MathDB

5

Part of 1996 Irish Math Olympiad

Problems(2)

dissection

Source: Ireland 1996

7/1/2009
Show how to dissect a square into at most five pieces in such a way that the pieces can be reassembled to form three squares of (pairwise) distinct areas.
geometry unsolvedgeometry
game

Source: Ireland 1996

7/1/2009
The following game is played on a rectangular chessboard 5×9 5 \times 9 (with five rows and nine columns). Initially, a number of discs are randomly placed on some of the squares of the chessboard, with at most one disc on each square. A complete move consists of the moving every disc subject to the following rules: (1) (1) Each disc may be moved one square up, down, left or right; (2) (2) If a particular disc is moved up or down as part of a complete move, then it must be moved left or right in the next complete move; (3) (3) If a particular disc is moved left or right as part of a complete move, then it must be moved up or down in the next complete move; (4) (4) At the end of a complete move, no two discs can be on the same square. The game stops if it becomes impossible to perform a complete move. Prove that if initially 33 33 discs are placed on the board then the game must eventually stop. Prove also that it is possible to place 32 32 discs on the boards in such a way that the game could go on forever.
combinatorics unsolvedcombinatorics