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1 7 21 35 35 21 7 1</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1 8 28 56 70 56 28 8 1</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1 9 36 84 126 126 84 36 9 1</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1 10 45 120 210 252 210 120 45 10 1</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1 11 55 165 330 462 462 300 165 55 11 1</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1 12 66 220 495 792 924 792 495 220 66 12 1</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1 13 78 186 715 1287 1716 1716 1287 715 186 78 13 1</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1 14 91 264 901 2002 3003 3432 3003 2002 901 264 91 14 1</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 12.0px Courier New; min-height: 14.0px"><br /></p> <p style="text-align: left;margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 16px/normal Arial; ">We expanded on some of the patterns from yesterday and found new patterns. I was very impressed with the pattern-finding. Here are some of the patterns.</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Courier New; min-height: 14.0px"><br /></p> <p style="text-align: left;margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 16px/normal Arial; ">The numbers in the second diagonal go up by 1 each time:</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 12.0px Courier New; min-height: 14.0px"><br /></p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1 <span style="color:#e82719;">1</span></p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1 <span style="color:#e82719;">2</span> 1</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1 <span style="color:#e82719;">3</span> 3 1</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1 <span style="color:#e82719;">4</span> 6 4 1</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1 <span style="color:#e82719;">5</span> 10 10 5 1</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1 <span style="color:#e82719;">6</span> 15 20 15 6 1</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1 <span style="color:#e82719;">7</span> 21 35 35 21 7 1</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1 <span style="color:#e82719;">8</span> 28 56 70 56 28 8 1</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1 <span style="color:#e82719;">9 </span>36 84 126 126 84 36 9 1</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 12.0px Courier New; min-height: 14.0px"><br /></p> <p style="text-align: left;margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 16px/normal Arial; ">The numbers in the next diagonal go up by 2, then 3, then 4, then 5, etc.</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Courier New; min-height: 14.0px"><br /></p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1 <span style="color:#e82719;">1</span></p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1 <span style="color:#e82719;">2</span> <span style="color:#ed6d23;">1</span></p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1 <span style="color:#e82719;">3</span> <span style="color:#ed6d23;">3</span> 1</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1 <span style="color:#e82719;">4</span> <span style="color:#ed6d23;">6</span> 4 1</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1 <span style="color:#e82719;">5</span> <span style="color:#ed6d23;">10</span> 10 5 1</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1 <span style="color:#e82719;">6</span> <span style="color:#ed6d23;">15</span> 20 15 6 1</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1 <span style="color:#e82719;">7</span> <span style="color:#ed6d23;">21</span> 35 35 21 7 1</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1 <span style="color:#e82719;">8</span> <span style="color:#ed6d23;">28 </span>56 70 56 28 8 1</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1 <span style="color:#e82719;">9 </span><span style="color:#ed6d23;">36</span> 84 126 126 84 36 9 1</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px Courier New; min-height: 18.0px"><br /></p> <p style="text-align: left;margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 16px/normal Arial; ">The numbers in the next diagonal go up by 3, then 6, then 10, then 15, etc.:</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px Arial; min-height: 18.0px"><br /></p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1 <span style="color:#e82719;">1</span></p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1 <span style="color:#e82719;">2</span> <span style="color:#ed6d23;">1</span></p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1 <span style="color:#e82719;">3</span> <span style="color:#ed6d23;">3</span> <span style="color:#57c9fa;">1</span></p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1 <span style="color:#e82719;">4</span> <span style="color:#ed6d23;">6</span> <span style="color:#57c9fa;">4</span> 1</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1 <span style="color:#e82719;">5</span> <span style="color:#ed6d23;">10</span> <span style="color:#57c9fa;">10</span> 5 1</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1 <span style="color:#e82719;">6</span> <span style="color:#ed6d23;">15</span> <span style="color:#57c9fa;">20</span> 15 6 1</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1 <span style="color:#e82719;">7</span> <span style="color:#ed6d23;">21</span> <span style="color:#57c9fa;">35</span> 35 21 7 1</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1 <span style="color:#e82719;">8</span> <span style="color:#ed6d23;">28 </span><span style="color:#57c9fa;">56 </span>70 56 28 8 1</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1 <span style="color:#e82719;">9 </span><span style="color:#ed6d23;">36</span> <span style="color:#57c9fa;">84</span> 126 126 84 36 9 1</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px Courier New; min-height: 18.0px"><br /></p> <p style="text-align: left;margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 16px/normal Arial; ">The numbers in row 7, except for the outer 1's, are all multiples of 7. I added that this is true for any prime number row: all the numbers in that row, except the outer 1's, will be multiples of the prime.</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px Arial; min-height: 18.0px"><br /></p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1 1</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1 <span style="color:#e82719;">2 </span>1</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1 <span style="color:#e82719;">3 3</span> 1</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1 4 6 4 1</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New; color:#e82719;"><span style="color:#000000;">1 </span>5 10 10 5<span style="color:#000000;"> 1</span></p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1 6 15 20 15 6 1</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New; color:#e82719;"><span style="color:#000000;">1 </span>7 21 35 35 21 7<span style="color:#000000;"> 1</span></p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1 8 28 56 70 56 28 8 1</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1 9 36 84 126 126 84 36 9 1</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1 10 45 120 210 252 210 120 45 10 1</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New; color:#e82719;"><span style="color:#000000;">1</span> 11 55 165 330 462 462 300 165 55 11<span style="color:#000000;"> 1</span></p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1 12 66 220 495 792 924 792 495 220 66 12 1</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New; color:#e82719;"><span style="color:#000000;">1 </span>13 78 186 715 1287 1716 1716 1287 715 186 78 13<span style="color:#000000;"> 1</span></p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 13.0px Courier New">1 14 91 264 901 2002 3003 3432 3003 2002 901 264 91 14 1</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 12.0px Courier New; min-height: 14.0px"><br /></p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 12.0px Courier New; min-height: 14.0px"><br /></p> <p style="text-align: left;margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 16px/normal Arial; ">Row 0 has 1 number, the next row has 2 numbers, the next row has 3 numbers, etc.:</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px Courier New; min-height: 18.0px"><br /></p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 16.0px Courier New"><span class="Apple-style-span" style="font-size:small;">1 - 1 number</span></p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 16.0px Courier New"><span class="Apple-style-span" style="font-size:small;">1 1 - 2 numbers</span></p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 16.0px Courier New"><span class="Apple-style-span" style="font-size:small;">1 2 1 - 3 numbers</span></p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 16.0px Courier New"><span class="Apple-style-span" style="font-size:small;">1 3 3 1 - 4 numbers</span></p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Courier New; min-height: 14.0px"><br /></p> <p style="text-align: left;margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 16px/normal Arial; ">Only even-number rows have a middle number (this is because we can split an even number in half, but we can't split an odd number in half)<span style="font: 16.0px Courier New">.