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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.