From Tetris to Sudoku: where the 9x9 block puzzle comes from

A 9x9 block puzzle is a blend of three older ideas. The pieces are polyominoes, shapes made of squares joined edge to edge, which a mathematician named and counted in the 1950s. The habit of clearing lines comes from Tetris, which appeared in 1984. The board, nine rows and nine columns split into nine 3x3 boxes, is the Sudoku grid. Each ancestor leaves numbers you can check.

Polyominoes: the pieces

Solomon Golomb introduced the word polyomino in 1953, and later wrote the book Polyominoes about them. A polyomino is a set of squares joined along whole edges. Counted as free shapes, where turning and flipping do not make a new shape, the numbers for 1 to 6 squares are small and then grow quickly:

The 12 pentominoes were a popular puzzle subject, and the usual story is that they inspired Tetris. Alexey Pajitnov, working in Moscow, built the first Tetris in 1984, and his pieces are the tetrominoes, one size down from the pentominoes. The name comes from the Greek prefix for four.

Five tetrominoes, seven pieces

There are only 5 free tetrominoes: the straight bar (I), the square (O), the T, the L-and-J pair and the S-and-Z pair. But Tetris deals 7 pieces, because L and J are mirror images, as are S and Z, and a falling piece can be turned but never flipped over. Counting shapes that may be turned but not flipped gives 7, called one-sided. Count shapes that may be neither turned nor flipped and you get 19:

• I: 2 orientations (flat or upright).
• O: 1.
• T, L and J: 4 each, which is 12.
• S and Z: 2 each, which is 4.

That adds to 2 + 1 + 12 + 4 = 19. The distinction matters in a block puzzle where pieces arrive in a fixed orientation. A piece that cannot be turned is effectively one of the 19, not one of the 7.

The counts follow the rule from the post on holes and islands: a piece with an h by w bounding box fits in (9 − h + 1) × (9 − w + 1) positions. A bar is 1 by 4, so 9 × 6 = 54. The other pieces sit in a 2 by 3 or 3 by 2 box, so 8 × 7 = 56. The column totals 1,068 placements across all 19 oriented shapes.

Tetris: a 10x20 well and only rows

Tetris plays in a well 10 cells wide and 20 high, so 200 cells. Pieces fall under gravity, and the only thing that clears is a full horizontal row of 10. There are 20 rows to clear and nothing else. Two facts are worth noticing:

• 200 is a multiple of 4, but a single row of 10 is not. So no row can be filled with tetrominoes alone without spilling into the next one.
• Clearing four rows at once needs the straight bar, because only it is long enough to reach four rows in a single piece.

The drop-block puzzles: no gravity, no clock

Later puzzles of the 10x10 kind, such as 1010!, kept the polyomino pieces but changed the rules. You are given a few pieces at a time and place them anywhere they fit. Nothing falls, so there is no timing to learn. Rows and columns both clear, and a 10x10 board has 100 cells and 20 lines, 10 rows and 10 columns.

That change alters the arithmetic. With gravity, a piece lands wherever the stack lets it. With free placement, every legal position is available, which is why the placement counts above matter and why a single misplaced block can remove many options. The skill moves from reflexes to planning.

Sudoku: the 9x9 board

Sudoku began as Number Place in Dell puzzle magazines in 1979, with the grid credited to Howard Garns. The Japanese publisher Nikoli took it up and gave it the name Sudoku in 1984. The grid is 9x9: 81 cells, nine rows, nine columns and nine 3x3 boxes, so 27 groups. In Sudoku each group must contain the digits 1 to 9 once.

A block puzzle on the same board turns that into a filling rule: complete a row, a column or a box and it clears. The post on the 9x9 grid counts what that means: each of the 81 cells belongs to exactly one row, one column and one box, and a single placement can free up to 21 cells. The 3x3 boxes are the part Tetris never had.

One more number: 81 is not a multiple of 4

Tetromino pieces have four squares each, so they can cover only a multiple of 4 cells. A 10x10 board of 100 cells could in principle be covered by 25 of them. A 9x9 board of 81 cannot be covered exactly by any number of four-square pieces: at least one cell always stays empty.

Where this leaves the player

Knowing the ancestry tells you which Tetris habits to keep. Keep the empty space connected, and stay aware of which shapes you cannot place. Drop the ones that depended on gravity and the clock. And use the box: it is the one line-clearing group Tetris never offered.

A good place to try this is Big Ben Blocks: Block Puzzle, which uses a 9x9 grid where rows, columns and 3x3 boxes clear, with untimed Classic play.