99 children stand in a circle, each initially having a ball. Every minute, each child with a ball throws their ball to one of the two neighbors; if two balls land with the same child, one of these balls is lost irretrievably. What is the minimum time after which only one ball may remain with the children?
Number the children and balls clockwise from 1 to 99.

Example. Suppose children 1 and 2 toss the first ball to each other. The other balls with odd numbers are always thrown counterclockwise until they reach the second child, who discards them (this happens after an odd number of minutes, and at that moment he also receives the first ball again). Balls with even numbers are thrown clockwise until they reach the first child, who discards them (after an even number of minutes, the first ball returns to him). It is easy to see that after 98 throws, all balls except the first one will be discarded.

Estimation. It is more convenient for us to assume that the balls do not get lost but stick together (retaining all numbers) and are collected at the end by the first child. Suppose there is a way to collect all the balls with him in $n$ minutes. If $n$ is odd, the first ball has made a full circle (otherwise, it returns to the first child only after an even number of minutes), so $n \geq 99$. If $n$ is even, the second ball has made a full circle minus one step (otherwise, it reaches the first child only after an odd number of minutes), so $n \geq 98$.

## Answer

After 98 minutes.