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Math Kangaroo practice

Name: ______________________Date: ____________5 questions · 25 points

1.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
2.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
3.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
4.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
5.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

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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 — 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.
2.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.
3.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.
4.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.
5.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.

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