Let (m,n,N) be a triple of positive integers. Bruce and Duncan play a game on an m\times n array, where the entries are all initially zeroes. The game has the following rules.
∙ The players alternate turns, with Bruce going first.
∙ On Bruce's turn, he picks a row and either adds 1 to all of the entries in the row or subtracts 1 from all the entries in the row.
∙ On Duncan's turn, he picks a column and either adds 1 to all of the entries in the column or subtracts 1 from all of the entries in the column.
∙ Bruce wins if at some point there is an entry x with ∣x∣≥N.
Find all triples (m,n,N) such that no matter how Duncan plays, Bruce has a winning strategy. gamegame strategywinning strategycombinatorics