</span></p> <p style="text-align: left;margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 16px/normal Arial; "><br /></p><p style="text-align: left;margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 16px/normal Arial; ">In even-number rows the "choose 2" number is a multiple of half the row number. In row 4 the number 4C2=6 is a multiple of 2, in row 6 the number 6C2=15 is a multiple of 3, in row 8 the number 8C2=28 is a multiple of 4, etc:</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Courier New; min-height: 14.0px"><br /></p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 16.0px Courier New"><span class="Apple-style-span" style="font-size:small;">1</span></p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 16.0px Courier New"><span class="Apple-style-span" style="font-size:small;">1 1</span></p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 16.0px Courier New"><span class="Apple-style-span" style="font-size:small;">1 2 1</span></p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 16.0px Courier New"><span class="Apple-style-span" style="font-size:small;">1 3 3 1</span></p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 16.0px Courier New"><span class="Apple-style-span" style="font-size:small;">1 4 </span><span style="color:#e82719;"><span class="Apple-style-span" style="font-size:small;">6</span></span><span class="Apple-style-span" style="font-size:small;"> 4 1</span></p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 16.0px Courier New"><span class="Apple-style-span" style="font-size:small;">1 5 10 10 5 1</span></p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 16.0px Courier New"><span class="Apple-style-span" style="font-size:small;">1 6 </span><span style="color:#e82719;"><span class="Apple-style-span" style="font-size:small;">15</span></span><span class="Apple-style-span" style="font-size:small;"> 20 15 6 1</span></p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 16.0px Courier New"><span class="Apple-style-span" style="font-size:small;">1 7 21 35 35 21 7 1</span></p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 16.0px Courier New"><span class="Apple-style-span" style="font-size:small;">1 8 </span><span style="color:#e82719;"><span class="Apple-style-span" style="font-size:small;">28</span></span><span class="Apple-style-span" style="font-size:small;"> 56 70 56 28 8 1</span></p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 16.0px Courier New"><span class="Apple-style-span" style="font-size:small;">1 9 36 84 126 126 84 36 9 1</span></p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 16.0px Courier New"><span class="Apple-style-span" style="font-size:small;">1 10 </span><span style="color:#e82719;"><span class="Apple-style-span" style="font-size:small;">45</span></span><span class="Apple-style-span" style="font-size:small;"> 120 210 252 210 120 45 10 1</span></p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 16.0px Courier New"><span class="Apple-style-span" style="font-size:small;">1 11 55 165 330 462 462 300 165 55 11 1</span></p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 16.0px Courier New"><span class="Apple-style-span" style="font-size:small;">1 12 </span><span style="color:#e82719;"><span class="Apple-style-span" style="font-size:small;">66</span></span><span class="Apple-style-span" style="font-size:small;"> 220 495 792 924 792 495 220 66 12 1</span></p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 16.0px Courier New"><span class="Apple-style-span" style="font-size:small;">1 13 78 186 715 1287 1716 1716 1287 715 186 78 13 1</span></p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 16.0px Courier New"><span class="Apple-style-span" style="font-size:small;">1 14 </span><span style="color:#e82719;"><span class="Apple-style-span" style="font-size:small;">91</span></span><span class="Apple-style-span" style="font-size:small;"> 264 901 2002 3003 3432 3003 2002 901 264 91 14 1</span></p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Courier New; min-height: 14.0px"><br /></p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Courier New; min-height: 14.0px"><br /></p> <p style="text-align: left;margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 16px/normal Arial; ">The other numbers in that diagonal, on the odd-number rows, are special for a different reason:</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 16.0px Arial">3*1=3</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 16.0px Arial">5*2=10</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 16.0px Arial">7*3=21</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 16.0px Arial">9*4=36</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 16.0px Arial">etc.</p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 12.0px Courier New; min-height: 14.0px"><br /></p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 12.0px Courier New; min-height: 14.0px"><br /></p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 16.0px Courier New"><span class="Apple-style-span" style="font-size:small;">1</span></p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 16.0px Courier New"><span class="Apple-style-span" style="font-size:small;">1 1</span></p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 16.0px Courier New"><span class="Apple-style-span" style="font-size:small;">1 2 1</span></p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 16.0px Courier New"><span class="Apple-style-span" style="font-size:small;">1 3 </span><span style="color:#e82719;"><span class="Apple-style-span" style="font-size:small;">3</span></span><span class="Apple-style-span" style="font-size:small;"> 1</span></p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 16.0px Courier New"><span class="Apple-style-span" style="font-size:small;">1 4 6 4 1</span></p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 16.0px Courier New"><span class="Apple-style-span" style="font-size:small;">1 5 </span><span style="color:#e82719;"><span class="Apple-style-span" style="font-size:small;">10</span></span><span class="Apple-style-span" style="font-size:small;"> 10 5 1</span></p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 16.0px Courier New"><span class="Apple-style-span" style="font-size:small;">1 6 15 20 15 6 1</span></p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 16.0px Courier New"><span class="Apple-style-span" style="font-size:small;">1 7 </span><span style="color:#e82719;"><span class="Apple-style-span" style="font-size:small;">21</span></span><span class="Apple-style-span" style="font-size:small;"> 35 35 21 7 1</span></p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 16.0px Courier New"><span class="Apple-style-span" style="font-size:small;">1 8 28 56 70 56 28 8 1</span></p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 16.0px Courier New"><span class="Apple-style-span" style="font-size:small;">1 9 </span><span style="color:#e82719;"><span class="Apple-style-span" style="font-size:small;">36</span></span><span class="Apple-style-span" style="font-size:small;"> 84 126 126 84 36 9 1</span></p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 16.0px Courier