A pile of walnuts is shared by 3 friends in turn. The first takes 1/3 of the pile plus 1 more walnut. The second takes 1/3 of what is left plus 1 more, and the third takes 1/3 of what is then left plus 1 more. Every share is a whole number of walnuts, and 13 walnuts remain at the end. How many walnuts were in the pile at the start?

Learn / Library / Brainteasers

Sharing walnuts

BrainteasersMedium

A pile of walnuts is shared by $3$ friends in turn. The first takes $\frac{1}{3}$ of the pile plus $1$ more walnut. The second takes $\frac{1}{3}$ of what is left plus $1$ more, and the third takes $\frac{1}{3}$ of what is then left plus $1$ more. Every share is a whole number of walnuts, and $13$ walnuts remain at the end. How many walnuts were in the pile at the start?

Checking an answer needs JavaScript. The hint and the solution open without it.

Hint

Each turn leaves $\frac{2}{3}$ of the pile minus $1$ ; reverse that.

Solution

If a pile of $p$ loses $\frac{p}{3} + 1$ , then $\frac{2 p}{3} - 1$ remains, so a remainder $r$ came from $p = \frac{3 ( r + 1 )}{2}$ .
Before the third friend there were $3 \times \frac{13 + 1}{2} = 21$ walnuts.
Before the second there were $3 \times \frac{21 + 1}{2} = 33$ .
At the start there were $3 \times \frac{33 + 1}{2} = 51$ walnuts.

Answer: $51$

Next question Back to the library