M&M Math

This M&M math is from Steve Heller, I adapted some of it for younger kids. Get some M&Ms. (I used coins since we have lots of pennies but no M&Ms around the house).

For a 6-year old:

• Make a filled in square. (She made a 3x3 square).
• How many pennies are there in the square? How many rows? How many columns? (She wanted to know which way do rows go?)
• Can you make a smaller square? (She made a 2x2 square). How many pennies? How many rows? How many columns?
• Can you make a smaller square? (She thought for a minute and put a single penny down. And laughed.) One row, one column, and one penny.
• Can you make an even smaller square? (She thought maybe not. I pointed at an empty space on the table. She laughed and said "zero rows, zero columns, zero pennies.

For the 11-year old.

• Make a sequence of triangles (the first one contains 1, the second 3, the third 6, the fourth 10). How much do you add to get from one to the next?
• If you take two triangles of base 4 (10 pennies each) and put them together what do you get (a rectangle that is 5 by 4).
• Can you see way to use that to calculate the number of pennies in a triangle of base size 10? (After some thinking he came up with that two of them were 10x11, so one of them would be 55.)
• Make a sequence of squares. How much do you add to get from one to the next. (You add 3 then 5 then 7 then 9...)
• So the sum of odd numbers from 1 to somewhere is a square.

For the 15-year olds: (They didn't actually touch the pennies. Nooo.. They can do it in their heads...)

• You remember what a pythogorean triple is? (Three integers a,b,c such that a²+b²=c².)
• How many are there? (Lots).
• Can you show there are an infinite number? (Well...)
• Consider a square of side length n and one of side length n+1. What's the difference in their sizes? (2n+1).
• Can 2n+1 be square? (They now jumped ahead to the conclusion. Every odd number when squared is odd. So the first triple is 2n+1 = 1, so you get 0,1,1 (that's cheating!) The second is 2n+1=9, so you get 3,4,5. The third is when 2n+1=25 so you get 5,12,13. (Dana chimed in that you should have those memorized). The next one is 2n+1=49 so you get 7,24,25.
• In general for any integer k, let l=2k+1, then l² is an odd square. The resulting pythogorean triple is l,(l²-1)/2, and (l²+1)/2.

dollar auction

I tried to run a dollar auction at dinner yesterday. At first the kids (13-year-olds) wouldn't bite. Finally they started bidding. Then when it was going badly, one of them bid everything in her bank account. I had to quit since I couldn't take all that money away. She claimed victory. Then they wanted to do it again. They offered a penny, and then split the dollar.

St. Petersberg Paradox

A few days ago we explored the St. Petersberg Paradox.

Here's how I stated it: We'll play a game involving coin flips, and I'll play you depending on how the coins flip. You have to decide how much you are willing to pay to play the game.

A warmup game: I flip a coin, and if it comes up heads I give you $1, tails I give you $2. How much to play?

Both 13-year olds immediately said "up to $1.50".

The real game: I flip a coin. If it comes up heads I give you $1. If it comes up tails, I flip again. If it comes up heads this time, I give you $2. If it comes up tails, I flip again. If it comes up heads this time I give you $4. In general if I flip k tails in a row I pay you 2^k dollars.

My 13-year-old boy thought for a minute and growled at me, complaining that the game was worth an infinite amount of money, and that no such game could really exist. He said he should be willing to pay any amount. My 13-year-old girl (at a different time) thought about at and found it much less upsetting, concluded that it had infinite value and offered $5 to play the game. (We didn't actually play it.)

We ran a few simulations, and found that it's really hard to get that $5 back even if you play many times.

Analysis: The game has no expected value (it's worth an infinite amount of money), but since I only have a finite amount of money in my pocket it's not really worth that much to play against me as the casino. Even if I had a lot of money, it takes a long time to recover a large payments because it requires events that are unlikely to ever be seen. For example, if you paid $100 to play this game, you wouldn't expect to come out ahead until we had played about 2¹⁰⁰ games, because it requires a sequence of 100 tails in a row to cover all the accumulated losses.

zero-knowledge sudoku

Today we did zero-knowledge sudoku.

1-x-x^2

The inverse of 1-x² can be calcuated two ways.

First way: We know 1/(1-y)=(1+y+y²+y³+...) so substitute y=2x, and it comes out.

Second way: Do it directly.

1. The constant coefficent must be 1.
2. The x coefficient must 2.
3. The x² coefficient must 4.
4. The x³ coefficient must 8.

We did a couple more examples of computing an inverse. Then we went to an interesting one.

What is the multiplicative inverse of 1-x-x².

It turns out to be 1+x+2x²+3x³+5x⁴+8x⁵+13x⁶...

The coefficients are the fibonacci series!

1/(1-x)

Treating polynomials as abstract symbolics entities (rather than as a formula into which you substitute a real number for x. Q: What do you get when you multiply (1-x) by (1+x+x²+x³+...) ? A: You get 1. Thus 1/(1-x)=(1+x+x²+x³+...).

5-minute math for a 3-year old

My 3-year old child wanted to do five-minute math. I said, OK, I grabbed the marker, and she said "write a 3". So I wrote a 3. Then she said "write a 2". So I wrote a 2. We spent five minutes with her dictating numbers and I filled up the white board.