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1.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
2.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
3.What is the units digit of 7 multiplied by itself 2026 times, that is 7 to the power 2026?[5]
Ait cannot be found without a calculator
B1
C3
D7
E9
4.A drawer holds 10 red socks, 10 blue socks and 10 green socks, all mixed up in the dark. How many socks must you take out to be sure of having a matching pair?[5]
A2
B3
C4
D11
E31
5.A number is 4 more than a third of itself. Which of the choices is that number?[5]
A3
B6
C8
D12
E18
6.The numbers 1 to 9 are placed in a row in some order. Can every neighbouring pair add up to an odd number?[5]
AYes, and many orders work
BYes, but only one order works
CNo, there are too many odd numbers
DNo, nine places is an odd number of places
EOnly if the row starts with an even number
7.A circle of radius 5 is drawn inside a square of side 10, touching all four sides. What fraction of the square is outside the circle, to the nearest tenth?[5]
Text description of the figure
A square with a circle drawn inside it, the circle touching the midpoint of each of the four sides. The region inside the square but outside the circle is shaded.
A0.1
B0.2
C0.3
D0.5
E0.8
8.A and B are different digits. The two-digit number AB added to the two-digit number BA gives 132. What is A + B?[5]
A3
B6
C11
D12
E13
9.Five children sit in a row. Two of them are twins who insist on sitting next to each other. In how many orders can the five sit?[5]
A12
B24
C48
D60
E120
10.Five beads of five different colours are threaded on a circular bracelet. Two bracelets count as the same if one can be turned or flipped into the other. How many different bracelets are there?[5]
A12
B20
C24
D60
E120
11.Ten different whole numbers, each bigger than zero, add up to 100. What is the largest that the biggest of them can be?[5]
A45
B50
C55
D91
E100
12.A room is 8 m long, 4 m wide and 2 m high. An ant walks across the floor and the walls from one bottom corner to the far top corner. What is the shortest distance in metres?[5]
Text description of the figure
The floor of a room, 8 metres by 4 metres, drawn flat, with one 8-metre-by-2-metre wall unfolded upwards from its far edge. A straight line joins the ant's starting corner on the floor to the opposite corner on the unfolded wall.
A9.2
B10
C11
D12
E14
13.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
14.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
15.You are 20 minutes into a 75-minute paper of 30 questions, still on question 9, and four minutes in with no progress. What is the best move?[5]
AKeep going — four minutes are already invested
BAnswer it with your best guess, mark it, and move on
CSkip it, leave it blank, and come back at the end
DGo back and re-check questions 1 to 8
EJump to question 30 and work backwards
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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, 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.
2.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.
3.E — 9KA-0037Last-digit behavior of products and powers
AI assumed a number this large has no reachable last digit and gave up on the cycle.
BI found the cycle 7, 9, 3, 1 but used a remainder of 0 when the remainder is actually 2.
CI found the cycle but counted its positions starting at zero, shifting my answer along by one.
DI assumed the units digit of any power of 7 is always 7.
4.C — 4KA-0040The pigeonhole principle
AI answered with the lucky case, where the first two socks happen to match.
BI used the number of colours as my answer without adding one for the sock that must repeat.
DI worked from the number of socks of each colour instead of the number of colours.
EI took the whole drawer, guaranteeing a pair but far more socks than are needed.
5.B — 6KA-0045Back-solve from the answer choices
AI tested the smallest choice first and stopped at the first one that felt about right.
CI read a third of itself as three times itself and solved the wrong equation.
DI reversed the comparison, finding a number whose third is 4 less than itself in the wrong direction.
EI picked the largest choice because dividing by three made me expect a big starting number.
6.A — Yes, and many orders workKA-0049Parity of numbers (odd/even behavior)
BI found one arrangement that works and assumed a problem this fiddly could only have one answer.
CI saw five odd numbers against four even ones and called that a mismatch, when a row of nine needs exactly five of one and four of the other.
DI blamed the length of the row rather than checking how many odd and even numbers it has to hold.
EI decided the first number settles it without checking that starting even would need five even numbers, and only four exist.
7.B — 0.2KA-0052Area by subtraction (shaded regions)
AI estimated by eye from the four corner pieces without comparing them to the whole square.
CI used a diameter of 5 instead of a radius of 5 when working out the circle's area.
DI assumed the circle covers half the square because it touches all four sides.
EI found the fraction covered by the circle and gave that instead of the fraction outside it.
8.D — 12KA-0060Operation puzzles and cryptarithms
AI added the digits of 132 instead of working out what A and B must be.
BI found A + B correctly and then halved it, as if the question wanted one digit.
CI spotted the 11 in the working and gave that instead of the sum it multiplies.
EI found a pair of digits by trial and misadded them by one.
9.C — 48KA-0070Counting with restrictions
AI treated the twins as one child and forgot they can also swap places with each other.
BI glued the twins together into a single block but then forgot that the block can be arranged with the others in more ways than I counted.
DI halved the 120 total, assuming the twins are together in exactly half the arrangements.
EI counted every arrangement of five children and forgot the twins' condition entirely.
10.A — 12KA-0071Overcount, then correct
BI divided the 120 arrangements by 6 rather than by the 10 movements that leave a bracelet looking the same.
CI allowed for turning the bracelet but forgot it can also be flipped over.
DI halved the 120 for flipping but forgot that turning also gives the same bracelet.
EI counted every arrangement in a line, as if the bracelet had a fixed first bead.
11.C — 55KA-0078Extremal / worst-case counting
AI found the smallest possible total of the other nine and gave that instead of what is left.
BI assumed the biggest number could be at most half the total.
DI made the other nine numbers all equal to 1, forgetting that they have to be different from each other.
EI ignored the other nine numbers entirely, as if the biggest could take the whole total.
12.B — 10KA-0082Shortest paths on and around shapes
AI measured the straight line through the middle of the room, but the ant has to stay on the surfaces.
CI unfolded the room but folded out the short wall rather than the one that makes the shortest route.
DI chose an unfolding that gives a longer straight line and did not compare it with the others.
EI walked along the edges of the room, 8 then 4 then 2, instead of cutting straight across the surfaces.
13.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.
14.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.
15.B — Answer it with your best guess, mark it, and move onKA-0126Time triage: now, later, or guess
AI let the time already spent decide, but that time is gone whatever I do next.
CI moved on but left a blank, which scores nothing if I never get back to it.
DI spent time re-reading work I had no reason to doubt, while 21 unseen questions waited.
EI assumed the hardest questions are the best use of time, when the unseen easy ones are worth the same each.