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Math Kangaroo practice

Name: ______________________Date: ____________5 questions · 25 points

1.A 4 by 4 board has two opposite corner squares removed, leaving 14 squares. Each domino covers exactly two squares that share an edge. Can 7 dominoes cover the board?[5]
Text description of the figure
A 4 by 4 chessboard-style grid coloured in alternating light and dark squares, with the top-left and bottom-right squares removed. Both removed squares were the same colour.
- AYes, and there is exactly one way
- BYes, and there are several ways
- CNo, because 14 is not divisible by 2
- DIt depends which corners are removed
- ENo, because the colours do not balance
2.One of nine identical-looking coins is slightly heavier. Using only a balance, what is the smallest number of weighings that is certain to find it?[5]
- A1
- B2
- C3
- D4
- E8
3.The numbers 1 to 10 stand in a row. A move swaps two neighbours. After exactly 45 moves, can the row be back in its starting order?[5]
- AYes, always
- BYes, if the swaps are chosen well
- CNo, because 45 is not a multiple of 10
- DNo, because the number of moves is odd
- EIt depends which numbers are swapped
4.Four children stand in a line. Ana is not first. Bo stands directly behind Ana. Cal is last. Who is first?[5]
- AAna
- BBo
- CCal
- DDee
- Eit cannot be decided
5.The numbers 1 to 10 are written on a board. You repeatedly rub out any two of them and write down their positive difference instead, until a single number is left. What can be said about that final number?[5]
- AIt is always odd
- BIt is always even
- CIt can be either odd or even
- DIt is always zero
- EIt is always 1

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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.E — No, because the colours do not balanceKA-0034Coloring arguments
- AI assumed that because 14 is even, seven dominoes must fit somehow.
- BI tried a few arrangements, got most of the board covered, and assumed a full covering existed.
- CI claimed the right answer for the wrong reason, since 14 is in fact divisible by 2.
- DI thought removing a different pair of opposite corners could change the colour balance, but opposite corners always share a colour.
2.B — 2KA-0094Weighing and balance puzzles
- AI assumed one weighing could separate nine possibilities, but it has only three outcomes.
- CI split the coins into halves each time, which wastes the balance's third outcome.
- DI weighed the coins one against another in pairs rather than in groups.
- EI compared each coin with a known good one in turn, which always works but is nowhere near the fewest.
3.D — No, because the number of moves is oddKA-0097Parity arguments
- AI assumed enough moves can undo anything, without asking what each move preserves.
- BI tried a few sequences that nearly worked and assumed a better choice would finish the job.
- CI reached for the number of items rather than for what a single swap actually changes.
- EI thought the choice of swaps could change the outcome, when every swap has the same effect.
4.D — DeeKA-0136Ordering and ranking from clues
- AI ignored the very first clue, which rules Ana out of the front directly.
- BI put Bo at the front, but Bo must have Ana immediately in front of him.
- CI forgot that Cal is fixed at the back of the line.
- EI stopped once two arrangements seemed possible, without testing whether the second one actually fits every clue.
5.A — It is always oddKA-0167Parity arguments
- BI checked the parity of the count of numbers rather than of their sum.
- CI tried two examples, got different answers, and concluded nothing is forced.
- DI assumed differences must shrink all the way to nothing.
- EI found one sequence of moves ending at 1 and assumed every sequence must.

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