The Mathematics Behind the Seven Shapes

Why exactly seven, what happens with five squares, and the tiling problems that follow.

The seven pieces are not a design choice. They are a fact about geometry, and the field they come from is genuinely interesting.

Polyominoes

A polyomino is a shape made by joining unit squares edge to edge. Two squares gives one shape, the domino. Three gives two shapes. Four gives seven, which are the pieces in this game — the tetrominoes.

The count depends on what you treat as identical. If reflections count as the same shape, there are five distinct tetrominoes. If, as in this game, a shape and its mirror image are different pieces, there are seven: S and Z are mirror images of one another, as are L and J.

Five squares

Pentominoes, the twelve shapes made from five squares, were the original inspiration. Twelve pieces is too many to track and too much for a 1984 minicomputer to handle at speed.

They have their own extensive recreational-mathematics literature, largely concerning tiling: the twelve pentominoes have a combined area of sixty squares, which is exactly a 6 by 10 rectangle, a 5 by 12, a 4 by 15 or a 3 by 20. Working out how many distinct ways each can be tiled is a classic computational problem.

The counts grow fast

Beyond five squares the numbers climb steeply, and there is no known simple formula for the number of distinct polyominoes of a given size. It is computed by exhaustive enumeration, and pushing that count further has been a recurring computational challenge for decades.

Tiling and the game

The connection is direct. A perfect clear is a tiling problem: can these pieces exactly fill this rectangle with nothing left over? The check most players use — is the number of filled cells a multiple of ten — is a parity argument of exactly the kind that shows up throughout polyomino mathematics.

Why it is worth knowing

Because it explains why the game feels the way it does. Seven pieces is not a balancing decision somebody could have made differently. It is what four squares give you, and everything downstream — the seven-bag, the shape of a clean stack, the awkwardness of S and Z — follows from that.

Read next

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• The Research on Blocks and Intrusive MemoriesSeveral groups have tested whether playing a visuospatial game after a distressing event reduces later flashbacks. The findings are interesting and preliminary.
• The Song Everyone Associates With Falling BlocksA nineteenth-century Russian folk tune about a pedlar, and how it became the most recognisable video game music ever written.
• Why This Particular Puzzle Is So Hard to Put DownClear feedback, a failure you can always attribute to yourself, and no ending. A design that keeps offering one more attempt.
• Plan in Sequences, Not PlacementsA piece that is awkward alone is often fine as the middle of a three-piece plan.
• Combo Pressure: Why Small Attacks Can Beat Big OnesContinuous garbage denies your opponent the quiet piece they need to survive.