The story of tetrominoes

A tetromino is a shape made by joining four squares edge to edge. There are only five ways to do it. Can you take one of each of the five and fit them together into a rectangle with no gaps? No, never, and you can tell without trying a single arrangement.

Count the colors of a checkerboard

Using each of the five shapes once covers 4 × 5 = 20 squares. A rectangle of 20 squares would be 1 × 20, 2 × 10 or 4 × 5. Color any of these like a checkerboard, light and dark. Each has 10 of each color.

Now look at the pieces. The straight piece, the square piece, the step piece and the L piece each cover two light and two dark squares, however you turn them. But the T piece covers three of one color and one of the other. Add them all up and you get 11 of one color and 9 of the other, never 10 and 10. So no rectangle is possible.

This site's program confirms the counts for every orientation: the T gives 3 to 1, and the rest give 2 to 2. It also searched every arrangement on the 4 × 5 and 2 × 10 boards and found no solution. Press “Show checkerboard” on the puzzle page to see the picture.

Counting colors proves something impossible without any trial and error. And if you use two T pieces? Put one with three light squares and the other with three dark ones, and the imbalance cancels. That is why the “Two sets” boards on this site can be completed.

Five, seven or nineteen

How many tetrominoes there are depends on how you count. If turning and flipping give the same shape, there are 5. If a mirror image counts as different but turning does not, there are 7, because only the L and the S have mirror images that differ. If every turned and flipped version counts separately, there are 19. The program counts the orientations of the five shapes on this site as 2 + 1 + 4 + 4 + 8 = 19.

The shapes are often named after letters they resemble. The same shapes are used in a famous falling-block video game.

Where the name comes from

The mathematician Solomon W. Golomb coined the word “polyomino” for shapes made of joined squares. It echoes “domino”, the two-square case, with a prefix for the number of squares: tetromino for four, pentomino for five. He is said to have introduced the names in 1953. In 1954 he published the first paper on polyominoes, “Checker Boards and Polyominoes”.

How to think about it

• Start at corners and edges. Few pieces fit a corner, so the choices narrow quickly.
• Count each pocket of empty squares. A pocket cut off from the rest, whose size is not a multiple of 4, can never be filled.
• Count the checkerboard colors. Especially when a T is involved, check whether the light and dark squares can balance.
• Use flips. The L and the S gain extra orientations when flipped over.

Stuck? The Hint button on the puzzle page shows a place to put a piece so the puzzle can still be finished from your current layout. “Watch search” shows the computer trying and backing up.