Problem 18 · 2023 Math Kangaroo
Stretch
Logic & Word Problems
work-backwarddivisibility
Robert and Sonja play a game. Taking turns, each player removes 1, 2, 3, 4 or 5 cards from the pile. Whoever takes the last card loses. There are 10 cards on the pile and it is Robert’s turn. How many cards should he leave for Sonja so that he is certain to win?
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Answer: C — 7
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Hint 1 of 2
Since taking the last card loses, you want to leave your opponent stuck taking it.
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Hint 2 of 2
Work backwards: leaving 1 card is a loss for the other player, and so is any number that is one more than a multiple of 6.
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Approach: find the losing positions and leave the opponent on one
- Whoever is forced to take the very last card loses, so leaving exactly 1 card hands the loss to your opponent.
- Because each turn removes 1 to 5 cards, you can always reply to keep the pile at the next “bad” number, which are 1, 7, 13, … (one more than a multiple of 6).
- From 10 cards Robert takes 3, leaving 7 — a losing position for Sonja, whatever she does next.
- So he should leave 7 cards, answer C.
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