MathDB

Alice and Bob play the following game. To start, Alice arranges the numbers

1,2,\ldots,n

in some order in a row and then Bob chooses one of the numbers and places a pebble on it. A player's turn consists of picking up and placing the pebble on an adjacent number under the restriction that the pebble can be placed on the number

k

at most

k

times. The two players alternate taking turns beginning with Alice. The first player who cannot make a move loses. For each positive integer

n

, determine who has a winning strategy.

Problem Statement