Problem 3
On an infinite chessboard, start with pieces in an by block, one per square. A move jumps horizontally or vertically over an adjacent occupied square to the unoccupied square immediately beyond, removing the jumped piece. Find those values of for which the game can end with only one piece remaining.
Step 2 of 4: Exclude multiples of 3
In plain words
The three colors move in lockstep, but a lone piece cannot have equal parity in all colors.
Detailed analysis
Because a jump removes two pieces and adds one, each of the three color parities is toggled. Initially, when n=3q, each color has 3q^2 pieces: the parity vector is either (0,0,0) or (1,1,1). A final single piece has one 1 and two 0s, impossible in either case.