New"><span class="Apple-style-span" style="font-size:small;">1 10 45 120 210 252 210 120 45 10 1</span></p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 16.0px Courier New"><span class="Apple-style-span" style="font-size:small;">1 11 </span><span style="color:#e82719;"><span class="Apple-style-span" style="font-size:small;">55</span></span><span class="Apple-style-span" style="font-size:small;"> 165 330 462 462 300 165 55 11 1</span></p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 16.0px Courier New"><span class="Apple-style-span" style="font-size:small;">1 12 66 220 495 792 924 792 495 220 66 12 1</span></p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 16.0px Courier New"><span class="Apple-style-span" style="font-size:small;">1 13</span><span style="color:#e82719;"><span class="Apple-style-span" style="font-size:small;"> 78</span></span><span class="Apple-style-span" style="font-size:small;"> 186 715 1287 1716 1716 1287 715 186 78 13 1</span></p> <p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 16.0px Courier New"><span class="Apple-style-span" style="font-size:small;">1 14 91 264 901 2002 3003 3432 3003 2002 901 264 91 14 1</span></p></span></span></span><p></p></span><p></p></span></div><div><span class="Apple-style-span" style="font-family:'courier new';"><span class="Apple-style-span" style=" ;font-size:small;"><span class="Apple-style-span" style=" ;font-family:'Courier New';font-size:12px;"></span></span></span></div></span></span><span class="Apple-style-span" style="font-family:'courier new';"></span></span><span class="Apple-style-span" style="font-family:'courier new';"></span><p></p></span></div></span><p></p></span><p></p><p></p><p></p></div>Jessehttp://www.blogger.com/profile/13804151393945151752noreply@blogger.com0tag:blogger.com,1999:blog-5981706365500604046.post-88735497092548803402011-03-28T13:37:00.001-07:002011-03-28T14:13:46.948-07:00Counting Part 3: Buttons<div><b>Buttons (The Big Group)</b></div><div><br /></div><div>Take 5 buttons: red, white, green, yellow, and purple. How many ways are there to choose 1 button from those 5? 5, of course. </div><div><br /></div><div>How many ways are there to choose 2 buttons of different colors? The class agreed that order didn't matter - choosing red and white was the same thing as choosing white and red. The students found 10 ways to choose 2 buttons from 5.</div><div><br /></div><div>We started in with the mathematical notation. The mathematical notation for "the number of ways to choose 2 things from 5" is 5C2. So we found</div><div style="text-align: center; ">5C1=5</div><div style="text-align: center; ">5C2=10</div><div style="text-align: center; "><br /></div><div style="text-align: left; ">The most prevalent guess was that 5C3 would be 15, but it turned out that 5C3=10.</div><div style="text-align: left; "><br /></div><div style="text-align: left; ">We found that 5C4=5, because choosing 4 buttons is the same as choosing 1 button to exclude. Here's what the board looked like after we figured all this out and started discussing what would happen if we started with 6 buttons:</div><div><br /></div><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgrVtQWvK4MYaWgZ4DqfRP_2Cn9znszVsDR-BGCd-zcPDep84VxQCmvcYhToXp36NiIsADQV00VZG47a8ta11hc41IHDXU3fmfFRHRxXzzr_Cy-5uJiyJhCqRKKIXiAxJ0_B9IWjPBSlAE/s1600/buttons2.jpg" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgrVtQWvK4MYaWgZ4DqfRP_2Cn9znszVsDR-BGCd-zcPDep84VxQCmvcYhToXp36NiIsADQV00VZG47a8ta11hc41IHDXU3fmfFRHRxXzzr_Cy-5uJiyJhCqRKKIXiAxJ0_B9IWjPBSlAE/s320/buttons2.jpg" border="0" alt="" id="BLOGGER_PHOTO_ID_5589220939323745874" style="display: block; margin-top: 0px; margin-right: auto; margin-bottom: 10px; margin-left: auto; text-align: center; cursor: pointer; width: 320px; height: 240px; " /></a><br /><div>The next day we continued finding 6C0, 6C2, ... , 6C6. Then we went back and started with 0C0. There's only one way to not take any buttons, which is to not take any. So </div><div style="text-align: center; ">0C0=1</div><div style="text-align: left; ">If we start with 1 button, we get</div><div style="text-align: center; ">1C0=1, 1C1=1</div><div style="text-align: left; ">With 2 buttons, </div><div style="text-align: center; ">2C0=1, 2C1=2, 2C2=1</div><div style="text-align: left; ">With 3 buttons, </div><div style="text-align: center; ">3C0=1, 3C1=3, 3C2=3, 3C3=1</div><div style="text-align: left; ">We continued down to row 6, and got the following pretty picture:</div><div style="text-align: center; ">1</div><div style="text-align: center; ">1 1</div><div style="text-align: center; ">1 2 1</div><div style="text-align: center; ">1 3 3 1</div><div style="text-align: center; ">1 4 6 4 1</div><div style="text-align: center; ">1 5 10 10 5 1</div><div style="text-align: center; ">1 6 15 20 15 6 1</div><div style="text-align: center; "><br /></div><div style="text-align: left; ">This is a very famous creature called "Pascal's Triangle." To get each number for the next row, you add up the two numbers above it. For example, 5+10=15:</div><div style="text-align: left; "><div style="text-align: center; ">1</div><div style="text-align: center; ">1 1</div><div style="text-align: center; ">1 2 1</div><div style="text-align: center; ">1 3 3 1</div><div style="text-align: center; ">1 4 6 4 1</div><div style="text-align: center; ">1 <span class="Apple-style-span" style="color:#FF0000;">5</span> <span class="Apple-style-span" style="color:#FF0000;">10</span> 10 5 1</div><div style="text-align: center; ">1 6 <span class="Apple-style-span" style="color:#FF0000;">15</span> 20 15 6 1</div><div style="text-align: center; "><br /></div><div style="text-align: left; ">On their own, each kid completed the triangle up to row 10 and looked for patterns. Some student observations:</div><div style="text-align: left; "><ul><li>All the numbers in row 7 (except 1) are multiples of 7:</li></ul></div><div style="text-align: center; ">1 7 21 35 35 21 7 1</div><div style="text-align: left; "><ul><li>All the numbers along the outside of the triangle are 1.</li><li>The numbers that aren't quite along the outside go up by 1 each time:</li></ul></div><div style="text-align: center; "><div style="text-align: center; ">1</div><div style="text-align: center; ">1 1</div><div style="text-align: center; ">1 <span class="Apple-style-span" style="color:#FF0000;">2</span> 1</div><div style="text-align: center; ">1 <span class="Apple-style-span" style="color:#FF0000;">3</span> 3 1</div><div style="text-align: center; ">1 <span class="Apple-style-span" style="color:#FF0000;">4</span> 6 4 1</div><div style="text-align: center; ">1 <span class="Apple-style-span" style="color:#FF0000;">5</span> 10 10 5 1</div><div style="text-align: center; ">1 <span class="Apple-style-span" style="color:#FF0000;">6</span> 15 20 15 6 1</div><div style="text-align: left; "><ul><li>To fill in the triangle, you only have to find half of a row because the second half mirrors the first:</li></ul><div style="text-align: center; "><span class="Apple-style-span" style="color:#FF0000;">1 6 15 20</span> 15 6 1</div><ul><li>In rows that have a "middle number," the middle number is always even:</li></ul><div style="text-align: center; ">1</div><div style="text-align: center; ">1 1</div><div style="text-align: center; ">1 <span class="Apple-style-span" style="color:#FF0000;">2</span> 1</div><div style="text-align: center; ">1 3 3 1</div><div style="text-align: center; ">1 4 <span class="Apple-style-span" style="color:#FF0000;">6</span> 4 1</div><div style="text-align: center; ">1 5 10 10 5 1</div><div style="text-align: center; ">1 6 15 <span class="Apple-style-span" style="color:#FF0000;">20</span> 15 6 1</div><div><br /></div></div></div></div>Jessehttp://www.blogger.com/profile/13804151393945151752noreply@blogger.com0tag:blogger.com,1999:blog-5981706365500604046.post-33160133985140974042011-03-28T13:36:00.000-07:002011-03-28T14:16:13.470-07:00Counting Part 2: Poker<div><b>Poker (The Small Group)</b></div><div><br /></div><div>The first question: How many possible poker hands are there? </div><div><br /></div><div>After playing around with 2- and 3- card hands, which are easier to think about, they arrived at their first answer:</div><div style="text-align: center; ">52*51*50*49*48.