From polyominoes to block puzzles: a short history
Every piece in a block puzzle is a polyomino: a shape made of squares joined edge to edge. The word is only a little over seventy years old, but it named shapes that puzzle makers had played with for decades, and it leads, by a fairly direct line, to Tetris and to the drag-and-drop block games on today's phones, Pebble Blocks among them. Here is that line, with a few of the numbers that make these shapes so rich.
Golomb gives the shapes a name
In 1953 Solomon W. Golomb, then a graduate student at Harvard, coined the word polyomino as an extension of domino: two squares make a domino, three a tromino, four a tetromino, five a pentomino. His paper "Checker Boards and Polyominoes" appeared in the American Mathematical Monthly in 1954, and his book Polyominoes followed in 1965, with a second edition in 1994. Martin Gardner's columns in Scientific American then carried the shapes to a much wider public.
The shapes were older than the name. Henry Dudeney's book The Canterbury Puzzles (1907) contains the earliest known puzzle that uses a complete set of pentominoes, and many early results appeared in the magazine Fairy Chess Review between 1937 and 1957 as "dissection problems".
A neat theorem of Golomb's shows how much structure these shapes hide. Remove any one square from an 8 by 8 board, and the other 63 can always be covered by L-shaped trominoes. The proof splits the board into four quarters, places one tromino in the middle so that each of the three untouched quarters loses one square, and then repeats the trick inside every quarter, down to boards of 2 by 2.
Twelve pentominoes
There are exactly 12 pentominoes, if a shape and its turned or mirrored copies count as one. Golomb named them after the letters they resemble: F, I, L, N, P, T, U, V, W, X, Y and Z. Together they cover 60 squares, which invites the classic challenge of packing all twelve into a rectangle. Counting each answer once, however the whole rectangle is turned or flipped, the numbers of solutions are:
| Rectangle | Solutions |
|---|---|
| 6 by 10 | 2,339 |
| 5 by 12 | 1,010 |
| 4 by 15 | 368 |
| 3 by 20 | 2 |
The 6 by 10 count was first worked out in 1960 by C. Brian Haselgrove and Jenifer Haselgrove. We checked it with our own exact-cover search, which found 9,356 placements, four for each of the 2,339 solutions. Two years before the Haselgroves, Dana Scott used a computer to show that the twelve pieces can cover an 8 by 8 board with a 2 by 2 hole in the centre in exactly 65 ways. His program was one of the first uses of backtracking in a computer, the search method later made famous in teaching by the eight queens puzzle.
Tetris adds gravity
In 1984 Alexey Pajitnov, working at the Dorodnitsyn Computing Centre of the Soviet Academy of Sciences in Moscow, began writing a game on an Electronika 60 computer. It was inspired by a pentomino game from his childhood, but it used the seven tetrominoes, pieces of four squares, and took its name from tetra, Greek for four, and tennis, his favourite sport.
Tetris made three choices that still define falling-block games. Gravity pulls each piece down, so the game sets the pace. You can rotate pieces as they fall. And only full rows clear. Rotation matters for the maths: since any piece can be turned, the seven tetrominoes cover every case. If turning were not allowed, the same shapes would need 19 different pieces, one for every way they can face.
Block puzzles without gravity
The drag-and-drop block puzzle keeps the polyominoes and the clearing, and changes almost everything else. There is no gravity and no clock, so you can think as long as you like. Pieces cannot be rotated: you place each one exactly as it is dealt. And both rows and columns clear, so the board empties in two directions. A 2014 hit from the Turkish studio Gram Games used a 10 by 10 board and helped bring the style to phones.
Those changes turn a reflex game into a planning game. In Pebble Blocks you get three pieces at a time on an 8 by 8 board, and three more arrive only when all three are placed. The game ends when none of the pieces left in your tray fits anywhere. Because pieces never turn, every orientation is its own piece. Of the 37 shapes, 19 are exactly those 19 fixed tetrominoes; there are four separate small corners; and the big 3 by 3 corner, in all four of its directions, is the pentomino Golomb called V.
Why random pieces and free choice work
A block puzzle deals you pieces you did not choose, then hands every other decision to you: which piece goes first, where it lands, which lines to keep open. The randomness keeps each game different. The choice means that a lost game can usually be traced back to a move you could have made better, which gives you something to try next time.
Good play is mostly risk management. The big pieces are the dangerous ones: the 3 by 3 square needs nine empty cells in a block, and a bar of five needs a clear run of five. Keep room for them and the small pieces will nearly always find a home. Pebble Blocks also deals with some care. Its dealer tries many random sets and prefers sets whose three pieces can all be placed, leaning towards sets that allow a clear when the board is crowded. Small pieces are a little more common early on, and big ones turn up more often as your score grows.
The scoring rewards the same patience. One cleared line is worth 80 points, two lines at once 240 and three about 480, and clears on consecutive moves are multiplied by the combo number. Emptying a busy board adds 300 more. For tactics on the board itself, see our block puzzle strategy guide.