A Medley of Mathematics

Mind-reading card trick

2011-06-07T08:32:00.000-07:00

Alice 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.

This trick works because of two facts:

When Alice draws a 5-card hand from a deck of cards, she must get at least two cards of the same suit.
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).

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.

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.

To tell Bob how much to count, Alice hands Bob her three remaining cards in the appropriate order:

low medium high - count up 1 step

low high medium - count up 2

medium low high - count up 3

medium high low - count up 4

high low medium - count up 5

high medium low - count up 6

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.

Here are a couple of examples.

Example 1

Alice draws: A clubs, 4 diamonds, 5 hearts, 6 spades, 7 spades.

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.

Her remaining cards are A clubs (low), 4 diamonds (medium), 5 hearts (high). She shows Bob these cards in this order.

Bob knows he must step up 1 from the 6 of spades, so the secret card must be the 7 of spades.

Example 2

Alice draws: K hearts, 4 hearts, 2 clubs, 2 diamonds, J spades.

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.

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

J spades (high), 2 clubs (low), 2 diamonds (medium)

Bob knows to step up 5 from the king of hearts, so the secret card must be the 4 of hearts.

Cryptology - the final foray

2011-06-06T09:57:00.001-07:00

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?

On week six I handed them these rules:

If there are prizes, they must be divided evenly.
Cooperate! Help each other! Also, make sure everyone gets to help.
Leave the room as you found it.

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.

Then I handed them the first clue and stood back.

1. Easiest first! XQGHU WKH WUDVK FDQ

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:

2. The mystery starts with AAA. AIHFACJQMBJOLYW

"Starts with AAA" indicated that they needed to use an Enigma machine with all its rotors initially set to A.

The next clue was found on the BOTTOMBOOKSHELF.

3. VMVVPNPMFGQVGGQCETDI - Jesse

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.

The next clue was found under the MIDDLELUNCHROOMTABLE.

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.

This is very unhelpful! However, when they held one of those funny-looking index cards over it, the cut-out spaces revealed this message:

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.

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.

Catching up on the blog!

2011-06-06T09:32:00.001-07:00

In which we build all the Platonic solids and do one real mathematical proof:

With the big group, we spent a lot of time playing with zome tools to build the Platonic solids. Platonic solids are convex regular polyhedra. Here's what that mouthful means:

A polyhedron is a 3-D object, as opposed to a polygon, which is a 2-D object.
Regular 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.
A polygon or polyhedron is convex 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.

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.

We talked about what a proof 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 geometric proof of this claim. Most of the reasonable listeners were convinced.

In which we do calculus:

With the small group, I had gotten some requests to do calculus, so we did.

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.

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 betterexplained.com.

We had a refresher of what "slope" means. I like the phrase

"slope equals rise over run"

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

(x,x^2) and (x+delta x, (x+delta x)^2)

and talking about what happens as delta x gets smaller and smaller.

In which we do origami:

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.

Creating Hydrogen and Oxygen from Water

2011-05-13T10:03:00.000-07:00

Renata blogged about the electrolysis demonstration Zach did for her class. Here are his directions for a version of that experiment that the kids can do at home.

Simple Electrolysis Experiment

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.

Here are the items you will need.

1: Glass container (a jam jar works well here).
2: Two pencils
3: A piece of cardboard slightly bigger than the glass.
4: Two pieces of thin electrical wire about 8-10 inches in length.
5: Electrical tape.
6: Epsom Salt (this is NOT table salt. Ask your parents!)
7: A 9 volt battery.

Experiment steps:

1: Remove the erasers from the pencils and sharpen both ends.

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.

3: Fill the glass container about 3/4 of the way with water.

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.

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.

6: Finally, attach the other end of the wire to the 9 volt battery with the electrical tape to help hold them in place.

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
bubbles is the Oxygen.

That's it! You have successfully split hydrogen and oxygen from water.

Cryptology - Enigma Machine

2011-05-11T10:54:00.000-07:00

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!

We watched this video 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!

We made our own Enigma machines out of paper and tape. Here's the Enigma machine construction in progress:

Using the engima machine to decode a message:

There's a nice article here about how the Engima machine works and about some of the things that helped the Allies figure out what was going on.

Cryptology - The Alphabet Clock

2011-05-11T09:46:00.001-07:00

In classes 3 and 4 we learned about modular arithmetic and the one-time pad.

Modular arithmetic 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:

7+11 = 6 mod 12

(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.

For cryptology, we use a clock with the 26 letters on it. These correspond to the numbers from 0 to 25.

On the alphabet clock,

0 and 26 both mean A

1 and 27 both mean B

25 and -1 both mean Z

and so on.

To use this clock to encrypt a message, we need some starting plaintext and a key. A key 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.

Let's use CHICKEN as the plaintext and MOOFAZA as the key.

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.

plaintext   CHICKEN   2    7    8    2   10  4  13
key         MOOFAZA  12   14   14    5   0   25  0
sum                  14   21   22    7   10  29 13

Then we translate the numbers back into letters. On the alphabet clock, 29 and 3 are both D.

14 21 22 7 10 29 13
O  V  W  H  K  D  N

We send the ciphertext OVWHKDN.

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:

ciphertext OVWHKDN 14 21 22 7 10 29 13
key        MOOFAZA 12 14 14 5  0 25  0
difference          2  7  8 2 10  4 13

Finally we translate those numbers back into letters:

2 7 8 2 10 4 13
C H I C K  E  N

If the key is totally random and as long as the message, we have what's called a one-time pad. 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.

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!

