Firstly, if we can color the table when $n=k$, then we can color it when $n=m<k$ because we can erase last columns. $i)m=2,$ all $n$ positive integers. We can color like $BWBWBWBW...$ $WBWBWBWB...$ $ii)m=3, n=1,2,3,4,5,6$ There are $2^3=8$ different colorings. $WWWWBBBB$ $BWWBBBWW$ $WWBBBWWB$ Assume that $n>6$ Then one of $WWW$ or $BBB$ will exist (WLOG $BBB$). Then $BBW,BWB,WBB$ don't exist. Contradiction So $n \leq 6$ Example for $6$ and less. $BWBBWW$ $WBWBBW$ $BWWWBB$ $iii)m=4,n=1,2,3,4,5,6$ It's obvious that $n \leq 6$ because by erasing the last row, we turn back to $m=3$. Example for $n=6$ and less. $WWWBBB$ $WBBWBW$ $BBWBWW$ $BWBWWB$