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Math Kangaroo practice

Name: ______________________Date: ____________3 questions · 15 points

1.Six friends sit around a circular table. Two seatings count as the same if one can be turned into the other by rotating the table. How many genuinely different seatings are there?[5]
- A120
- B720
- C36
- D60
- E24
2.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
3.An ordinary six-sided die is rolled three times. What is the probability that at least one roll shows a six?[5]
- A91/216
- B1/2
- C125/216
- D3/216
- E1/6

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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.A — 120KA-0166Counting up to symmetry
- BI counted every arrangement in a row, so each circular seating got counted once for every rotation.
- CI divided 720 by 20 or some other number instead of by the 6 rotations.
- DI divided by 12, treating reflections as identical too, though the question only allows rotations.
- EI fixed two people rather than one, dividing by an extra factor.
2.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.
3.A — 91/216KA-0168Complementary counting
- BI added 1/6 three times, which counts the overlapping cases more than once and would exceed 1 for seven rolls.
- CI worked out the probability of NO six and forgot to subtract it from 1.
- DI found the probability of three sixes instead of at least one.
- EI gave the probability for a single roll and ignored that there are three.

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