1997 Putnam A2

Players $1,2,3,\ldots,n$ are seated around a table, and each has

a single penny. Player 1 passes a penny to player 2, who then passes

two pennies to player 3. Player 3 then passes one penny to Player 4,

who passes two pennies to Player 5, and so on, players alternately

passing one penny or two to the next player who still has some

pennies. A player who runs out of pennies drops out of the game and

leaves the table. Find an infinite set of numbers $n$ for which some

player ends up with all $n$ pennies.

玩家 $1,2,3,\ldots,n$ 围坐在桌子周围，每个人都有

一分钱。玩家 1 将一分钱传给玩家 2，然后玩家 2 又将硬币传给玩家

两便士给玩家 3。玩家 3 然后将一便士传递给玩家 4，

谁将两便士传递给玩家 5，依此类推，玩家轮流

将一两便士传递给下一个还有一些钱的玩家

便士。耗尽所有硬币的玩家退出游戏并

离开桌子。找到一组无限的数字$n$，其中一些

玩家最终获得所有 $n$ 便士。