more packing

Suppose we fill a 20x20x20 cube with 2x2x1 cuboids. (Packed horizontally/vertically with no gaps or overlaps.)

Explain why it's possible to draw a straight line through the interior of the cube which does not meet the interiors of any of the cuboids.

because a line has no other dimention than length?

Originally posted by mazziewag
because a line has no other dimention than length?

Well, you're right a line is only one dimensional, but that doesn't mean to say it can't pierce one of the 2x2x1 cuboids.

Okay, this one's quite tricky, so I'll give an easier version of essentially the same problem...

Suppose we cover a 6x6 square with 18 2x1 tiles. Why must there always be a straight "fault-line" running across the tiling from one side to another?

Answer: think about the five horizontal lines and five vertical lines which cut the 6x6 square into single cells. Each of these lines must cut through an even number of 2x1 tiles...0 or 2 or 4 or 6 etc. This is because the line splits the 6x6 area into two rectangles of even area.

Now suppose, for a contradiction, that none of the ten lines mentioned above go right through the 6x6 square without cutting a 2x1 tile in half. Then they each hit at least two 2x1 tiles, and hence there are at least 20 tiles in the tiling. This is a contradiction as we have only used 18.

We have thus shown that one of the horizontal or vertical lines must miss all 2x1 tiles, and this gives us the required fault.