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Math Kangaroo practice

Name: ______________________Date: ____________1 questions · 5 points

1.An 8 by 8 chessboard has two opposite corner squares removed, leaving 62 squares. Each domino covers exactly two squares that share an edge. Can the 62 squares be covered exactly by 31 dominoes?[5]
- ANo, because the two removed corners share a colour, so 32 squares of one colour remain and only 30 of the other
- BYes, because 62 is even and 31 dominoes cover exactly 62 squares
- CNo, because 62 is not divisible by 4
- DYes, but only if the dominoes may be placed diagonally
- ENo, because the board is no longer rectangular

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Answer key — Math Kangaroo practice

Each wrong choice carries the thinking that produces it. When a student picks C, this is what they were doing.

1.A — No, because the two removed corners share a colour, so 32 squares of one colour remain and only 30 of the otherKA-0176Coloring arguments
- BI checked that the counts match but a matching count does not make a covering possible.
- CI invented a divisibility condition; dominoes cover two squares, so only divisibility by 2 could matter.
- DI changed the rules rather than testing them; the question says dominoes cover squares sharing an edge.
- EI appealed to the shape, but plenty of non-rectangular regions can be tiled by dominoes.

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