Ken Jennings, the 74-time Jeopardy! champion, has a blog.
Today he wrote about polyominoes, inspired by the question of how many pieces there would be in Tetris if it were played with pieces made of five or six squares instead of the canonical four. He's not a mathematician, and he finds it surprising that there's no nice formula for the number of polyominoes with n cells. I suppose it is kind of surprising to someone with no mathematical training; by this point in my life I've gotten used to the fact that things just don't work that way.
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3 comments:
Yeah, my intuitive sense is to be amazed if there were a nice formula for that. Once you see that septomino with a hole in it, you realize how screwed you are on the idea of counting them in any elegant way.
What gets me is how a trivia buff of his stature, even though not a "mathy" person, apparently had never heard of pentominoes, a fairly regular staple in the JH/HS math manipulatives department.
Another interesting point is that he is treating reflections as trivial when, in fact, the game of Tetris does not. I understand the idea. Just an interesting parallel. Would there be a more elegant way of counting them for arbitrary N if non-trivial reflections (i.e. the two 'Z'-shaped pieces) were BOTH counted, making there 3 triominoes, 6 tetrominoes, 18(?) pentominoes, etc?
Sean, from what I understand, the problem is basically just as hard if you make the change you suggest. (But sometimes that sort of change does make problems easier.)
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