Problem 14 · 2012 Math Kangaroo
Hard
Algebra & Patterns
substitutionsum-constraint
Tom and Mary play a game with a coin. When the coin shows heads, Mary wins and Tom must give her two sweets. When the coin shows tails Tom wins and Mary must give him three sweets. After 30 throws of the coin they each have the same number of sweets as they had at the start of the game. How often has Tom won?
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Answer: B — 12
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Hint 1 of 2
Let h be heads and t tails; the totals must end where they started.
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Hint 2 of 2
Tom's net change is −2 per head and +3 per tail, and it must be zero.
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Approach: set net change to zero with a fixed number of throws
- In 30 throws let h be heads (Mary wins, Tom loses 2) and t tails (Tom wins 3), with h + t = 30.
- For the counts to return to the start, Tom's net is 0: −2h + 3t = 0, i.e. 2h = 3t.
- Solving with h + t = 30 gives h = 18, t = 12, so Tom won (the tails) 12 times (B).
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