The Pentominoes

The Pentominoes (The ultimate Neuro-Training game), geometric shapes formed by joining five equal-sized squares edge to edge, were formally defined and popularized by American mathematician Solomon W. Golomb in 1953.

Martin Gardner brought them to the broader public in his October 1965 Mathematical Games column in Scientific American. 

The Pentominoes influenced the creation of the famous video game Tetris.

A standard pentomino puzzle is to tile a rectangular box with the pentominoes, i.e. cover it without overlap and without gaps. Each of the 12 pentominoes has an area of 5 unit squares, so the box must have an area of 60 units. Possible sizes are 6×10, 5×12, 4×15 and 3×20.

The 6×10 case was first solved in 1960, there are exactly 2,339 solutions, the 5×12 box has 1010 solutions, the 4×15 box has 368 solutions, and the 3×20 box has just 2 solutions.