Problem 24 · 2021 Math Kangaroo
Stretch
Logic & Word Problems
caseworksum-constraint
Three boys played a “Word” game in which each wrote down 10 words. A boy scored 3 points for a word if neither of the other boys had it, and 1 point if exactly one of the other boys also had it. No points were given for a word all three boys had. When they added up their scores, all three were different. Sam had 19 points, the smallest score, and James had the highest. How many points did James score?
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Answer: E — 25
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Hint 1 of 2
Each of a boy's 10 words earns him 3 (his alone), 1 (shared with exactly one other), or 0 (all three have it), so any boy's score is one of \(3u + s\) where \(u + s \le 10\).
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Hint 2 of 2
A word shared by two boys gives each of them 1 point, so a shared word adds points symmetrically; track the total points across all three boys and use that Sam is fixed at the minimum 19.
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Approach: bound the top score, then exhibit a triple that reaches it
- Write each boy's score as \(3u + s\): \(u\) words his alone (3 each), \(s\) words shared with exactly one other boy (1 each), and \(u + s \le 10\), so every score is between 0 and 30.
- Sam is the minimum at 19, and the three scores are distinct, so James (the max) is more than 19; trying James \(= 25\) means his words are eight unique \((24)\) plus one shared \((1)\), using \(8+1=9\) of his 10 words.
- The shared words are also counted for whoever shares them, and a consistent set of 10-word lists can be built giving the three totals \(19,\,21,\,25\) (all different, Sam lowest, James highest), while no arrangement makes James's total exceed 25 once Sam is pinned at 19 and all three differ.
- So James scored \(25\) points, choice (E).
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