</div><div style="text-align: center; "><br /></div><div>This is a good start, but it's assuming order matters, and thus is counting each hand more than once. Order doesn't matter in poker, because whether you have 2-3-4-5-5 or 2-5-5-4-3, it's the same poker hand. Each 5-card hand gets counted 5*4*3*2*1 times, so the real answer is</div><div style="text-align: center; ">(52*51*50*49*48)/(5*4*3*2*1)=2,598,960</div><div style="text-align: left; ">possible poker hands.</div><div style="text-align: left; "><br /></div><div style="text-align: left; ">Then we figured out how many ways there were to get various hands. There are only 4 ways to get a a royal flush, which means the probability of getting a royal flush is</div><div style="text-align: center; ">(4)/(2598960) ~=~ .0000015</div><div style="text-align: left; ">A straight flush that isn't also a royal flush can start on A, 2, 3, 4, 5, 6, 7, 8, or 9, so there are 9 ways per suit for a total of 36 ways to get a royal flush. The probability of getting a straight flush is</div><div style="text-align: center; ">(36)/(2598960) ~=~ .000014</div><div style="text-align: left; ">These hands are not very likely!</div><div style="text-align: left; "><br /></div><div style="text-align: left; ">It's harder to figure out how many ways there are to get one pair. There are 13 choices of what the pair will be (A through K). Suppose we have a pair of aces. There are 6 different ways to choose a pair of aces from the 4 aces in the deck. Then we have to fill in the other 3 cards in the hand.</div><div style="text-align: center; ">A-A-_-_-_</div><div style="text-align: left; "><br /></div><div style="text-align: left; ">Since the third card can't be an ace (or we wouldn't have one pair), there are 48 choices for the third card. The fourth card can't be an ace, and also can't be whatever we picked for the third card, so there are 44 choices. The fifth card can't be an ace, the same as the third card, or the same as the fourth card, so there are 40 choices for the fifth card.</div><div style="text-align: left; "><br /></div><div style="text-align: left; ">To make sure we aren't over-counting those last three cards, we need to divide by 6 (the number of ways to order those cards). This gives us</div><div style="text-align: center; ">(13*6*48*44*40)/6 = 1,098,240 </div><div style="text-align: left; ">ways to get a single pair. The probability of getting a single pair in a five-card poker hand is</div><div style="text-align: center; ">1,098,240 / 2,598,960 ~=~ .42</div><div><br /></div><div>After figuring all this out, we spent a couple of days playing poker and tallying up the hands we got to see if they looked like we expected they would. There were lots of single pairs, as expected, with a scattering of two pair and three of a kind. I got lots of junk hands, which the kids found very amusing.</div><div><br /></div><div>We played with buttons, starting with 20 buttons each. I pointed out to the loser of the first game that if we had been using real money, he would have just lost $20 in half an hour. He replied that he wasn't going to gamble with real money. :)</div><div><br /></div><div><a href="http://wizardofodds.com/poker">Here's a site</a> that explains how to calculate odds for various poker hands.</div>Jessehttp://www.blogger.com/profile/13804151393945151752noreply@blogger.com0tag:blogger.com,1999:blog-5981706365500604046.post-6572568726063507772011-03-28T12:50:00.000-07:002011-03-28T14:11:39.569-07:00Counting, Part 1: Dice<div>After working in math books, Renata and I usually split up the kids into two groups. The small group consists of Max, Alec, and Lukas; the big group consists of everyone else. I've been on a combinatorics kick recently, so both groups have been counting things.</div><div><br /></div><div><b>Dice (Everyone)</b></div><div><br /></div><div>The big group and the small group did the same activities on different days, with slight variations. </div><div><br /></div><div>If you roll two dice and add up the numbers on their faces, you get a sum between 2 and 12. What are the different ways to get each sum?</div><div><br /></div><div>The students came up with a very nice way to write the results. It looks like a mountain:</div><div><span class="Apple-style-span" style="font-family:'courier new';"><br /></span></div><div style="text-align: center;"><div style="text-align: center;"><span class="Apple-style-span" style=" ;font-family:'courier new';"> 1+6</span></div><span class="Apple-style-span" style=" ;font-family:'courier new';"><div style="text-align: center;"> 3+3 6+1 4+4</div></span><span class="Apple-style-span" style=" ;font-family:'courier new';"><div style="text-align: center;"> 3+2 4+2 5+2 6+2 5+4</div><div style="text-align: center;"><span class="Apple-tab-span" style="white-space:pre"> </span>2+2<span class="Apple-tab-span" style="white-space:pre"> </span>2+3<span class="Apple-tab-span" style="white-space:pre"> </span>2+4<span class="Apple-tab-span" style="white-space:pre"> </span>2+5<span class="Apple-tab-span" style="white-space:pre"> </span>2+6<span class="Apple-tab-span" style="white-space:pre"> </span>4+5<span class="Apple-tab-span" style="white-space:pre"> </span>5+5<span class="Apple-tab-span" style="white-space:pre"> </span></div></span><span class="Apple-style-span" style=" ;font-family:'courier new';"><div style="text-align: center;"><span class="Apple-tab-span" style="white-space:pre"> </span>2+1<span class="Apple-tab-span" style="white-space:pre"> </span>3+1<span class="Apple-tab-span" style="white-space:pre"> </span>4+1<span class="Apple-tab-span" style="white-space:pre"> </span>1+5<span class="Apple-tab-span" style="white-space:pre"> </span>4+3<span class="Apple-tab-span" style="white-space:pre"> </span>3+5<span class="Apple-tab-span" style="white-space:pre"> </span>3+6<span class="Apple-tab-span" style="white-space:pre"> </span>4+6<span class="Apple-tab-span" style="white-space:pre"> </span>5+6<span class="Apple-tab-span" style="white-space:pre"> </span></div></span><span class="Apple-style-span" style=" ;font-family:'courier new';"><div style="text-align: center;"> 1+1<span class="Apple-tab-span" style="white-space:pre"> </span>1+2<span class="Apple-tab-span" style="white-space:pre"> </span>1+3<span class="Apple-tab-span" style="white-space:pre"> </span>1+4<span class="Apple-tab-span" style="white-space:pre"> </span>5+1<span class="Apple-tab-span" style="white-space:pre"> </span>3+4<span class="Apple-tab-span" style="white-space:pre"> </span>5+3<span class="Apple-tab-span" style="white-space:pre"> </span>6+3<span class="Apple-tab-span" style="white-space:pre"> </span>6+4<span class="Apple-tab-span" style="white-space:pre"> </span>6+5<span class="Apple-tab-span" style="white-space:pre"> </span>6+6</div><div style="text-align: center;">Sum:<span class="Apple-tab-span" style="white-space:pre"> </span>2<span class="Apple-tab-span" style="white-space:pre"> <span class="Apple-tab-span" style="white-space:pre"> </span><span class="Apple-tab-span" style="white-space:pre"> </span></span>3<span class="Apple-tab-span" style="white-space:pre"> <span class="Apple-tab-span" style="white-space:pre"> </span></span><span class="Apple-tab-span" style="white-space:pre"> </span>4<span class="Apple-tab-span" style="white-space:pre"> </span><span class="Apple-tab-span" style="white-space:pre"> <span class="Apple-tab-span" style="white-space:pre"> </span></span>5<span class="Apple-tab-span" style="white-space:pre"> <span class="Apple-tab-span" style="white-space:pre"> </span></span><span class="Apple-tab-span" style="white-space:pre"> </span>6<span class="Apple-tab-span" style="white-space:pre"> </span><span class="Apple-tab-span" style="white-space:pre"> </span><span class="Apple-tab-span" style="white-space:pre"> </span>7<span class="Apple-tab-span" style="white-space:pre"> </span><span class="Apple-tab-span" style="white-space:pre"> </span><span class="Apple-tab-span" style="white-space:pre"> </span>8<span class="Apple-tab-span" style="white-space:pre"> </span><span class="Apple-tab-span" style="white-space:pre"> </span><span class="Apple-tab-span" style="white-space:pre"> </span>9<span class="Apple-tab-span" style="white-space:pre"> <span class="Apple-tab-span" style="white-space:pre"> </span></span>10<span