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Boxes and balls

Source: Olympiad

January 20, 2018
combinatoricsnumber theory

Problem Statement

Let nn be a positive integer, the players A and B play the following game: we have nn balls with the numbers of 1,2,3,4,....,n1, 2, 3, 4,...., n this balls will be in two boxes with the symbols ∏\prod and ∑\sum. In your turn, the player can choose one ball and the player will put this ball in some box, in the final all the balls of the box ∏\prod are multiplied and we will get a number PP, after this all the balls of the box ∑\sum are added up and we will get a number QQ(if the box ∏\prod is empty P=1P = 1, if the box ∑\sum is empty Q=0Q = 0). The player(s) play alternately, player A starts, if P+QP + Q is even player A wins, otherwise player B wins.
a)If n=6n= 6, which player has the winning strategy??? b)If n=2012n = 2012, which player has the winning strategy???
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