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1.Toma spent 4 euros, then spent half of what was left, and now has 6 euros. How much did she start with?[3]
A10
B14
C16
D20
E24
2.Suppose it is true that some birds cannot fly. Which of these must also be true?[3]
ANo birds can fly
BAll birds can fly
CMost birds cannot fly
DExactly one bird cannot fly
ENot all birds can fly
3.Ana, Bo and Cal each keep a different pet: a cat, a dog and a fish. Ana does not keep the cat. Bo keeps neither the cat nor the fish. Who keeps the cat?[4]
AAna
BBo
CCal
Dit cannot be decided
Eeither Ana or Cal
4.Five children queue up. Ana is somewhere ahead of Bo. Cal is behind Bo but ahead of Dee. Eve is last. Who is second in the queue?[4]
AAna
BBo
CCal
DDee
Eit cannot be decided
5.Two apples balance three pears. One apple weighs 90 grams. How many grams does one pear weigh?[4]
A45
B60
C90
D135
E180
6.Ana says: "Bo and I are both liars." Each child is either always truthful or always a liar. Who is telling the truth?[4]
AAna only
BBo only
Cboth of them
Dneither of them
Eit cannot be decided
7.All the red boxes are heavy. Box X is not heavy. What must be true?[4]
ABox X is red
BBox X is not red
CBox X is blue
DSome red boxes are light
ENothing can be said about box X
8.A number is doubled, then 6 is added, then the result is halved. The answer is 11. What was the original number?[4]
A4
B8
C11
D16
E22
9.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
10.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
11.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
12.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
13.Can a 10 by 10 board be covered exactly by T-shaped tiles of four squares each, with no gaps and no overlaps?[5]
AYes, and it is straightforward
BYes, but the arrangement is fiddly
CNo, because 100 is not a multiple of 4
DNo, because the two colours cannot balance
EIt depends how the tiles are turned
14.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
15.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.C — 16KA-0050Working backwards
AI added the 4 and the 6 and stopped, forgetting to undo the halving.
BI doubled the 6 first and then added the 4, reversing the steps in the wrong order.
DI doubled the total at the end rather than doubling only the amount left after the first spend.
EI doubled twice because spending half felt like it needed undoing more than once.
2.E — Not all birds can flyKA-0146Reading a logical statement precisely
AI read some cannot as meaning none can, which is a far stronger claim.
BI chose a statement that directly contradicts the one I was given.
CI read some as meaning a majority, when it only guarantees at least one.
DI read some as meaning exactly one, when it means at least one.
3.C — CalKA-0091Elimination grids
AI used Bo's clue and forgot the very first clue, which rules Ana out directly.
BI read Bo's clue as telling me what Bo has rather than what Bo does not have.
DI stopped after using each clue once, without going back to see what the ticks had ruled out.
EI applied Bo's clue but never came back to Ana's, so I left two people in the running.
4.B — BoKA-0092Ordering and ranking from clues
AI gave the child who is first rather than the one who is second.
CI placed Cal directly after Ana, reading ahead of Dee as meaning immediately ahead.
DI built the order backwards from the last position instead of forwards from the first.
EI assumed the clues left several orders open without checking that they chain into a single one.
5.B — 60KA-0093Weighing and balance puzzles
AI halved the apple's weight, as if one apple balanced two pears.
CI assumed a pear must weigh the same as an apple because the two sides balance.
DI multiplied the apple's weight by three halves the wrong way round, making the pear heavier.
EI gave the total weight of the two apples rather than the weight of one pear.
6.B — Bo onlyKA-0096Truth-tellers and liars
AI took Ana's statement at face value without checking whether a truthful person could say it.
CI decided nobody was lying without testing Ana's statement against that assumption.
DI believed Ana's statement even after concluding she was a liar, which is what a liar's statement cannot be.
EI gave up when Ana's statement looked circular, instead of testing one assumption all the way through.
7.B — Box X is not redKA-0099Reading a logical statement precisely
AI read the rule backwards, as if being heavy were what makes a box red.
CI assumed not red must mean one particular other colour, when the rule says nothing about which.
DI contradicted the rule I was given rather than applying it to box X.
EI decided one fact could not settle anything, without testing what would follow if X were red.
8.B — 8KA-0100Working backwards
AI undid the operations in the order they were written instead of in reverse order.
CI gave the final answer back, assuming the three steps cancelled each other out.
DI doubled the 11 and subtracted the 6 but forgot the final halving that undoes the doubling.
EI undid only the halving and stopped there without touching the other two steps.
9.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.
10.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.
11.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.
12.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.
13.D — No, because the two colours cannot balanceKA-0101Coloring arguments
AI checked that 100 divides by 4 and treated that as proof that a covering exists.
BI assumed a covering must exist somewhere and that I simply had not found it yet.
CI gave a reason that is not even true, since 100 really is a multiple of 4.
EI thought orientation could rescue it, but every turn of a T covers the same mixture of colours.
14.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.
15.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.