Remember that Tetris game, with that delightful music copied from the Borat television show? Well in Tetris there are 7 different types of shapes: all the possible ways of combining four blocks (up to rotation, but not reflection). Is it possible to put all these 7 shapes together to form a rectangle? You must use each shape once and only once. Justify your answer, but make it as simple as possible.
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Anonymous2011-02-06 20:59
No, you can't. Because you have seven four-block shapes, the resulting rectangle has to have an area of 28 blocks. That's either a rectangle of 2x14 or one of 4x7.
The 2x14 one is out easily. Even if you ignore the "I" piece, there aren't enough pieces that can be positioned to satisfy squarable corners.
The 4x7 presents an interesting condition. It's easy to arrange blocks symmetrically to produce the rectangle but the triangle block throws everything off and there's no easy way to square it. I can't completely justify the answer I have, yet, but I think the triangle block is what throws the whole system off. I'm going with "by virtue of being a singleton that has an odd non-1 side, unlike the other two non-mirrored pieces." There's also an observed difficulty in, as before, squaring all the corners without creating gaps.