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TOT 103 1985 Autumn J7 S7 super-chess, 30 x 30 board, 20 different pieces

Source:

August 26, 2019
game strategygamecombinatorics

Problem Statement

(a)The game of "super- chess" is played on a 30×3030 \times 30 board and involves 2020 different pieces. Each piece moves according to its own rules , but cannot move from any square to more than 2020 other squares . A piece "captures" another piece which is on a square to which it has moved. A permitted move (e.g. mm squares forward and nn 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 2020 starting squares. (ii) It is possible to arrange all 2020 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 100×100100 \times 100 board and involves 2020 different pieces. Each piece moves according to its own rules , but cannot move from any square to more than 2020 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. mm squares forward and nn squares to the right) may vary, depending on the piece's position . Prove that one can arrange all 2020 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