EB: Cryptology

2011-04-20T11:35:00.000-07:00

The first class we learned about the Caesar Cipher and other shift ciphers, as well as learning some useful cryptology words.

Plaintext - the meaningful English message
Ciphertext - what you actually send; the secret coded message
Encrypt - turn the plaintext into ciphertext
Decrypt - turn the ciphertext back into plaintext

The students paired off and sent their partners encrypted messages to decrypt.

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!

Here's the first one:

MJQQT, HQFXX.

HTSLWFYZQFYNTSX TS GWJFPNSL YMNX HNUMJW!

- OJXXJ

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.

For the next message, I didn't give them any helpful formatting.

KXNDRSCYXOKVCYFOBIXSMO

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.

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)!

VCPT PT E DPYYRHRNV VMSR FY ZFDR

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".

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.

From there, they got the answer!

The picture shows, from left to right:

The ciphertext-to-plaintext translation (we used this more for the first two puzzles)
The ciphertext, with plaintext underneath
A list of common 2-letter English words

More Killer Sudoku

2011-04-19T08:56:00.000-07:00

Yesterday and today we worked on this puzzle. 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!

Origami and Pascal's Triangle

2011-04-14T08:49:00.000-07:00

The 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 boxes. 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).

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:

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

If we sum the numbers in each row, we get doubles:

Sum

1 1

1 1 2

1 2 1 4

1 3 3 1 8

1 4 6 4 1 16

1 5 10 10 5 1 32

If we sum the numbers on each finite diagonal, we get the Fibonacci numbers. I think this is really cool.

Sum

1 1

1 1 1

2 1 2 1

3 1 3 3 1

5 1 4 6 4 1

8 1 5 10 10 5 1

Pascal's Triangle and Tetrahedrons

2011-03-31T11:51:00.000-07:00

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.

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).

The diagonal of Pascal's triangle that contains the numbers 1, 3, 6, 10, 15, etc. consists of the triangular numbers.

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:

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 zometools. I had never played with these before. Now I want buckets of zometools. So much fun!

We all built tetrahedra, myself included.

Lydia built a tetrahedron tree.

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.

He very kindly let Noah put on the last one:

The tetrahedral numbers are 1, 4, 10, ... which just happen to be the next diagonal in Pascal's Triangle. We have the natural numbers, the triangular numbers, and the tetrahedral numbers:

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

1 7 21 35 35 21 7 1

1 8 28 56 70 56 28 8 1

1 9 36 84 126 126 84 36 9 1

1 10 45 120 210 252 210 120 45 10 1

1 11 55 165 330 462 462 300 165 55 11

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.

Patterns in Pascal's Triangle

2011-03-30T07:38:00.000-07:00

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.

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

1 7 21 35 35 21 7 1

1 8 28 56 70 56 28 8 1

1 9 36 84 126 126 84 36 9 1

1 10 45 120 210 252 210 120 45 10 1

1 11 55 165 330 462 462 300 165 55 11 1

1 12 66 220 495 792 924 792 495 220 66 12 1

1 13 78 186 715 1287 1716 1716 1287 715 186 78 13 1

1 14 91 264 901 2002 3003 3432 3003 2002 901 264 91 14 1

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.

The numbers in the second diagonal go up by 1 each time:

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

1 7 21 35 35 21 7 1

1 8 28 56 70 56 28 8 1

1 9 36 84 126 126 84 36 9 1

The numbers in the next diagonal go up by 2, then 3, then 4, then 5, etc.

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

1 7 21 35 35 21 7 1

1 8 28 56 70 56 28 8 1

1 9 36 84 126 126 84 36 9 1

The numbers in the next diagonal go up by 3, then 6, then 10, then 15, etc.:

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

1 7 21 35 35 21 7 1

1 8 28 56 70 56 28 8 1

1 9 36 84 126 126 84 36 9 1

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.

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

1 7 21 35 35 21 7 1

1 8 28 56 70 56 28 8 1

1 9 36 84 126 126 84 36 9 1

1 10 45 120 210 252 210 120 45 10 1

1 11 55 165 330 462 462 300 165 55 11 1

1 12 66 220 495 792 924 792 495 220 66 12 1

1 13 78 186 715 1287 1716 1716 1287 715 186 78 13 1

1 14 91 264 901 2002 3003 3432 3003 2002 901 264 91 14 1

Row 0 has 1 number, the next row has 2 numbers, the next row has 3 numbers, etc.:

1 - 1 number

1 1 - 2 numbers

1 2 1 - 3 numbers

1 3 3 1 - 4 numbers

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).

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:

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

1 7 21 35 35 21 7 1

1 8 28 56 70 56 28 8 1

1 9 36 84 126 126 84 36 9 1

1 10 45 120 210 252 210 120 45 10 1

1 11 55 165 330 462 462 300 165 55 11 1

1 12 66 220 495 792 924 792 495 220 66 12 1

1 13 78 186 715 1287 1716 1716 1287 715 186 78 13 1

1 14 91 264 901 2002 3003 3432 3003 2002 901 264 91 14 1

The other numbers in that diagonal, on the odd-number rows, are special for a different reason:

3*1=3

5*2=10

7*3=21

9*4=36

etc.

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

1 7 21 35 35 21 7 1

1 8 28 56 70 56 28 8 1

1 9 36 84 126 126 84 36 9 1

1 10 45 120 210 252 210 120 45 10 1

1 11 55 165 330 462 462 300 165 55 11 1

1 12 66 220 495 792 924 792 495 220 66 12 1

1 13 78 186 715 1287 1716 1716 1287 715 186 78 13 1

1 14 91 264 901 2002 3003 3432 3003 2002 901 264 91 14 1

Counting Part 3: Buttons

2011-03-28T13:37:00.001-07:00

Buttons (The Big Group)

Take 5 buttons: red, white, green, yellow, and purple. How many ways are there to choose 1 button from those 5? 5, of course.

