## 285. Math Puzzle $2 / 89$

Heinz, Klaus, and Hans-Peter played together with marbles. At the beginning of the game, the ratio of the number of their marbles was 2:3:5, and at the end of the "competition" it was 1:2:5. The same order is always assumed.

One of the three young men ended up losing two marbles to the others.

Which boy is it? How many marbles did the other two win or lose during the game?
Assume: at the beginning of the game, Heinz had $2 n$, Klaus had $3 n$, and Hans-Peter had $5 n$ marbles, totaling $10 n$.

Assume: by the end of the game, the three had $1 m$, $2 m$, and $5 m$ marbles; totaling $8 m$ marbles.

From $10 n = 8 m$ it follows that $n = 4$ and $m = 5$, since $n$ and $m$ must be natural numbers,

This results in a marble count at the start/end of the game for Heinz of 8/5, for Klaus of 12/10, and for Hans-Peter of 20/25.

During the game, Klaus lost two and Heinz lost three marbles to Hans-Peter.