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I am only a mathematical amateur, but have been bothered by a problem for a long time. In the game Tetris, you have figures made by squares and there exist five really unique figures which cannot not be made congruent by mirroring or rotation.

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Now my question is, why do exactly five figures exist? Is there a formula how to calculate the number of unique figures for the general case (x figures with y equal sides)? The requirement is that all elements of the figure are connected with at least one other element on a whole border.

I would be thankful for answers and links.

AGuyCalledGerald
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    The term you want to research is "polyomino": "a plane geometric figure with one or more squares joined edge to edge". A four-square Tetris figure is a "tetromino", just as a two-square figure is a "domino". The Wikipedia article is a good starting place for information on counting and such; there's a wealth of "recreational mathematics" about polyominos, since the figures give rise to lots of puzzles. – Blue Apr 04 '14 at 09:52
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    Not fully relevant; Arthur C Clarke's novel ("Imperial Earth"?) has a character playing around with Pentaminoes. There are 12 of them, with a total area of 60 sq. units. The book has a solution to fitting them as a jigsaw puzzle into a rectangle of dimensions $5\times 12$ or $6\times 10$. – P Vanchinathan Apr 04 '14 at 11:03
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    Check out this paper: Enumeration of symmetry classes of convex polyominoes in the square lattice by Pierre Leroux, Etienne Rassart, Ariane Robitaille... – draks ... Apr 04 '14 at 12:38

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For all the squares to be connected, we need to consider three cases:-

($1$) $4$ squares in $1$ row: $1$ way = $\large {4 \choose 0}$

($2$) $3$ squares in $1$st row, $1$ square in $2$nd row : $2$ ways = $\large {3 \choose 1}-1$ (the minus $1$ is due to obtaining a shape that is $180^\circ$ rotation of an existing shape - the L shape)

($3$) $2$ squares in $1$st row, $2$ squares in $2$nd row : $2$ ways = $\large {2 \choose 2}+1$ - the $1$ term represents the maximum shift of one of the rows relative to the other such that all squares remain connected, and there are no repeated shapes due to a $180^\circ$ rotation.

Thus there are $5$ unique figures.

Alijah Ahmed
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