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.

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

5C1=5

5C2=10

The most prevalent guess was that 5C3 would be 15, but it turned out that 5C3=10.

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:

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

0C0=1

If we start with 1 button, we get

1C0=1, 1C1=1

With 2 buttons,

2C0=1, 2C1=2, 2C2=1

With 3 buttons,

3C0=1, 3C1=3, 3C2=3, 3C3=1

We continued down to row 6, and got the following pretty picture:

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

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:

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

On their own, each kid completed the triangle up to row 10 and looked for patterns. Some student observations:

All the numbers in row 7 (except 1) are multiples of 7:

1 7 21 35 35 21 7 1

All the numbers along the outside of the triangle are 1.
The numbers that aren't quite along the outside go up by 1 each time:

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

To fill in the triangle, you only have to find half of a row because the second half mirrors the first:

1 6 15 20 15 6 1

In rows that have a "middle number," the middle number is always even:

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

Counting Part 2: Poker

2011-03-28T13:36:00.000-07:00

Poker (The Small Group)

The first question: How many possible poker hands are there?

After playing around with 2- and 3- card hands, which are easier to think about, they arrived at their first answer:

52*51*50*49*48.

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

(52*51*50*49*48)/(5*4*3*2*1)=2,598,960

possible poker hands.

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

(4)/(2598960) ~=~ .0000015

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

(36)/(2598960) ~=~ .000014

These hands are not very likely!

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.

A-A-_-_-_

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.

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

(13*6*48*44*40)/6 = 1,098,240

ways to get a single pair. The probability of getting a single pair in a five-card poker hand is

1,098,240 / 2,598,960 ~=~ .42

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.

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. :)

Here's a site that explains how to calculate odds for various poker hands.

Counting, Part 1: Dice

2011-03-28T12:50:00.000-07:00

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.

Dice (Everyone)

The big group and the small group did the same activities on different days, with slight variations.

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?

The students came up with a very nice way to write the results. It looks like a mountain:

1+6

3+3 6+1 4+4

3+2 4+2 5+2 6+2 5+4

2+2 2+3 2+4 2+5 2+6 4+5 5+5

2+1 3+1 4+1 1+5 4+3 3+5 3+6 4+6 5+6

1+1 1+2 1+3 1+4 5+1 3+4 5+3 6+3 6+4 6+5 6+6

Sum: 2 3 4 5 6 7 8 9 10 11 12

Ways: 1 2 3 4 5 6 5 4 3 2 1

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. Look at the pretty mountains.

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).

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.

1+_+_=6 _+_=5 4 ways

We can do the same sort of thing for the other possible values of the first die.

2+_+_=6 _+_=4 3 ways

3+_+_=6 _+_=3 2 ways

4+_+_=6 _+_=2 1 way

This means there are 4+3+2+1=10 ways for three dice to sum to 6.

Using this shortcut, I think the small group counted how many ways you could make each possible sum with 4 dice!

Bagels

2011-03-28T12:39:00.000-07:00

We made a mathematically correct breakfast! 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.

The well-earned reward:

Villas, Cars, and Interest Rates

2011-01-25T13:54:00.000-08:00

The 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.

Last week they were enjoying being "Unfair Banks" where they charged 99% interest on loans. Not realistic, but funny.

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 website selling French Villas. Things are expensive in the south of France - the one they wanted to buy cost 27,500,000 Euros. Yes, you read that correctly.

We pulled out the currency converter 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%.

If you took out a $37 million loan to buy this house, ONE DAY of interest would cost you approximately

(.00013)(37,000,000)=$4810.

That's enough to buy a car. Probably a used car, but still a car.

Probability and Falling Down

2011-01-20T08:18:00.000-08:00

On Thursday we explored probability using ideas I got from a UMS workshop with Dr. Schaffer and Mr. Stern, founders of Math Dance.

First we warmed up by doing things like "make a shape with two hands on the floor but no feet."

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!

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.

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.

Although there was a lot of falling down, yes, we really were doing math!

Problem-solving: How do you get two hands on the floor and no feet? How do you make these different shapes with your body?
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.
Collaboration: How do we all run around like horses in a small room without crashing into each other?

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.

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.

Killer Sudoku, cont.

2011-01-13T11:18:00.001-08:00

We finished the Killer Sudoku! Here it is, in all its glory:

And here's Saul posing with the finished puzzle:

Perimeter and City Blocks

2011-01-12T10:14:00.000-08:00

At some point in your life you were probably asked a question like this: what is the perimeter of this shape?

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.

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.

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.

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).

The upper and lower path each go 24 blocks E and 15 blocks N, so the total perimeter is

2(24+15)=2(39)=78.

Warning: this trick is just for shapes made up of two paths that only travel East and North!

My first math publication!

2011-01-11T08:54:00.000-08:00

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!

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.

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.

My favorite part is the way 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. :)

Killer Sudoku

2011-01-05T12:29:00.000-08:00

"Killer Sudoku" 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).

Today we started work on this puzzle. 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!

A General Thank-You

2011-01-04T12:17:00.000-08:00

Thank you to all the families who chipped in for the gift card! I'm having fun deciding how best to use it at Bellanina. :)

I hope everyone had a nice holiday season, and wish you all the best for 2011!

