Killer Sudoku

"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

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

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

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

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

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

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.