(a)The game of "super- chess" is played on a 30×30 board and involves 20 different pieces. Each piece moves according to its own rules , but cannot move from any square to more than 20 other squares . A piece "captures" another piece which is on a square to which it has moved. A permitted move (e.g. m squares forward and n squares to the right) does not depend on the piece 's starting square . Prove that
(i) A piece cannot cap ture a piece on a given square from more than 20 starting squares.
(ii) It is possible to arrange all 20 pieces on the board in such a way that not one of them can capture any of the others in one move. (b) The game of "super-chess" is played on a 100×100 board and involves 20 different pieces. Each piece moves according to its own rules , but cannot move from any square to more than 20 other squares. A piece "captures" another piece which is on a square to which it has moved. It is possible that a permitted move (e.g. m squares forward and n squares to the right) may vary, depending on the piece's position .
Prove that one can arrange all 20 pieces on the board in such a way that not one of them can capture any of the others in one move.( A . K . Tolpygo, Kiev)PS. (a) for Juniors , (b) for Seniors game strategygamecombinatorics