Jesse

Coloring

2011-01-04T07:45:00.000-08:00

New year, new math. I'm tired of origami (the three-headed dragon took a very long time), so we're moving on.

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.

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.

The challenge: given a map, how many colors do you need to be able to color it following the rules? Use the smallest number of colors you can get away with.

Here's an example:

We can't color this following the rules with only 1 color, or with 2. What about 3? Maybe we need 4, or 5.

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.

Adventures in Origami

2010-12-02T13:00:00.001-08:00

I've been on an origami kick recently, and encouraging the habit in the kids.

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.

Origami EB

During this six-week class we made paper cranes, warblers, and Sonobe Units 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!

Three-Headed Dragons

Lukas, Alec, and Max are currently working on the three-headed dragon 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.

Area and Volume

Start with a small square piece of paper, side length s. The area of a big square piece of paper with side length 2s is 4 times the area of the small paper, as we can see by putting 4 pieces of origami paper next to each other.

If we make a cube with small pieces of paper (side length s), we get a small cube. If we make a cube with big pieces of paper (side length 2s), 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.

Infinity, parts 3 & 4

2010-05-13T08:22:00.000-07:00

Infinity, part 3

We talked about different types of numbers. First we have the counting numbers:

1,2,3,4,5,...

Then we include 0:

0,1,2,3,4,5,...

Then we include negative counting numbers to get the integers:

...,-5,-4,-3,-2,-1,0,1,2,3,4,5,...

This gets us a "numberline" that looks like a bunch of disconnected dots. If we include rational numbers (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 irrational numbers to fill in those infinitely many holes that we can't see. The real numbers are the rational and irrational numbers.

The integers are discrete: that is, there are big gaps between them. The rational numbers and the real numbers are continuous: between any two rational (or real) numbers, there is another rational (or real) number.

That was quite enough heavy thinking for one day, so we made Mobius strips.

Infinity part 4

An Infinity Bigger than Omega

We finally got to an infinity bigger than omega. First we did some warm-ups. I wrote down three 3-digit numbers:

123

234

235

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:

123

234

235

One possible answer is

278

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.

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 Cantor's Diagonal Argument.

Arc Length

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.

Zeno's Paradox

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.

Cookies!

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?

Infinity, part 5

2010-05-13T08:06:00.000-07:00

The first thing talked about today was why "infinity minus 1" doesn't make sense. Look at a list of numbers:

1,2,3,4,5,6,7,...

Adding 1 to a number means moving one number to the right in the list. Subtracting 1 from a number means moving one number to the left in the list. If we try to move one number to the left of infinity, where do we end up? We would have to land on a finite number, but which finite number? Gah! "Infinity minus 1" is undefined, because there is no answer that makes sense.

Then we talked about limits. The "limit" of a process can be thought of as what you end up with if you could actually get to the end of time. Here are several examples.

Take a cookie. Eat half the cookie. Now eat half the cookie that remains. And eat half again. Continue this for all time. The limit is no cookie left!
Inscribe a triangle in a circle. Then a square. Then a pentagon. Then a hexagon. Continue. The limit is a circle.
Take a line segment. Draw dots to divide the line segment into thirds. Then erase the middle third (but keep the dots). With the two remaining line segments, draw dots to divide into thirds, then erase the middle third (but keep the dots). Continue. The limit is the Cantor Set.
This one's easy to see, but hard to explain: the Koch Snowflake. And here's a good edible approximation.
Draw a square. Divide into ninths, and remove the middle ninth. Divide each remaining square into ninths and remove the middle ninth. Continue. The limit is the Sierpinski Carpet. We didn't make Sierpinski cookies today, but I'm eyeing that for a future project.
If you take the idea of the Cantor set (1 dimension) and the Sierpinski Carpet (2 dimensions) and go up to 3 dimensions, you get the Menger Sponge. Someone said this made them dizzy - I said it makes me a bit dizzy, too!

My favorite part of the class was when the kids figured out, all on their own, that the Cantor set contains no lines but infinitely many dots. Wow :)

We also had a discussion about how many sides a circle has. 0? 1? infinitely many? 2 (an inside and an outside)?

Infinity, parts 1 & 2

2010-04-23T12:52:00.000-07:00

Infinity, part 1

Last week in "Approaching Infinity," we talked about words. "Finite" comes from the root "fin," which means "end" in French. "Finite" means having an end. "INfinite" means not having an end, or being endless. We also brainstormed words meaning "really big": big, large, gigantic, enormous, etc.. Some words that go along with "infinity" are "endless," "forever," and "eternity."

We identified some of the finite things in the room: a paperclip, our bodies, a pencil. We made the infinity symbol out of human bodies:

I told a story about a mountain made of sand and a bird that carries away one grain of sand every thousand years. When the mountain is gone, eternity has barely even started. Here's a fun version of the story as told by Neil Gaiman.

We started set theory. A set is a container that holds things. Some examples:

{1,2,3}

{nose, Peter, tornado}

We learned how to draw the funny "curly brackets" that surround the set. We also looked at the infinite set of all the counting numbers:

{1,2,3,...}

The "..." part means we keep going forever. We listed some other numbers in this set, like 20, and a googleplex, and one hundred, to make sure all the kids understood the "..." part. This set is named "omega," for which I will write w:

w={1,2,3,...}.

I ended class with the following questions to think about for homework:

Which is bigger, w or {1,2,3,...,w}?
For infinite sets, what does "bigger" even mean?

Infinity, part 2

A set is a container. Today we made sets - that is, containers that could hold numbers or other things. The sets included a boat, some origami, and several drawings. Sets can contain sets!