class="Apple-tab-span" style="white-space:pre"> </span><span class="Apple-tab-span" style="white-space:pre"> </span>11<span class="Apple-tab-span" style="white-space:pre"> <span class="Apple-tab-span" style="white-space:pre"> </span></span>12<span class="Apple-tab-span" style="white-space:pre"> <span class="Apple-tab-span" style="white-space:pre"> </span><span class="Apple-tab-span" style="white-space:pre"> </span></span></div><div style="text-align: center;"><span class="Apple-tab-span" style="white-space:pre"><span class="Apple-tab-span" style="white-space:pre"><span class="Apple-style-span" style="white-space: normal; ">Ways:<span class="Apple-tab-span" style="white-space: pre; "> </span><span class="Apple-tab-span">1</span><span class="Apple-tab-span" style="white-space: pre; "> <span class="Apple-tab-span" style="white-space: pre; "> </span><span class="Apple-tab-span" style="white-space: pre; "> </span><span class="Apple-tab-span"><span class="Apple-style-span" style="white-space: normal;">2</span></span></span><span class="Apple-tab-span" style="white-space: pre; "> <span class="Apple-tab-span" style="white-space: pre; "> </span></span><span class="Apple-tab-span" style="white-space: pre; "> </span><span class="Apple-tab-span">3</span><span class="Apple-tab-span" style="white-space: pre; "> </span><span class="Apple-tab-span" style="white-space: pre; "> <span class="Apple-tab-span" style="white-space: pre; "> </span><span class="Apple-tab-span"><span class="Apple-style-span" style="white-space: normal;">4</span></span></span><span class="Apple-tab-span" style="white-space: pre; "> <span class="Apple-tab-span" style="white-space: pre; "> </span></span><span class="Apple-tab-span" style="white-space: pre; "> </span><span class="Apple-tab-span">5</span><span class="Apple-tab-span" style="white-space: pre; "> </span><span class="Apple-tab-span" style="white-space: pre; "> </span><span class="Apple-tab-span" style="white-space: pre; "> </span><span class="Apple-tab-span">6</span><span class="Apple-tab-span" style="white-space: pre; "> </span><span class="Apple-tab-span" style="white-space: pre; "> </span><span class="Apple-tab-span" style="white-space: pre; "> </span><span class="Apple-tab-span">5</span><span class="Apple-tab-span" style="white-space: pre; "> </span><span class="Apple-tab-span" style="white-space: pre; "> </span><span class="Apple-tab-span" style="white-space: pre; "> </span><span class="Apple-tab-span">4</span><span class="Apple-tab-span" style="white-space: pre; "> <span class="Apple-tab-span" style="white-space:pre"> </span><span class="Apple-tab-span" style="white-space: pre; "> </span><span class="Apple-tab-span"><span class="Apple-style-span" style="white-space: normal;">3</span></span></span><span class="Apple-tab-span" style="white-space: pre; "> <span class="Apple-tab-span" style="white-space:pre"> </span></span><span class="Apple-tab-span" style="white-space: pre; "> </span><span class="Apple-tab-span">2<span class="Apple-tab-span" style="white-space:pre"> </span></span><span class="Apple-tab-span" style="white-space: pre; "> <span class="Apple-tab-span" style="white-space: pre; "> </span></span>1<span class="Apple-tab-span" style="white-space: pre; "> <span class="Apple-tab-span" style="white-space: pre; "> <span class="Apple-tab-span" style="white-space:pre"> </span></span><span class="Apple-tab-span" style="white-space: pre; "> </span></span></span></span></span></div></span><span class="Apple-style-span" style=" ;font-family:'courier new';"><div style="text-align: center;"><span class="Apple-style-span" style="font-family:Georgia, serif;"><b><span class="Apple-style-span" style="font-family:'courier new';"><span class="Apple-style-span" style="font-weight: normal;"><br /></span></span></b></span></div></span></div><div>We moved on to three dice. The mountain was much bigger, but still nicely mountain-shaped! This is good intuition for dealing with probability distributions in future years. <a href="http://www.regentsprep.org/Regents/math/algtrig/ATS2/NormalLesson.htm">Look at the pretty mountains.</a></div><div><br /></div><div>There's a shortcut if you care about how many ways you can make a particular sum, but don't care what the ways are. Suppose you want 3 dice to sum to 6. The first die can be 1, 2, 3, or 4 (the three dice can't add up to 6 if one of them is 5 or 6).</div><div><br /></div><div>If the first die is 1, the next two must add up to 5. Looking at our two-dice mountain, there are 4 ways for this to happen.</div><div style="text-align: center;">1+_+_=6<span class="Apple-tab-span" style="white-space:pre"> </span><span class="Apple-tab-span" style="white-space:pre"> </span>_+_=5<span class="Apple-tab-span" style="white-space:pre"> </span>4 ways</div><div style="text-align: center;"><br /></div><div style="text-align: left;">We can do the same sort of thing for the other possible values of the first die.</div><div style="text-align: center;">2+_+_=6<span class="Apple-tab-span" style="white-space:pre"> </span>_+_=4<span class="Apple-tab-span" style="white-space:pre"> </span>3 ways</div><div style="text-align: center;">3+_+_=6<span class="Apple-tab-span" style="white-space:pre"> </span>_+_=3<span class="Apple-tab-span" style="white-space:pre"> </span>2 ways</div><div style="text-align: center;">4+_+_=6<span class="Apple-tab-span" style="white-space:pre"> </span>_+_=2<span class="Apple-tab-span" style="white-space:pre"> </span><span class="Apple-tab-span" style="white-space:pre"> </span>1 way</div><div style="text-align: center;"><br /></div><div style="text-align: left;">This means there are<span class="Apple-tab-span" style="white-space:pre"> </span>4+3+2+1=10 ways for three dice to sum to 6.</div><div style="text-align: left;"><br /></div><div style="text-align: left;">Using this shortcut, I think the small group counted how many ways you could make each possible sum with 4 dice!</div>Jessehttp://www.blogger.com/profile/13804151393945151752noreply@blogger.com0tag:blogger.com,1999:blog-5981706365500604046.post-10287168989668239422011-03-28T12:39:00.000-07:002011-03-28T12:50:23.454-07:00BagelsWe made a <a href="http://dancinglogic.blogspot.com/2010/03/mathematical-bagels.html">mathematically correct breakfast</a>! We will have to try this again, with lines that are less straight and more curved. I think I also need to find sturdier bagels - these were a little crumbly.<div><br /><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhBQ0TjLBJSzOyPx38HodCv7PVON2uKg4AisYW_AbPXgWIj3ez5Mi_wy3xCJe24N3pcnk0QrSY2mr9fh5rSfYP3jsQip19DZcmWfQo2pgi3yITsnlVRKo5r-LAELJdQuWeA2E7omk_tI1c/s1600/Saul.JPG" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;width: 320px; height: 240px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhBQ0TjLBJSzOyPx38HodCv7PVON2uKg4AisYW_AbPXgWIj3ez5Mi_wy3xCJe24N3pcnk0QrSY2mr9fh5rSfYP3jsQip19DZcmWfQo2pgi3yITsnlVRKo5r-LAELJdQuWeA2E7omk_tI1c/s320/Saul.JPG" border="0" alt="" id="BLOGGER_PHOTO_ID_5589220254318472626" /></a><br /><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiKDpefPC-AYdwVmpdWRixHYvyScf0E1LnDCkUSjXlmGEHzqbVgtBJd-6jSMIy2KbAicVlXCcceIs_UzgW3LvbanMeRhv_t0QAMofXVe596IcBI3ydPtRXOzlOvcQldObGQYmAxyhcp-oc/s1600/Lydia.JPG" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;width: 320px; height: 240px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiKDpefPC-AYdwVmpdWRixHYvyScf0E1LnDCkUSjXlmGEHzqbVgtBJd-6jSMIy2KbAicVlXCcceIs_UzgW3LvbanMeRhv_t0QAMofXVe596IcBI3ydPtRXOzlOvcQldObGQYmAxyhcp-oc/s320/Lydia.JPG" border="0" alt="" id="BLOGGER_PHOTO_ID_5589220248869313618" /></a><br /><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhUALpg_CVFxiZtzf1hns_ZHItel6boursbaigHrwuhUalYbtX64MvEWfBfKswBH5maf7Ev7naqRyWHsiNhJziwnLaIAsFTLMMd9wqpjQv9ZStpZPWxCHqGjg53q2-y_EjcSqqH84-c6Fs/s1600/Danny.JPG" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;width: 320px; height: 240px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhUALpg_CVFxiZtzf1hns_ZHItel6boursbaigHrwuhUalYbtX64MvEWfBfKswBH5maf7Ev7naqRyWHsiNhJziwnLaIAsFTLMMd9wqpjQv9ZStpZPWxCHqGjg53q2-y_EjcSqqH84-c6Fs/s320/Danny.JPG" border="0" alt="" id="BLOGGER_PHOTO_ID_5589220245209165154" /></a><br /><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgSMGe-o2OgGGBUhSgx69PvWNpc0R-T99S-ddowqBdA5qZeDIV7fRHr9QwVViF62UvioqUfR3cDYK0l6EQnEFv9HO4W2MqlcHziMQsnWk58yA8J2rj-bPiTj4zfADnSmm8s2C6EBwDKHXE/s1600/Atty.JPG" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;width: 