Hint: Combinatorics Q1

First of all, how many cubes are there in the brick? (Hint: 5) What does this imply about cubes which we might fill (hint: their side length must be a multiple of 5). So the smallest we could hope to fill is ?? and the next smallest is ??

Working with the smallest case you should find that it seems to be difficult to get it to work out -- something always gets missed or sticks out. Think about the domino problem again. Another way to express the fact that it can't be done is to say: "There are 32 white squares in the truncated board, so we need 32 dominoes. However, there are only 62 squares all together so we could only use 31 dominoes". How many bricks would we use if we could fill the smallest cube? Can we find a bigger set of unit cubes each of which requires its own brick?

Even if you haven't shown the smallest case doesn't work, you might get frustrated and move on to the next case. Now we come to the issue that the bricks are awkward shapes -- so perhaps we should think about making more convenient super-bricks out of groups of the original bricks (e.g. bricks shaped like real bricks!) If we can get a superbrick whose dimensions divide that of the cube we'd be done ...