Along the way, it was discovered that if you hold paper and an origami crane over the heating vent, the air makes the paper become "magnetic" and stick to the bird:

We looked at the sets

{1,2,3,4,5,...}

{2,4,6,8,10,...}

and agreed that these sets are the same size, since you can make the numbers "hold hands" (Stanley's expression): 1 and 2 hold hands, 2 and 4 hold hands, and so on, until all the numbers get matched up. Similarly, the sets

{1,2,3,4,5,...,w}

{1,2,3,4,5,...}

are the same size: have w hold hands with 1, then 1 holds hands with 2, and so on, until everybody gets matched up!

{w,1,2,3,4,5,...}

{1,2,3,4,5,...}

The blue squiggly line on the board is inspired by Escher's endless stair.

We also tried to make the endless stair out of bodies. There was mixed success.

Scientific Notation

2010-03-29T13:29:00.001-07:00

Renata posted in her blog about this fun animation she found about the scale of the universe.

She wanted the kids to understand what was actually going on with the numbers, so we spent most of last week talking about scientific notation and what it really means to multiply by 10.

Using exclamation points as units, here's 1:

Here's 10:

!!!!!!!!!!

And here's 100:

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Each time we multiply by 10, we really get a lot more stuff!

Scientific notation uses these facts:

10^0=1

10^1=10

10^2=100

10^(-1)=.1

10^(-2)=.01

With scientific notation, instead of writing a distance as

320 meters

we would write it as

3.2 x 10^2 meters.

This helps us get a sense of how big (or small) things really are, in relation to distances we have some sort of feel for. 2 meters is relatively easy to think about. 20 meters might be a little harder to think about - but 20 meters is really 2 x 10^1 meters, which is 2 meters repeated 10 times.

We had a great time measuring out our own scale of the universe on the sidewalk, and filling in things like the length of our math class (with and without teachers).

Here's Stanley writing the scientific notation:

10^(-1) and 10^(-2) are pretty small!

More Partitions

2010-03-29T13:25:00.000-07:00

Last week I visited Mrs. Carpenter's class and we played the partition game I did with Elaine's class in January. If we have 5 kids, what are the different ways we can group them?

5=4+1=3+2=2+2+1=1+1+1+1+1=2+1+1+1=3+1+1

Measuring Elaine's Classroom

2010-03-10T08:06:00.000-08:00

On Tuesday, I was with Elaine's class. They've been working on measuring - gallons, quarts, pints, cups, etc.. I decided to do something a little more physical: using parts of ourselves as the measuring tools. We measured tables in handlengths (learning the word "perimeter" as we did so), the length of the room in foot-lengths, and the length of the room in person-lengths!

For the last one, we worked in pairs. One person laid down on the floor, and the other held a ruler to mark where their head had been, so the lying-down person could get up and lie down again with their feet where their head had been. I'm sorry I don't have pictures, but I was sort of busy helping remember numbers and measuring the room myself! For most of the kids, the room was about 7 kid-lengths. The room was only about 5 Jesse-lengths.

Mathematical Bagels

2010-03-08T12:39:00.000-08:00

Earlier this year, Joanna sent me a link to a page that explains how to construct a mathematically correct breakfast. With careful cutting, one can turn a bagel into two linked bagel halves!

The activity as described involves drawing on a bagel with a permanent marker. Unfortunately, if you do that you can't eat the bagel. I figured that by now there must be such a thing as edible markers, so I went hunting online and found some. With the aid of food markers, last Thursday we set about bagel-ing.

Most of the kids *almost* got it to work, but had one half that was just a little too thin and broke. We'll try this activity again later in the semester.

The quote of the day, possibly of the year, was from Alec: "The only thing better than food is math that is food."

Step 1: Meet the bagels, and decide which side is the "front."

Step 2: Draw on the bagels.

Step 3: Cut the bagels.

Ta-da!

(This one is mine:)

A Greek letter, the Golden Ratio, and more dancing

2010-02-18T07:40:00.000-08:00

Renata's Class:

Following the theme of logic, Renata had the kids playing a game to help them understand the idea of a "fair witness" - that is, one who can only speak things that are true. My main contribution to Renata's class this week was explaining why my tshirt was funny: the symbol in the cat's speech bubble is the Greek letter mu, pronounced "mew".

Susan's Class:

The Golden Ratio is a weird number sort of like pi. Its decimal expansion goes on forever without repeating, and it shows up everywhere. The Golden Ratio is approximately 1.62. It shows up in the proportions of the ideal body according to Leonardo DaVinci, in the Parthenon, and in the shape of credit cards. Sometimes people disagree on whether the Golden Ratio is part of something (such as the dimensions of the Great Pyramid, or the painting of the Mona Lisa).

We started with the idea of DaVinci's ideal dimensions. We paired the kids up and had them measure each other: first the distance from feet to bellybutton, then the distance from bellybutton to head. They divided the first distance by the second, and mostly got numbers between 1.5 and 1.8 - pretty close to the Golden Ratio!

I mentioned that credit cards are in the shape of a Golden Rectangle, and we started looking for other shapes with similar dimensions. The desks? The blackboard? A tissue box? Next time we need to measure other things besides the kids!

Elaine's Class:

More dancing today, with some jobs including the idea of "if and only if". I was impressed by Emi, who really got the idea that "if A then B" means "if A happens you have to do B, but you can also do B if A doesn't happen".

Implication and Dance

2010-02-11T07:45:00.000-08:00

In Elaine's class today, we danced. I gave each kid a piece of paper with a job on it. Some of the jobs were

"If someone touches you, fall down."

"Hug people (gently)."

