Problem 21 · 2017 Math Kangaroo
Stretch
Logic & Word Problems
sum-constraint
A bag contains only red and green marbles. If you take out any 5 marbles, at least one is red. If you take out any 6 marbles, at least one is green. What is the greatest possible number of marbles in the bag?
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Answer: C — 9
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Hint 1 of 2
'Any 5 include a red' limits how many greens there can be; 'any 6 include a green' limits the reds.
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Hint 2 of 2
Turn each rule into a cap: greens ≤ 4 and reds ≤ 5, then add.
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Approach: convert each guarantee into a maximum count
- If any 5 marbles always contain a red, there can be at most 4 greens (else 5 greens could be drawn).
- If any 6 marbles always contain a green, there can be at most 5 reds (else 6 reds could be drawn).
- So at most 4 green + 5 red = 9 marbles, and 4 green with 5 red satisfies both rules.
- The maximum is 9 (C).
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