Explained

Is there an easier way?

In short, yes. Working through all combinations is rather tedious, and error-prone when you work on larger numbers of nuts. There are two levels of looking at this problem, one is to recognize the pattern, and the other is to understand why that pattern exists.

So what's the pattern?

Let's look at the total number of nuts and total combinations for each again:

• 1 -- 1
• 2 -- 2
• 3 -- 3
• 4 -- 5
• 5 -- 8
• 6 -- 13
• 7 -- 21

See the pattern? Some of the more advanced 4th graders were able to figure it out, so this is a good question to pose them, for example, how many ways can the two squirrels get 10 nuts. And you can always extend this pattern for them to 8 or 9 to help them:

• 8 -- 34
• 9 -- 55

So if you haven't figured it out yet, the answer for 10 nuts is 89. The number of combinations is the sum of the two prior combinations.

Why does it work?

What's more interesting, at least to me, is to understand why this works, for two reasons -- one is to be able to solve similar problems (for example with 3 squirrels or with each squirrel taking more than 1 or 2 nuts at a time), but I think more importantly is to expand a child's thinking and approach toward math problems.

Let's look beyond simply the total combinations and break it down to the combinations that start with 1 or 2 (which is why earlier I included that approach):

• 1 nut -- 1 way
  • 1 -- 1 way
  • 2 -- 0 ways
• 2 nuts -- 2 ways
  • 1 -- 1 way
  • 2 -- 1 way
  • 3 nuts -- 3 ways
    • 1 -- 2 ways
    • 2 -- 1 way
  • 4 nuts -- 5 ways
    • 1 -- 3 ways
    • 2 -- 2 ways
  • 5 nuts -- 8 ways
    • 1 -- 5 ways
    • 2 -- 3 ways
  • 6 nuts -- 13 ways
    • 1 -- 8 ways
    • 2 -- 5 ways
  • 7 nuts -- 21 ways
  • 1 -- 13 ways
  • 2 -- 8 ways

So each time the total is basically the sum of all combinations starting with 1 or 2, of course. But the number of combinations that start with 1 is the same as the total number of combinations as the prior number of nuts. And the total that start with 2 is the same as two prior. In other words:

• 7 nuts -- 21 ways
  • 1 -- 13 ways, or equal to the total ways for 6 nuts
  • 2 -- 8 ways, or equal to the total ways for 5 nuts

Pretty cool, huh? (OK, I'm a nerd.) That's because once you take 1 nut from the pile of 7, the number of possible combinations is the same as for 6 nuts. And when you take 2 nuts from the pile of 7, the number of possible combinations is the same as for 5 nuts. So the reason that each is the sum of the two prior is because one squirrel takes 1 nut and the other takes 2 nuts.

So if you extend this thinking, if you have three squirrels, each taking 1, 2, and 3 nuts, then the total would be the sum of the three prior values. And if you have two squirrels each taking 2 and 3 nuts, then the total would be the sum of n-2 and n-3 prior values (instead of n-1 and n-2).

Final side-note -- this is basically a Fibonacci sequence, which has a closed-form solution for any value N, but I think that's a bit too advanced for 4th graders.