"Act like a Penguin."

Half the class did their jobs, while the other half guessed what the jobs were. Then they switched. We talked about how the rule

"If someone touches you, fall down"

doesn't say anything about what to do if no one touches you. You could fall down or not fall down - either one is ok!

If we wanted you to fall down when someone touched you, and only then, we would have to say so:

"Fall down if and only if someone touches you."

None of the jobs today involved "if and only if", so we'll probably have a future round including some "if and only if" jobs. The kids made suggestions of jobs, so we can use those too.

Truth Tables

2010-02-11T07:38:00.000-08:00

On Monday and Tuesday in Renata's class we learned about truth tables.

Since mathematicians are lazy, we abbreviate whole statements by single letters:

A "Maria has a purple pencil."

B "Alec has a blue pencil."

Then we use a truth table to examine what happens if each statement is true or false. The tilde (~) is used to mean "not" or "negation."

A B (A and B) (A or B) ~A

T T T T F

T F F T F

F T F T T

F F F F T

More on Logic

2010-02-03T08:25:00.000-08:00

It all began with babies and crocodiles, courtesy of Lewis Carroll.

Babies are illogical;
Nobody is despised who can manage a crocodile;
Illogical persons are despised.

Each of these statements can be written as an implication, a statement of the form "A implies B" or "if A, then B". As mathematicians, we abbreviate such a statement by A -> B. We can replace the statements about babies and crocodiles with implications.

baby -> illogical
manage crocodile -> not despised
illogical -> despised

From the statement A -> B we can conclude that its contrapositive (not A) -> (not B) must also be true, so we can replace our implication for statement 2 with

despised -> cannot manage crocodile.

Then we string the chains of implications together:

baby -> illogical -> despised -> cannot manage crocodile.

Turning this back into English, we conclude that babies cannot manage crocodiles.

After doing several of these puzzles, we had the kids come up with their own. Here are some of their puzzles:

All doughnuts are fattening.
0 calorie food is not fattening.
Paczkis are big doughnuts.
Meijer is selling 0 calorie Paczkis. Can this be true?

If you're cool you are a dude.
Luke is not a dude.
If you're not cool you can't get into the club.

All happy people go to Summers-Knoll.
All people at Summers-Knoll are smart.
Victor does not go to Summers-Knoll.

In the last puzzle, we can conclude that Victor does not go to Summers-Knoll. However, we can't conclude that Victor is not smart. From

SK -> smart

we can conclude the contrapositive:

not smart -> not SK

but we can NOT conclude

not SK -> not smart.

There could be smart people who don't go to Summers Knoll!

Logic Puzzles

2010-01-27T08:41:00.000-08:00

Today we worked on a puzzle Renata found in The Man Who Counted. Then I introduced some logic puzzles written by Charles Lutwidge Dodgson, better known as Lewis Carroll. Some of the puzzles may be found here.

Fibonacci Nim

2010-01-21T13:01:00.001-08:00

Last week we started playing a game that starts out with two players and a pile of stones. The players take turns taking away handfuls of stones, and whoever takes the last stone wins. The first player may take as many stones as they like, but not the whole pile. After that first move, each player may take up to twice as many stones as the last person took. Passing is not allowed.

A sample game might go like this: we start out with 10 stones. Player A takes 2 stones, leaving 8. Player B may take up to 4 stones, but decides to take only 3, leaving 5. Player A may now take up to 6 stones, so Player A takes the remaining 5 and wins the game.

Depending on the number of stones in the initial pile, either Player A can always win if they play brilliantly enough, or Player B can always win if they play brilliantly enough. After the kids figured out who should win depending on the number of stones, I revealed that the name of the game is Fibonacci Nim.

I won't say more than that, in case someone wants to play you in this game!

Partitions

2010-01-13T11:06:00.000-08:00

Today I visited Elaine's class and introduced the kids to partitions. A partition is a way of breaking up a number into a sum of numbers. For example, we can partition 3 in three different ways:

3=3+0

3=2+1

3=1+1+1.

In order to make things fun, we partitioned groups of kids. To partition the number 3, we had a group of 3 kids stand holding hands, and the other kids rearranged those 3 into a pair and a single, or three singles. We got all the way up to partitions of 6!

See Elaine's post for pictures!

Modular Origami and Fibonacci Numbers

2010-01-12T10:58:00.000-08:00

On Monday, Renata was out sick and I got to do math all by myself. First we worked in Singapore books, then we did modular origami. I taught those who hadn't learned yet how to make a Sonobe unit (less technically known as a "box side"). Six units can be assembled to make a cube. More units can be assembled to make other things.

We also made sure everyone was familiar with the Fibonacci sequence, which is the sequence of numbers

1,1,2,3,5,8,13,21,34,55,89,...

The first two numbers are predetermined. To get each number from the third number on, we add the previous two numbers. The next number if we continue the sequence above would be

55+89 = 144.

Today, Tuesday, we reviewed how to write numbers in base 3. It can be helpful to break the numbers up first - for example, to write 17 in base 3, first we notice that

17=9+6+2

and then we can write 17 in base 3 as

122.

We also broke numbers up into sums of Fibonacci numbers. The rules:

Use the biggest Fibonacci numbers you can
No Fibonacci number may be used more than once.

Some examples:

11=8+3

12=8+3+1

40=34+5+1

We wouldn't write 40=34+3+2+1, because we can write 5 instead of 3+2, and the first rule says we should use the biggest Fibonacci numbers we can. We also wouldn't write 12=8+2+2, because we're not allowed to use any Fibonacci number more than once.

