Library / UKMT Mathematical Olympiad for Girls /29 of 31

UKMT Mathematical Olympiad for Girls, 2013

Combinatorics Difficulty 3.0 AMC 10/12 Prove it United Kingdom

(a) An ant can move from any square on an $8 \times 8$ chessboard to an adjacent square. (Two squares are adjacent if they share a side).

The ant starts in the top left corner and visits each square exactly once. Prove that it is impossible for the ant to finish in the bottom right corner.

[You may find it helpful to consider the chessboard colouring.]

(b) A ladybird can move one square up, one square to the right, or one square diagonally down and left, as shown in the diagram, and cannot leave the board.

Is it possible for the ladybird to start in the bottom left corner of an $8 \times 8$ board, visit every square exactly once, and return to the bottom left corner?

Figure for this problem

We don't reproduce this publisher's solutions. Their own solution is here — work the problem first.

Want a route through all this instead of an archive? The track puts 2,604 problems in a working order, from Junior Challenge level to the IMO shortlist.

Source: UK Mathematics Trust, licensed © UK Mathematics Trust; past papers published free at ukmt.org.uk. Statement reproduced verbatim; metadata (topic, difficulty) added by this project. Solutions are the publisher's, linked not copied.