Problem 28 · 2021 Math Kangaroo
Stretch
Logic & Word Problems
Counting & Probability
caseworkcareful-counting
In a town there are 21 knights who always tell the truth and 2000 knaves who always lie. A wizard divided 2020 of these 2021 people into 1010 pairs. Every person in a pair described the other person as either a knight or a knave. As a result, 2000 people were called knights and 20 people were called knaves. How many pairs of two knaves were there?
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Answer: D — 995
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Hint 1 of 3
Work out what each type of pair (two knights, two knaves, mixed) makes the partners say.
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Hint 2 of 3
Only mixed pairs produce 'knave' answers, two each — that pins down the number of mixed pairs.
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Hint 3 of 3
Then use the 21 knights to back out the other pair types.
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Approach: classify pairs by the labels they generate
- In a same-type pair both say 'knight'; in a mixed pair both say 'knave'.
- The 20 'knave' calls come 2 per mixed pair, so there are 10 mixed pairs (using 10 knights and 10 knaves).
- With one knight left out, the other 10 knights form 5 knight-knight pairs; the remaining 1990 knaves form 995 knave-knave pairs.
- So the answer is D.
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