Counting, Part 1: Dice

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!