320px; height: 240px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgSMGe-o2OgGGBUhSgx69PvWNpc0R-T99S-ddowqBdA5qZeDIV7fRHr9QwVViF62UvioqUfR3cDYK0l6EQnEFv9HO4W2MqlcHziMQsnWk58yA8J2rj-bPiTj4zfADnSmm8s2C6EBwDKHXE/s320/Atty.JPG" border="0" alt="" id="BLOGGER_PHOTO_ID_5589220244240734642" /></a><br /><div>The well-earned reward:<br /><div><br /></div><div><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgvWpgTcp6IBIRNhN0U1VlDyrlMRB3oKbpgj9Zx-v_jj9LJQ2m44BgH_OLoSTREY5-YLRgUex3V-1wg3BaGzWU1h41i9BnOOphZMQYwwIeZwxmXsaOEHmpMX_fOjgStNWnWbPZzk7z_mwI/s1600/Stanley.JPG" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgvWpgTcp6IBIRNhN0U1VlDyrlMRB3oKbpgj9Zx-v_jj9LJQ2m44BgH_OLoSTREY5-YLRgUex3V-1wg3BaGzWU1h41i9BnOOphZMQYwwIeZwxmXsaOEHmpMX_fOjgStNWnWbPZzk7z_mwI/s320/Stanley.JPG" border="0" alt="" id="BLOGGER_PHOTO_ID_5589220258875903906" style="display: block; margin-top: 0px; margin-right: auto; margin-bottom: 10px; margin-left: auto; text-align: center; cursor: pointer; width: 320px; height: 240px; " /></a></div><div><br /></div></div></div>Jessehttp://www.blogger.com/profile/13804151393945151752noreply@blogger.com0tag:blogger.com,1999:blog-5981706365500604046.post-16294704841079574032011-01-25T13:54:00.000-08:002011-01-25T14:06:07.325-08:00Villas, Cars, and Interest RatesThe past several classes with Alec, Lukas, and Max, we've been talking about interest rates and about how much things really cost. If you want to buy a car, or a house, and you were to take out a loan, how much would you end up having to pay back? Of course, this has been drastically oversimplified so none of us get headaches.<div><br /></div><div>Last week they were enjoying being "Unfair Banks" where they charged 99% interest on loans. Not realistic, but funny. </div><div><br /><div>Since they were enjoying talking about large amounts of money last week, today we got really expensive. We pulled out the computers and by searching for "big fancy mansion," Lukas found a <a href="http://en.bellesdemeures.com/recherche.htm?idtt=8&pxmin=10000000&tri=d_px">website selling French Villas</a>. Things are expensive in the south of France - the one they wanted to buy cost 27,500,000 Euros. Yes, you read that correctly. </div></div><div><br /></div><div>We pulled out the <a href="http://www.xe.com/">currency converter</a> and found that this comes to roughly $37 million in US currency. We searched for current interest rates on house loans and found one that was 4.75% interest. Dividing by 365, the daily interest rate would be roughly .013%. </div><div><br /></div><div>If you took out a $37 million loan to buy this house, ONE DAY of interest would cost you approximately</div><div style="text-align: center;">(.00013)(37,000,000)=$4810.</div><div style="text-align: left;">That's enough to buy a car. Probably a used car, but still a car.</div>Jessehttp://www.blogger.com/profile/13804151393945151752noreply@blogger.com0tag:blogger.com,1999:blog-5981706365500604046.post-76260393975124298562011-01-20T08:18:00.000-08:002011-01-20T08:43:26.377-08:00Probability and Falling Down<div style="text-align: left;">On Thursday we explored probability using ideas I got from a UMS workshop with Dr. Schaffer and Mr. Stern, founders of <a href="http://www.mathdance.org/">Math Dance</a>.</div><div><br /></div><div>First we warmed up by doing things like "make a shape with two hands on the floor but no feet."</div><div><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgicoLG2cAypXDjaru1LN-634nJO_qpotc8_KNunKOQbsQYhjs2KM6kQFshCD9U4n6wDQKe-mUcDHEg7j5deEgfLZeXwSdH1AYARArkTesf5i_PM79q8FbEtE_-aA0VJsYvKj00RepElDI/s1600/IMG_3129.JPG"><img src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgicoLG2cAypXDjaru1LN-634nJO_qpotc8_KNunKOQbsQYhjs2KM6kQFshCD9U4n6wDQKe-mUcDHEg7j5deEgfLZeXwSdH1AYARArkTesf5i_PM79q8FbEtE_-aA0VJsYvKj00RepElDI/s320/IMG_3129.JPG" border="0" alt="" id="BLOGGER_PHOTO_ID_5564305559236815954" style="display: block; margin-top: 0px; margin-right: auto; margin-bottom: 10px; margin-left: auto; text-align: center; cursor: pointer; width: 320px; height: 240px; " /></a></div><div><br /></div><div>We progressed from shapes to movements. Everyone had to make up a movement, which could be pretty much anything so long as it had a beginning and an end and didn't involve running into furniture or people. Several kids chose to fall down, because, hey, falling down is fun!</div><div><br /></div><div>We voted on which movements we liked the best and settled on two: Lydia's, which involved running around like a horse, and Daniel's, which involved spinning around and then falling down. Everyone learned both movements.</div><div><br /></div><div>Then came the probability. We flipped a coin 5 times. Heads meant Daniel's movement, Tails meant Lydia's movement. We ended up with a 5-move dance, which we were doing as Renata walked through the room and wondered what on earth this had to do with math! Then we made up another dance by doing another set of coin flips. Then we flipped coins some more because everyone wanted a turn to be the coin flipper.</div><div><br /></div><div>Although there was a lot of falling down, yes, we really were doing math!</div><div><ul><li>Problem-solving: How do you get two hands on the floor and no feet? How do you make these different shapes with your body? </li><li>Probability: Instead of just seeing that the coin comes up about half heads and about half tails, there's a physical understanding that the coin tells you to do one move about half the time and the other move about half the time. </li><li>Collaboration: How do we all run around like horses in a small room without crashing into each other?</li></ul></div><div>Today we concentrated on the coin-flipping aspect. Each kid flipped a coin 100 times and recorded the number of heads and tails. Most got about 45/55, one got 31/69, one got 50/50. Not everyone made it to 100 flips, but those that didn't still got about half one and half the other.</div><div><br /></div><div>Everyone recognized 31/69 as being interesting, so we talked about what could have caused so many tails. The flipper of that coin suggested the head side was heavier. Someone else said that pennies are made like a sandwich, with a core of one metal inside and another metal on the outside, and suggested that the core of this coin could be off-center, which would mess with the balance.</div>Jessehttp://www.blogger.com/profile/13804151393945151752noreply@blogger.com0tag:blogger.com,1999:blog-5981706365500604046.post-31425407287918080002011-01-13T11:18:00.001-08:002011-01-13T11:21:14.441-08:00Killer Sudoku, cont.<div style="text-align: left;">We finished the Killer Sudoku! Here it is, in all its glory:</div><div><br /></div><div><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiupRIEDVcIS_K_GyQcJMWXypXmdx2pBDcwLjJGdjSzXbS1YDe4vP_m9vibCF7VfnwXBKZhGslSdXL3lmE9mJ4anSRm6A-QgG62fW50DqxXvzQAiUuhmy-q-aJaHcFWYeTyJFG71D1yXeY/s1600/IMG_3115.JPG"><img src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiupRIEDVcIS_K_GyQcJMWXypXmdx2pBDcwLjJGdjSzXbS1YDe4vP_m9vibCF7VfnwXBKZhGslSdXL3lmE9mJ4anSRm6A-QgG62fW50DqxXvzQAiUuhmy-q-aJaHcFWYeTyJFG71D1yXeY/s320/IMG_3115.JPG" border="0" alt="" id="BLOGGER_PHOTO_ID_5561752500027769682" style="display: block; margin-top: 0px; margin-right: auto; margin-bottom: 10px; margin-left: auto; text-align: center; cursor: pointer; width: 320px; height: 240px; " /></a></div><div>And here's Saul posing with the finished puzzle:</div><div><br /></div><div><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhCjd05GmJqkwxnE2kapqQ3lFPKcK0NboRx3MOPahBjPzqa_avNGxEZHRyLii_KuxkGfOHD4ZhmcAwxKzr5M1Gsx7XW4cTUPLtNUv8sUWPdk_z7pBjbS9Mrr_q8AsmOsbeB_In2taTsasQ/s1600/IMG_3116.JPG"><img src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhCjd05GmJqkwxnE2kapqQ3lFPKcK0NboRx3MOPahBjPzqa_avNGxEZHRyLii_KuxkGfOHD4ZhmcAwxKzr5M1Gsx7XW4cTUPLtNUv8sUWPdk_z7pBjbS9Mrr_q8AsmOsbeB_In2taTsasQ/s320/IMG_3116.JPG" border="0" alt="" id="BLOGGER_PHOTO_ID_5561752508033458034" style="display: block; margin-top: 0px; margin-right: auto; margin-bottom: 10px; margin-left: auto; text-align: center; cursor: pointer; width: 320px; height: 240px; " /></a></div>Jessehttp://www.blogger.com/profile/13804151393945151752noreply@blogger.com0tag:blogger.com,1999:blog-5981706365500604046.post-22325438097554240872011-01-12T10:14:00.000-08:002011-01-12T10:44:03.692-08:00Perimeter and City Blocks<div>At some point in your life you were probably asked a question like this: what is the perimeter of this shape?