Tours on a mutilated board

Intermediate Mathematics English

Also available: Deutsch

Created by Best · 17.07.2026 at 11:55 UTC

On a board with some squares removed, decide whether a rook can tour all remaining squares with single steps. Colour parity predicts the answer before you try. A single-step tour alternates colours, so a closed tour needs equal numbers of light and dark squares, and an open tour needs the two colours to differ by at most one, with endpoints of the right colours.

Remove squares and you change the colour balance. If the removed squares leave the counts too unequal, no tour exists, and you know it from counting alone. Parity converts a search into arithmetic.

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Tasks

Question 1

Visit every square of the mutilated 4x4 region (a4 and d4 removed) by unit rook steps, starting at a1.

Question 2

Before attempting a tour on a mutilated board, colour parity lets you:

choose which colour to remove

measure the board size

count the number of pieces

predict feasibility from the counts of light and dark squares

Question 3

On a 4 x 4 corner with a1 and a3 removed (both dark), what does parity tell you?

an open tour exists as long as it starts on a dark square

parity says nothing once squares have been removed

no tour exists at all, because the counts now differ by 2

a closed tour exists, but no open one

Card Info

Topic: Mathematics
Difficulty: Intermediate
Completed: 0 users

Creator

Best

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