The kids also came up with their own Fibonacci-type sequences, by deciding what they wanted the first two numbers to be, adding those to get the third number, and so on.

Here they are writing their sequences up on the board:

And here are the completed sequences:

Finally, here's Maria's origami "other thing," using 12 units:

Beyond Base 10

2010-01-07T09:39:00.000-08:00

This week we looked at numbers written in different bases.

We usually write numbers in base 10. In base 10, the string of digits

321

represents the sum of 3 hundreds, 2 tens, and 1 one. In base 10 the places are

..., ten thousands, thousands, hundreds, tens, ones.

Another way to say this is that the places are powers of ten:

..., 10^4, 10^3, 10^2, 10^1, 10^0.

We could also write numbers in base 2, in which case we're only allowed to use the digits 0 and 1. The places in base 2 are powers of two instead of powers of ten:

..., 2^4, 2^3, 2^2, 2^1, 2^0

or in other words,

..., sixteens, eights, fours, twos, ones

In base 2, the string of digits

1101

represents the sum of 1 eight, 1 four, 0 twos, and 1 one for a total of thirteen.

We could also write numbers in base 3, or base 4, or any base we like. Computers use base 2 a lot. They also use base 16, better known as hexadecimal. In base 16 we use the digits from 0 to 9, but we also use letters:

10 A

11 B

12 C

13 D

14 E

15 F

The place values in hexadecimal are powers of 16:

..., 4096, 256, 16, 1

Thus the string

10AF

represents the sum of 1 four-thousand ninety-six, 0 two-hundred fifty-sixes, 10 sixteens, and 15 ones for a total of 4271.

The class solved the following puzzle:

If only you, I, and DEAD people know hexadecimal, how many people know hexadecimal?

Cryptology

2010-01-05T10:34:00.000-08:00

In December we studied codes and properties of the English language. The kids worked in two "competing" groups.

Group 1 came up with this message:

HI STANLEY I LIKE PIZZA THIS CODE IS EASY.

They encoded their message with the classic Caesar Cipher (where each letter is shifted 3, so instead of A we write D, instead of B we write E, and so on). The coded message was:

KL VWDQOHB L OLNH SLCCD WKLV FRGH LV HDVB.

Group 2 broke the code by recognizing that "HI" is a very good word for starting a message, and then seeing that the whole alphabet had been shifted by three characters.

Then Group 2 came up with a message:

HELLO PEOPLE OF EARTH,

THE SKELETAL DEVIL HAS COME OUT OF THE EARTH, FOR I PROSPERO THE MAGICIAN HAVE LET HIM OUT! FOR LUNCH.

P.S. HE WANTS TO EAT YOU ALL.

They encoded this by a random letter substitution, so the coded message was

WEUUI NEINUE IX EGHVW,

VWE YLEUEVGU TERBU WGY QIDE IFV IX VWE EGHVW XIH B NHIYNEHI VWE DGPBQBGZ WGRE UEV WBD IFV XIH UFZQW NY WE KGZVY VI EGV OIF GUU!

Group 1 broke the code using some useful properties of the English language. They knew B should mean either A or I, since those are the only valid one-letter words in English. They also figured out that VWE was THE, since THE is one of the most commonly occurring three-letter words in English. Since the alphabet was all mixed up instead of shifted, this code was pretty tricky to break.

Along the way, I had the kids pick storybooks they were reading and count how many times different letters occurred. This gave everyone a feeling for how common different letters are. For example, T and E show up all the time, but J and Z and Q hardly show up at all. This can be useful in cracking a code - the most frequently occurring symbol probably doesn't mean Z, unless the message is about pizza!

2009-11-19T10:42:00.000-08:00

Here's some of what we've been up to in math recently.

Card Tricks

I start with a deck of cards. I make 6 piles of cards, face up. Then I turn each pile over, so they're face down now. The kids choose 3 piles to keep and give me back the other cards. Then they turn over the top cards on two of the piles. I perform some magic, and tell them the top card on the remaining pile. Of course this trick has to do with math. Maybe I'll reveal the details in a future post.

Measuring Slope

I made a slanted line, and the kids measured my slope. See Renata's post for the story and pictures.

Russell's Paradox

We talked about Russell's Paradox, the short version of which is

"The barber shaves every man who does not shave himself.

Who shaves the barber?"

A more detailed version, with fewer loopholes, is presented here. The kids were great at finding loopholes in the version I gave ("Maybe the barber's a woman!", "The barber is bald!").

Set Theory and Infinity

We talked about sets and about union, intersection, and set difference. Start with two sets,

{1,2,3,4,5} and {2,4,6,8}.

Their union is the set of stuff that's in one or both of the sets:

{1,2,3,4,5,6,8}

Their intersection is the set of stuff that's in BOTH sets:

{2,4}

The set difference {1,2,3,4,5}-{2,4,6,8} is everything in the first set but not the second set:

{1,3,5}

Union, intersection, and set difference can be represented with Venn Diagrams (at least when we're only working with 2 or 3 sets).

We revisited the idea of defining the numbers as sets, only this time we went further.

0={}

1={0}

2={0,1}

10={0,1,...,9}

100={0,1,...,99}

Any whole number can be defined as a set in this way. We can also define infinity as the set of all whole numbers, called w (the Greek letter omega):

w={0,1,2,...,100,...1000,...}

We can add 1 to any whole number, and get "the next" whole number (or set). We can also subtract 1 from any whole number (except 0) and get "the previous" whole number (or set). "Infinity minus 1" doesn't make sense, though, because there is no "previous" set. There is no set that comes immediately before w.

