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 1 of 4: Color the board
In plain words
A move changes the parity of all three color counts together.
Detailed analysis
Color square (i,j) by i+j modulo 3. Every legal jump uses three consecutive squares in one row or column, hence one square of each color.