Problem 27 · 2014 Math Kangaroo
Stretch
Logic & Word Problems
caseworksum-constraint
A group of 25 people is made up of knights, rascals and shilly-shalliers. The knights always tell the truth, the rascals are always untruthful, and the shilly-shalliers answer alternately truthfully and falsely (in either order). After the first question to everybody, “Are you a knight?”, 17 answered “Yes!”. After the second question, “Are you a shilly-shallier?”, 12 answered “Yes!”. After the third question, “Are you a rascal?”, 8 answered “Yes!”. How many knights are in this group?
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Answer: B — 5
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Hint 1 of 2
Work out how each type answers each question; note that knights and rascals both say 'yes' to 'are you a knight?'.
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Hint 2 of 2
Track the shilly-shalliers by their two alternating patterns and turn the three 'yes' counts into equations.
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Approach: translate each yes-count into an equation
- Split shilly-shalliers into those answering true-false-true and those answering false-true-false across the three questions.
- Question 3 ('are you a rascal?'): only the false-true-false shillies say yes, so that group has 8 people.
- Question 2 ('are you a shilly?'): rascals plus those same shillies say yes: r + 8 = 12, so r = 4.
- Question 1 ('are you a knight?'): knights, rascals and those shillies say yes: k + 4 + 8 = 17, so k = 5 knights.
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