MathDB
Turkey NMO 2010 1st Round - P36 (Combinatorics)

Source:

July 26, 2012

Problem Statement

Two players are playing a turn based game on a n×nn \times n chessboard. At the beginning, only the bottom left corner of the chessboard contains a piece. At each turn, the player moves the piece to either the square just above, or the square just right, or the diagonal square just right-top. If a player cannot make a move, he loses the game. The game is played once on each 6×76\times 7, 6×86 \times 8, 7×77 \times 7, 7×87 \times 8, and 8×88 \times 8 chessboard. In how many of them, can the first player guarantee to win?
<spanclass=latexbold>(A)</span> 1<spanclass=latexbold>(B)</span> 2<spanclass=latexbold>(C)</span> 3<spanclass=latexbold>(D)</span> 4<spanclass=latexbold>(E)</span> None <span class='latex-bold'>(A)</span>\ 1 \qquad<span class='latex-bold'>(B)</span>\ 2 \qquad<span class='latex-bold'>(C)</span>\ 3 \qquad<span class='latex-bold'>(D)</span>\ 4 \qquad<span class='latex-bold'>(E)</span>\ \text{None}