MathDB

1

Part of 1995 Bundeswettbewerb Mathematik

Problems(2)

game with 2 persons and 2 heaps of p and q stones

Source: 1995 German Federal - Bundeswettbewerb Mathematik - BWM - Round 1 p1

1/27/2020
A game is played with two heaps of pp and qq stones. Two players alternate playing, with AA starting. A player in turn takes away one heap and divides the other heap into two smaller ones. A player who cannot perform a legal move loses the game. For which values of pp and qq can AA force a victory?
combinatoricsgame strategygamewinning strategy
moving stones in lattice points, (a,b) ->(2a,b) or (a,2b), (a-b,b) or (a,b-a)

Source: 1995 German Federal - Bundeswettbewerb Mathematik - BWM - Round 2 p1

1/27/2020
Starting at (1,1)(1,1), a stone is moved in the coordinate plane according to the following rules: (i) From any point (a,b)(a,b), the stone can move to (2a,b)(2a,b) or (a,2b)(a,2b). (ii) From any point (a,b)(a,b), the stone can move to (a−b,b)(a-b,b) if a>ba > b, or to (a,b−a)(a,b-a) if a<ba < b. For which positive integers x,yx,y can the stone be moved to (x,y)(x,y)?
lattice pointslatticecombinatorics