Katamino
The game consists of 12 different pieces constructed of right angled blocks so each piece is made of 5 "squares". (Think Tetris pieces, but 5 squares instead of 4.)
An old version of the game had only 10 pieces; the completely straight 5 square and the 5 square "plus sign" were not included. Both versions also have a bunch of "filler" pieces of 1 or 2 squares.
The gameboard is constructed with a movable divider so one can take sets of 4 up to the whole set of 12 pieces and form them into a 5 block by X rectangle. (Where X = the number of wood blocks in your set.)
The two or three player strategy game is accomplished on a square board divided into 64 smaller squares. Players take turns to place a piece on the gameboard. The first player who cannot place a piece anymore loses. (Similar to Blokus)
The two player puzzle game mode is accomplished by dividing the board into two sections, each player chooses five blocks and are also given 4 small "filler" blocks of 1 and 2 squares. The first one to fit all their blocks perfectly into their half of the rectangle board, wins.