Clocks, origami, and sets

2009-10-22T07:35:00.000-07:00

The first week in October, we did clock arithmetic. On a clock, if you count up to twelve you get back to the beginning, so 12 is basically the same thing as zero. Weird things happen in clock arithmetic. For example, it's possible to multiply two non-zero numbers (2 and 6, for example) and get zero. This doesn't happen with regular integers and real numbers!

The last couple of weeks we've been doing "magic tricks" with a compass and straightedge. We cut a line segment in half without even knowing how long it was, and we cut an angle in half without knowing the measure of the angle. These magic tricks are done in high school geometry. Magic trick 3, however, usually isn't done in school at all: while it's provably impossible to cut an angle into thirds using only a compass and straightedge, it can be done with origami paper!

Today we did some set theory. We created all whole numbers using only the symbols {, }, and a comma. Here's how:

0={} is the empty set

1={{}}

2={{},{{}}}

3={{},{{}},{{},{{}}}}

and so on. Since it's hard to make sense out of all those brackets, we can write the numbers they represent instead. Then we get something prettier:

0={}

1={0}

2={0,1}

3={0,1,2}

and so on, and so on...

Areas, volumes, and dimensions

2009-10-01T07:24:00.000-07:00

This has been the week of finding areas. Renata's been working with the kids on finding the areas of funny shapes by decomposing them into rectangles and triangles and squares. I decided to approach areas from the point of view of units and dimension. If a square has side length 1 ft, what's the area of the square in square inches? Most of the kids said 144 very quickly, which made me happy (having had lots of college students who would have said 12). Then we went to volume, finding volumes in cubic feet and cubic inches.

Today we had a little time left at the end for more fun stuff, so we drew 4-dimensional cubes (and, in the case of one of the kids, a 3-dimensional pair of pants!).

Tilings and Symmetry

2009-09-24T07:31:00.000-07:00

This week we've been making tilings, also known as tessellations. A tiling is what you get when you lay a bunch of shapes edge-to-edge to cover a flat space. The shapes aren't allowed to overlap, and you aren't allowed to leave gaps.

I gave the kids stencils for squares and equilateral triangles whose sides were 1 inch long, and told them to either trace them onto paper to make a tiling, or cut out tiles from construction paper and glue them down to make a tiling. They could use the squares and triangles to make different shapes, like bigger triangles, or "house-shaped" pieces.

We got some very nice tilings, and also some nice pictures made of tiles that weren't tilings because they had overlapping tiles or gaps.

Today we used the tilings to talk about reflectional and rotational symmetry. Reflectional symmetry is when you can draw a line (called the "line of symmetry") through the middle of a picture and the two sides of the picture look the same. Rotational symmetry is when you can rotate the picture and it looks the same after rotation as it did before (for example, take a square and turn it one-quarter turn). We drew some lines of symmetry for squares and triangles, then looked at the types of symmetry in our tilings.

We also looked at pictures from this website, which has pictures of ancient tilings including ones that are Egyptian, Persian, Arabian, and Moresque. When we look at symmetries, all those tilings can be clumped into 17 groups. We talked about some of those groups and the funny names they have.

Discussion: What is Math?

2009-09-15T11:35:00.000-07:00

Today I said I wanted to talk about math. That didn't go over particularly well - the kids responded with things like "math is boring," "I don't like assessments," and "no fractions!"

We digressed to talking about our favorite numbers, which went over a little better. A couple of people liked the number "2" because it's a fun number to write. "-12" was a favorite, as was "1". The discussion got so animated that Renata had to get out a feather (for a talking stick) to keep order.

Then we went around the circle, with (almost) every kid giving an answer to the question "what is math?" Some fun thoughts that came out of this:

mathematicians don't even know that 1+1=2
math is "random," and the only order there is the order we put on it
math is about asking questions and answering them
the word that comes to mind is "progress," because without math we couldn't have computers or cars
do trees have anything to do with math?
in ancient times, didn't people have to know "how far" they could throw a rock, so they could successfully hunt animals?

Along the way we talked about the complex numbers (mathematicians wanted a number that you could square to get -1, so they made one up and called it i) and about mathematical logic (using the notation x arrow y to abbreviate the statement "if it is raining, then there are clouds in the sky").

Introduction

2009-09-15T11:12:00.000-07:00

Hello, world!

My name is Jessica Metcalf-Burton, but I go by Jesse (no "i"). I teach math to a bunch of kids at Summers-Knoll School. This blog is intended for parents to follow along and see what their kids are doing. Since I haven't met most of the parents yet, I figure I should introduce myself first.

I have a Ph.D. in mathematics from the University of Michigan, and undergraduate degrees in math and computer science from the University of Maryland. My first experience as a teaching assistant was my sophomore year of college, and I've pretty much been teaching ever since. At University of Michigan I've taught a simply ridiculous number of calculus courses, and over the last several years I've tutored everything from high school algebra and geometry through calculus.

When I'm not teaching math, I have a side business teaching swing dancing. I learned how to swing dance my first year of college, and it stuck. I also like baking, reading, traveling, traveling to Spain, attempting to speak Spanish, writing poetry, listening to music, playing music (on piano, djembe, recorder, bones, bodhran, or zills), making beaded jewelry, crocheting, trying other forms of dance besides swing, doing photography (preferably while traveling), and eating ice cream.

As the semester gets a little more underway, I'll be introducing some of the kids at Summers-Knoll to advanced math ideas that students usually don't get to until high school (if they're lucky) or college. I have a pretty long list of ideas already, and if anyone has fun math activities or topics they'd like to suggest, I welcome comments.