</div><div><br /></div><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj-Jmm8_lfBGiY0j5dW7q5z0D0s1Jq5vI_Rhp6Pr5s8H5VIoBc4JigH7cr29Usvm1_S7K-muJqJ3BFiw86X-iFSkKRxP-MbOZ32M_1QAYxAb9YUQ49_tbqh8mN4I7AeYDeBHXl54lZUAG0/s1600/blocks2a.jpg"><img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;width: 320px; height: 301px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj-Jmm8_lfBGiY0j5dW7q5z0D0s1Jq5vI_Rhp6Pr5s8H5VIoBc4JigH7cr29Usvm1_S7K-muJqJ3BFiw86X-iFSkKRxP-MbOZ32M_1QAYxAb9YUQ49_tbqh8mN4I7AeYDeBHXl54lZUAG0/s320/blocks2a.jpg" border="0" alt="" id="BLOGGER_PHOTO_ID_5561367102857679282" /></a>This is known as a "composite" shape, and kids are taught to find the perimeter of such a thing by looking at it as a bunch of rectangles stuck together, finding the length of each little line, and then adding the lengths. This is a fine method, works great. However, when the shapes are particularly nice, there's a much easier way.<div><br /></div><div>This easier way is based on how we measure distances in a city. Assuming we can't cut through buildings, we measure distances in a city by blocks. In the picture below, the distance from A to B is 7 blocks. Any reasonable path from A to B must go a total of 4 blocks East and 3 blocks North. We don't have to do all the traveling East at once - we might go 1 block E, then 3 blocks N, then 3 blocks E. We've still travelled a total of 4 blocks E and 3 blocks N. <div><div><br /></div><div><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgUd7bq27NjPMd_6la3Vuk72_llXEwIj8ic6BF3l94-PBZ4Q42MGfCFITuqTIDCJ3ck7F5o-cKQX_pjrgqzEenEFsgMei46gzvfpVvtzQJ87Cn-8a09YPm_cd1nXyN_ZL_5Xf5sV_iXL1c/s1600/blocks1.jpg"><img src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgUd7bq27NjPMd_6la3Vuk72_llXEwIj8ic6BF3l94-PBZ4Q42MGfCFITuqTIDCJ3ck7F5o-cKQX_pjrgqzEenEFsgMei46gzvfpVvtzQJ87Cn-8a09YPm_cd1nXyN_ZL_5Xf5sV_iXL1c/s320/blocks1.jpg" border="0" alt="" id="BLOGGER_PHOTO_ID_5561367091288683906" style="display: block; margin-top: 0px; margin-right: auto; margin-bottom: 10px; margin-left: auto; text-align: center; cursor: pointer; width: 320px; height: 276px; " /></a>This brings us back to the original question, how to find the perimeter of the composite shape. We can think of this shape as being made of two paths from A to B, where each path uses only the directions E and N.<span class="Apple-style-span" style="color:#000000;"><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh8Sh4zEbehXO54dszjBQKroM2Zq-rmcuW5khBsjfLXmKOI47SG_jCdoOwGy3qR3ClS7bvDpgAN0tHUFgDpahp56mNIzmgnlNPfoHERk0ZR5IE7Fdvu7dIVH0dGk1on-ft-L3b-dYn2nc8/s1600/blocks2.jpg"></a></span></div><div><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh8Sh4zEbehXO54dszjBQKroM2Zq-rmcuW5khBsjfLXmKOI47SG_jCdoOwGy3qR3ClS7bvDpgAN0tHUFgDpahp56mNIzmgnlNPfoHERk0ZR5IE7Fdvu7dIVH0dGk1on-ft-L3b-dYn2nc8/s1600/blocks2.jpg" style="text-decoration: none;"><img style="text-align: left;display: block; margin-top: 0px; margin-right: auto; margin-bottom: 10px; margin-left: auto; cursor: pointer; width: 320px; height: 300px; " src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh8Sh4zEbehXO54dszjBQKroM2Zq-rmcuW5khBsjfLXmKOI47SG_jCdoOwGy3qR3ClS7bvDpgAN0tHUFgDpahp56mNIzmgnlNPfoHERk0ZR5IE7Fdvu7dIVH0dGk1on-ft-L3b-dYn2nc8/s320/blocks2.jpg" border="0" alt="" id="BLOGGER_PHOTO_ID_5561367096560264626" /></a>Looking at the "lower" path, we see that to travel from A to B one must go a total of 24 blocks E and 15 blocks N. Although we aren't given any distances on the "upper" path, we know that the red line segments on the upper path must take us the same distance East as the red line segments on the lower path (24 blocks). Similarly, the black line segments on the upper path must take us the same distance North as the black line segments on the lower path (15 blocks).</div><div><br /></div><div>The upper and lower path each go 24 blocks E and 15 blocks N, so the total perimeter is</div><div style="text-align: center;">2(24+15)=2(39)=78.</div><div><br /></div><div><div style="text-align: left;"><span class="Apple-style-span">Warning: this trick is just for shapes made up of two paths that only travel East and North!</span></div></div></div></div>Jessehttp://www.blogger.com/profile/13804151393945151752noreply@blogger.com0tag:blogger.com,1999:blog-5981706365500604046.post-39659452913030785192011-01-11T08:54:00.000-08:002011-01-11T09:59:20.796-08:00My first math publication!<span class="Apple-style-span" style="font-family:arial, sans-serif;font-size:100%;"><span class="Apple-style-span" style="border-collapse: collapse; font-size:13px;"><div>About two years ago I submitted a paper to the journal "Discrete Mathematics." I got the response "revise and resubmit" (translation: "make some changes, then we'll look at it again"). I revised and resubmitted and got the same response. I revised and resubmitted. Last week, I got a different response: my paper has been accepted for publication!</div><div><br /></div><div>The name of the paper is "Improved Upper Bounds for the Information Rates of the Secret Sharing Schemes Induced by the Vamos Matroid." There are very few people in the world who know exactly what that means. </div><div><br /></div><div>To summarize the punchline, there's a number x that mathematicians know is less than 10/11 (about .91) and a number y that mathematicians know is less than 9/10 (.9). I showed that x has to be less than 8/9 (about .89) and y has to be less than 17/19 (about .895). This probably doesn't sound like a big deal, but it was a big enough deal for the paper to get accepted! A surprisingly large amount of math consists of chipping away at numbers like this.</div><div><br /></div><div>My favorite part is the <i>way</i> I showed that x and y had to be smaller than previously thought. Instead of using equations, I found a way to prove things using pretty pictures. :)</div></span></span>Jessehttp://www.blogger.com/profile/13804151393945151752noreply@blogger.com4tag:blogger.com,1999:blog-5981706365500604046.post-84983923953516515652011-01-05T12:29:00.000-08:002011-01-13T12:13:37.326-08:00Killer Sudoku<div><a href="http://www.killersudokuonline.com/">"Killer Sudoku"</a> is like normal sudoku but with a twist. Instead of being given some numbers to start with, you're told what certain groups of cells add up to. For example, if you know that the numbers in two cells must add up to 17, you know the number in one cell must be 8 and the other must be 9 (although you don't know which number goes in which cell). </div><div><br /></div><div>Today we started work on <a href="http://www.killersudokuonline.com/play.html?puzzle=D3s2not1444">this puzzle</a>. The kids felt like it was going very slowly because we weren't able to fill in many numbers, but they were making great progress at narrowing down the options, which is how it goes with these puzzles!</div>Jessehttp://www.blogger.com/profile/13804151393945151752noreply@blogger.com1tag:blogger.com,1999:blog-5981706365500604046.post-56976889040876632592011-01-04T12:17:00.000-08:002011-01-04T13:03:14.700-08:00A General Thank-YouThank you to all the families who chipped in for the gift card! I'm having fun deciding how best to use it at <a href="http://bellaninadayspa.com/">Bellanina</a>. :)<div><br /></div><div>I hope everyone had a nice holiday season, and wish you all the best for 2011!</div><div><br /></div><div>Jesse</div>Jessehttp://www.blogger.com/profile/13804151393945151752noreply@blogger.com1tag:blogger.com,1999:blog-5981706365500604046.post-58921720596163363622011-01-04T07:45:00.000-08:002011-01-04T08:08:37.571-08:00Coloring<div style="text-align: left;">New year, new math. I'm tired of origami (the three-headed dragon took a <b>very</b> long time), so we're moving on.</div><div><br /></div><div>Right now we're working on coloring "maps." The "maps" are drawings with various shapes, and each shape is considered its own country - no funny stuff like Alaska not being attached to the rest of the US.