[ Induction (miscellaneous). ]
$[$ Division with remainder $]$

For which $n>3$ can a set of weights with masses $1,2,3, \ldots, n$ grams be divided into three equal-mass piles?
Six weights, whose masses are six consecutive numbers, can easily be divided into three equal-mass piles.

## Solution

Let's call a number $n$ good if a set of weights with masses $1,2,3, \ldots, n$ can be divided into three equal-mass piles. For a good $n$, the total mass of the weights, which is $1 / 2 n(n+1)$, must be divisible by 3. Therefore, $n$ can only give remainders of 0 or 2 when divided by 3.

If the number $n$ is good, then the number $n+6$ is also good, since the weights with masses $n+1, n+2, \ldots, n+6$ can easily be divided into three equal-mass piles.

It is easy to check that the numbers $5,8,9$ are good: $1+4=2+3=5 ; 1+2+3+6=4+8=5+7 ; 1+2+3+4+5=6$ $+9=7+8$. That $6$ is a good number is shown above. Therefore, good numbers include all numbers of the form $3 k(k \geq 3)$ and $3 k+2(k \geq 1)$.

## Answer

For $n$ that are multiples of 3 or give a remainder of 2 when divided by 3.