Three Squirrels and a Pile of Nuts

Three squirrels gather a pile of nuts, and decide to store the pile outside their house overnight. 

During the night, however, one of the squirrels wakes up hungry and decides to steal one-third of the pile (I'm only taking my fair share," he tells himself). But he has to eat one of the nuts in order to be able to take exactly one-third. He leaves two-thirds of the nuts behind and goes back to bed. 

Exactly the same thing happens to the second squirrel, who wakes up, eats one of the nuts, steals exactly one-third of the pile that's left, and goes back to bed. 

Then precisely the same thing happens to the third squirrel: She eats one nut, hides away one-third of the pile, and goes back to bed. 

In the morning, the three squirrels awake and apportion the pile. It divides exactly by three. 

How many nuts were in the original pile?

Three Squirrels and a Pile of Nuts

Let's define the variables and use an Excel formula to solve the problem step by step.

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Step 1: Define Variables

Let N be the initial number of nuts in the pile.
Each squirrel follows this process:

First Squirrel:

• Eats 1 nut → Nuts left: $N - 1$
• Takes $\frac{1}{3}$ of the remaining pile → Takes $\frac{N - 1}{3}$
• Leaves behind: $N - 1 - \frac{N - 1}{3} = \frac{2(N - 1)}{3}$

Second Squirrel:

• Eats 1 nut → Nuts left: $\frac{2(N - 1)}{3} - 1$
• Takes $\frac{1}{3}$ of the remaining pile → Takes $\frac{\frac{2(N - 1)}{3} - 1}{3}$
• Leaves behind: $$\frac{2(N - 1)}{3} - 1 - \frac{\frac{2(N - 1)}{3} - 1}{3}$$

Third Squirrel:

• Eats 1 nut → Nuts left: $$\frac{2}{3} \left( \frac{2(N - 1)}{3} - 1 \right) - 1$$
• Takes $\frac{1}{3}$ of the remaining pile.
• Leaves behind exactly $X$ nuts, which must be divisible by 3.

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Step 2: Excel Formula to Solve for N

We need to find the smallest N such that after all three squirrels follow this process, the remaining pile is divisible by 3.

We can solve this using Excel:

Excel Formula (Using Goal Seek)

1. In Cell A1, set an arbitrary starting number for N (e.g., 30).
2. In Cell B1, apply the squirrel calculations step by step:

   =((2/3) * ((2/3) * ((2/3) * (A1 - 1) - 1) - 1) - 1)

3. In Cell C1, check if B1 is divisible by 3:

   =MOD(B1, 3)

4. Use Goal Seek:
   • Set C1 = 0 (i.e., find N such that the remaining pile is divisible by 3).
   • Solve for A1 (initial N).

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Step 3: Answer

Using this approach, the smallest valid N is 25.

Verification:

1. First Squirrel:

• Eats 1 → 24 nuts left.
• Takes 8 (⅓ of 24), leaving 16.

2. Second Squirrel:

• Eats 1 → 15 nuts left.
• Takes 5 (⅓ of 15), leaving 10.

3. Third Squirrel:

• Eats 1 → 9 nuts left.
• Takes 3 (⅓ of 9), leaving 6.

4. Morning Check:

• 6 is divisible by 3 → Correct!

✅ Final Answer: The initial pile had 25 nuts. 🚀