Originally posted by howardbradley
Imagine the 6x6x6 cube as composed of 27 2x2x2 cubes coloured in chess board fashion - ie no cubes of the same colour adjacent. There would be 5+4+5 (=14) of one colour and 4+5+4 (=13) of the other.
However, no matter how you place the 1x1x4 pieces they have to take up and equal amount of each colour. Clearly a contradiction.
Wait... no. Why imagine the cube as composed of 27 2x2x2 pieces? Especially when the given piece is 1x1x4? A 2x2x2 piece is 8 little (1x1x1) cubes, while the given 1x1x4 piece is 4 little cubes. By halving the number of pieces, you have a 50-50 chance of creating an odd number of pieces.
Clearly, you can't fit 54 1x1x4 pieces into a cube 6x6x6.
But the logic you describe has nothing to do with it.
Using similar logic, I just as easily could have said: Imagine the 6x6x6 cube as composed of 8 3x3x3 cubes colored in chess board fashion - ie no cubes of the same color adjacent. There would be 2+2 (=4) of one color and 2+2 (=4) of the other. No matter how you place the 1x1x4 pieces they have to take up an equal amount of each color. 4=4, so therefore, it is possible.
Clearly, this isn't the case.