Sum of cells on infinite board is even Let $a,b,c$ be pairwise relatively prime positive integers. In an infinite checker board (infinite in all directions), each cell contains an integer. The sum of the integers in any $a\times a$ square is even, in any $b\times b$ square is even, and in any $c\times c$ square also even. Does it always follow that all numbers on the board are even?
If $a=1$ then all the numbers must be even. For higher $a,b,c$, we probably have to take overlapping squares of sizes $a,b,c$ to deduce that some cells are even. But it is not clear which ones.

Yes, I think all numbers on the board must be even. It is probably already sufficient to check the contents of $3 \times 3$ squares. For the sum to be even, each $3 \times 3$ cell must contain $0, 2, 4, 6$ or $8$ odd numbers. Yet as you start devising patterns with $2, 4, 6$ or $8$ odd numbers, just moving the $3 \times 3$ cell up/down/left/right one row generally leads to conflicts. 
