tmua.fyi Paper 2 Challenge 2 · Question 15

Two squares at opposite ends of a diagonal are removed from an $8\times8$ board, as shown. Can the remaining $62$ squares be covered exactly by $31$ dominoes, each covering two squares that share an edge?

AYes, and an explicit covering can be written down.

BYes, because

62

is even.

CNo, because

62

is not divisible by

4

DNo, because no domino can be placed in a corner.

EYes, because

62=2\times31

FNo, because an odd number of rows is left uncovered.

GNo: the two removed squares are the same colour, so

32

squares of one colour and

30

of the other remain, while every domino covers one of each.

HIt depends on which pair of opposite corners is removed.

examday

tmua.fyi Paper 2 Challenge 2 · Q15