Problem 22 · 2023 Math Kangaroo
Stretch
Counting & Probability
Logic & Word Problems
careful-countingcasework
Snow White organises a chess tournament for the seven dwarfs, lasting several days. Every dwarf has to play every other dwarf exactly once. On Monday Grumpy plays 1 game, Sneezy plays 2, Sleepy 3, Bashful 4, Happy 5 and Doc 6 games. How many games does Dopey, the 7th dwarf, play on Monday?
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Answer: C — 3
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Hint 1 of 3
Each dwarf can play at most the other six, and the given counts 1, 2, 3, 4, 5, 6 are all different.
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Hint 2 of 3
Start from the extremes: the one who played 6 games played everybody, while the one who played only 1 must have played that same dwarf.
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Hint 3 of 3
Keep pairing the highest remaining count with the lowest, peeling off the players whose schedule is now complete.
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Approach: pair the largest count with the smallest, working inward
- Doc played 6 games, so he played everyone — including Grumpy, whose single game must therefore be against Doc only.
- Happy played 5: everyone except Grumpy (Grumpy is already finished), so Happy played the other five, including Dopey; Sneezy's 2 games are then Doc and Happy.
- Bashful played 4: he must be Doc, Happy, Sleepy and Dopey; Sleepy's 3 are Doc, Happy and Bashful.
- So Dopey was played by Doc, Happy and Bashful only — that is 3 games, option C.
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