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1.Ana says "Bo is lying." Bo says "Cal is lying." Cal says "Ana and Bo are both lying." How many of the three are telling the truth?[5]
A0
B1
C2
D3
Eit cannot be decided
2.Seven cups all stand upside down. In one move you must turn over exactly two cups. Can all seven ever stand the right way up?[5]
AYes, in 4 moves
BYes, in 7 moves
CYes, but it takes many moves
DIt depends which two cups you pick
ENo, it is impossible
3.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
4.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
5.One hundred apples are packed into twelve boxes. What is the largest number n for which you can always be sure that some box holds at least n apples?[5]
A8
B9
C10
D12
E100
6.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
7.The numbers 1 to 8 are on a board. A move rubs out two of them and writes their difference, larger minus smaller. After seven moves one number is left. Can it be 1?[5]
AYes, and there are many ways
BYes, but only one way
CNo, the last number is always even
DNo, the last number is always 0
EIt depends on the order of the moves
8.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
9.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
10.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.B — 1KA-0017Truth-tellers and liars
AI assumed everyone could be lying at once without checking that Cal's statement would then be true.
CI found one consistent truth-teller and added another without testing whether both could hold together.
DI assumed everyone was telling the truth without noticing that Ana's statement then contradicts Bo's.
EI gave up after one assumption led to a contradiction, instead of trying the other assumption.
2.E — No, it is impossibleKA-0028Parity arguments
AI found a sequence that turned over most of the cups and assumed the last one could be fixed somehow.
BI matched the number of moves to the number of cups without checking whether the target is reachable at all.
CI assumed that with enough moves any arrangement can be reached.
DI thought the choice of which cups to flip could change whether the target is reachable.
3.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.
4.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.
5.B — 9KA-0095The extremal principle
AI divided 100 by 12 and rounded down instead of up.
CI rounded the division to a convenient number without checking that a packing with a smaller maximum exists.
DI gave the number of boxes as the answer rather than a number of apples.
EI described the case where one box holds everything, which is possible but not guaranteed.
6.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.
7.C — No, the last number is always evenKA-0098Invariants and monovariants
AI found sequences ending in small numbers and assumed 1 was among the reachable ones.
BI assumed a hard-looking question must have exactly one answer.
DI found one sequence ending in 0 and assumed every sequence must end there.
EI thought the order could change the outcome, when the quantity that decides it never changes.
8.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.
9.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.
10.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.