(In reply to Solution
I was able to prove it using alternating 1X1X1 bricks of 4 different colors, arranged so that every 1X1X4 brick had to cover one cell of each color. But your proof is light-years better!
For one thing, it is absolutely clear using your method that the number of black and white bricks cannot be equal (since there are 125 in total). I had 1000 bricks, but it is a little tricky to prove that their numbers are unequal.
For another thing, your method clearly establishes an upper limit of not more than 248 that are achievable. My upper limit was not more than 249. (Since 248 is possible, that is the maximum achievable.)