1

Part of 2005 Bundeswettbewerb Mathematik

Problems(2)

Moving a dice across a chessboard (easy)

Source: Bundeswettbewerb Mathematik 2005, 1st Round, problem 1

In the centre of a

2005 \times 2005

chessboard lies a dice that is to be moved across the board in a sequence of moves. One move consists of the following three steps: - The dice has to be turned with an arbitrary side on top, - then it has to be moved by the shown number of points to the right or left - and finally moved by the concealed number of points upwards or downwards. The attained square is the starting square for the next move. Which squares of the chessboard can be reached in a finite sequence of such moves?

A game on a 100x100 chessboard

Source: German Mathematical Competition BWM 2005, 2nd round, problem 1

A

B

have one stone each on a

100 \times 100

chessboard. They move their stones one after the other, and a move means moving one's stone to a neighbouring field (horizontally or vertically, not diagonally). At the beginning of the game, the stone of

A

lies in the lower left corner, and the one of

B

in the lower right corner. Player

A

starts. Prove: Player

A

is, independently from that what

B

does, able to reach, after finitely many steps, the field

B

's stone is lying on at that moment.

floor functioncombinatorics proposedcombinatorics