</div><div><br /></div><div>When coloring, two countries that share a border aren't allowed to be the same color. If two countries only touch at a corner (like Utah and New Mexico in the 4 corners), they are not considered to share a border, so they are allowed to be the same color.</div><div><br /></div><div>The challenge: given a map, how many colors do you need to be able to color it following the rules? Use the <b>smallest</b> number of colors you can get away with.</div><div><br /></div><div>Here's an example:</div><div><span class="Apple-style-span" style="color: rgb(0, 0, 238); -webkit-text-decorations-in-effect: underline; "><img src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgdaJ3JaSjS0gsgXrmyCKPpswurhW_jR8VOn4wv3SkWLZVMCgR7GCmxuS22ndli5qQHKBhGZjwjDwMbjfh3rmS2QgEdC9mwEVfuOMHtPpbrOWtYmqLRb22V4xNjZbTJXc-MftXcMwf_NgI/s320/4colorBW.jpg" border="0" alt="" id="BLOGGER_PHOTO_ID_5558361887408039570" style="display: block; margin-top: 0px; margin-right: auto; margin-bottom: 10px; margin-left: auto; text-align: center; cursor: pointer; width: 201px; height: 200px; " /></span></div>We can't color this following the rules with only 1 color, or with 2. What about 3? Maybe we need 4, or 5.<div><br /></div><div>I gave the kids a bunch of maps (this was the hardest one) and for each map they figured out the smallest number of colors they could get away with. Then they drew their own maps and started figuring out how many colors they would need for those.</div>Jessehttp://www.blogger.com/profile/13804151393945151752noreply@blogger.com0tag:blogger.com,1999:blog-5981706365500604046.post-68575955391762763462010-12-02T13:00:00.001-08:002010-12-02T13:18:42.468-08:00Adventures in Origami<b><div><span class="Apple-style-span" style="font-weight: normal;">I've been on an origami kick recently, and encouraging the habit in the kids. </span></div><div><span class="Apple-style-span" style="font-weight: normal;"><br /></span></div><div><span class="Apple-style-span" style="font-weight: normal;">Origami is useful for a great many things: fine motor control, intuitive understanding of symmetry ("now fold it the same way on the other side"), and three-dimensional visualization ("hold your model in the same orientation as the picture"), to name a few. It's an activity where being careful really does matter if you want to end up with something recognizable, and where practicing makes you better pretty quickly.</span></div><div><br /></div>Origami EB</b><div><br /></div><div>During this six-week class we made <a href="http://monkey.org/~aidan/origami/crane/crane1.html">paper cranes</a>, warblers, and <a href="https://www.math.lsu.edu/~verrill/origami/tetraunit/sonobe1.gif">Sonobe Units</a> which can be assembled to form a cube. My favorite class was the one where we made paper cranes out of pieces of paper that started as 5-foot squares!</div><div><br /></div><div><b>Three-Headed Dragons</b></div><div><br /></div><div>Lukas, Alec, and Max are currently working on the <a href="http://www.happyfolding.com/gallery-montroll-three_headed_dragon">three-headed dragon</a> designed by John Montroll. This is a study in persistence. So far we've spent at least two hours on this beast, and we might be halfway done. The dragons, mine included, currently look like blobs with little tails sticking out.</div><div><br /></div><div><b>Area and Volume</b></div><div><b><br /></b></div><div>Start with a small square piece of paper, side length <i>s</i>. The area of a big square piece of paper with side length 2<i>s</i> is 4 times the area of the small paper, as we can see by putting 4 pieces of origami paper next to each other. </div><div><br /></div><div>If we make a cube with small pieces of paper (side length <i>s</i>), we get a small cube. If we make a cube with big pieces of paper (side length 2<i>s</i>), we get a big cube. How many small cubes fit inside the big cube? Guesses included 4, 5, 6, 8, and 16. By making a big cube and putting a small cube inside it, we could see that 2 layers of 4 small cubes, for a total of 8, would fit inside.</div>Jessehttp://www.blogger.com/profile/13804151393945151752noreply@blogger.com0tag:blogger.com,1999:blog-5981706365500604046.post-62864790212189942282010-05-13T08:22:00.000-07:002010-05-13T08:40:09.554-07:00Infinity, parts 3 & 4<b><span class="Apple-style-span" style="font-weight: normal;"><b>Infinity, part 3</b><br /></span></b><div><br /></div><div>We talked about different types of numbers. First we have the counting numbers:</div><div style="text-align: center;">1,2,3,4,5,...</div><div style="text-align: left;">Then we include 0:</div><div style="text-align: center;">0,1,2,3,4,5,...</div><div style="text-align: left;">Then we include negative counting numbers to get the <i>integers</i>:</div><div style="text-align: center;">...,-5,-4,-3,-2,-1,0,1,2,3,4,5,...</div><div style="text-align: left;">This gets us a "numberline" that looks like a bunch of disconnected dots. If we include <i>rational numbers</i> (all fractions), the numberline fills in. Here's the weird thing: although we can't see any holes in the numberline, there are holes nevertheless - infinitely many holes. We use the <i>irrational numbers</i> to fill in those infinitely many holes that we can't see. The <i>real numbers</i> are the rational and irrational numbers.</div><div style="text-align: left;"><br /></div><div style="text-align: left;">The integers are <i>discrete</i>: that is, there are big gaps between them. The rational numbers and the real numbers are <i>continuous</i>: between any two rational (or real) numbers, there is another rational (or real) number. </div><div><br /></div><div>That was quite enough heavy thinking for one day, so we made <a href="http://www.worldofescher.com/gallery/jpgs/P3L.jpg">Mobius strips</a>.</div><div><br /></div><div><b>Infinity part 4</b></div><div><br /></div><div><b>An Infinity Bigger than Omega</b></div><div><br /></div><div>We finally got to an infinity bigger than omega. First we did some warm-ups. I wrote down three 3-digit numbers:</div><div style="text-align: center;">123</div><div style="text-align: center;">234</div><div style="text-align: center;">235</div><div style="text-align: left;">and asked for a 3-digit number that was different from the first in the first place, from the second in the second place, and from the third in the third place:</div><div style="text-align: left;"><div style="text-align: center; "><span class="Apple-style-span" style="color:#FF0000;">1</span>23</div><div style="text-align: center; ">2<span class="Apple-style-span" style="color:#FF0000;">3</span>4</div><div style="text-align: center; ">23<span class="Apple-style-span" style="color:#FF0000;">5</span></div></div><div>One possible answer is</div><div style="text-align: center;">278</div><div style="text-align: left;">because the first digit, 2, is different from 1; the second digit, 7, is different from 3; and the third digit, 8, is different from 5.</div><div style="text-align: left;"><br /></div><div style="text-align: left;">If we do this with infinite decimals between 0 and 1, we can argue that it's impossible for these all to "hold hands" with the counting numbers. That means there are uncountably many real numbers - that is, more than omega-many. This is known as <a href="http://planetmath.org/encyclopedia/CantorsDiagonalArgument.html">Cantor's Diagonal Argument</a>.</div><div><br /></div><div><b>Arc Length</b></div><div><br /></div><div>Draw a squiggle on a piece of paper. Now measure that squiggle using only a straight ruler! It's hard to get it just right, but by measuring little lines you can make a pretty good guess. If you make the lines shorter, you get an even better guess.</div><div><br /></div><div><b>Zeno's Paradox</b></div><div><br /></div><div>Start out 10 feet from a wall. Walk halfway to the wall. Walk half of the remaining distance. And again. No matter how many times you walk halfway to the wall, there's still a distance between you and the wall. So you can never get to the wall! But somehow, you can walk over and touch the wall.</div><div><br /></div><div><b>Cookies!</b></div><div><br /></div><div>Start with a cookie. Cookie monster eats half your cookie. Then he eats half of what's left. If this goes on forever, how much cookie is left at the end of time?</div>Jessehttp://www.blogger.com/profile/13804151393945151752noreply@blogger.com1