Problem 20 · 2011 Math Kangaroo
Hard
Logic & Word Problems
casework
The brothers Gerhard and Günther pass on information about the members of their chess club. Gerhard says: “All members of our club are male, with five exceptions.” Günther says: “In every group of six members there are at least four female members.” How many members does the chess club have?
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Answer: B — 7
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Hint 1 of 2
'Five exceptions' means exactly five female members.
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Hint 2 of 2
If every group of six must hold at least four females, no group of six can contain three males — so there are at most two males.
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Approach: bound the males, then make the group-of-six condition meaningful
- There are exactly 5 female members. Any six members must include ≥4 females, so at most 2 males.
- For the 'every group of six' statement to be about more than the whole club, there must be more than six members.
- So 5 females + 2 males = 7 members.
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