Problem 13 · 2017 AMC 8
Easy
Logic & Word Problems
sum-constraint
Peter, Emma, and Kyler played chess with each other. Peter won 4 games and lost 2 games. Emma won 3 games and lost 3 games. If Kyler lost 3 games, how many games did he win?
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Answer: B — 1 win.
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Hint 1 of 2
You're not told how many games were played — and you don't need to be. Every single game produces exactly one winner and one loser, so across everyone, total wins must equal total losses.
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Hint 2 of 2
Counting principle: when each event creates one of each kind, the two totals are forced equal. Add up the losses, set wins to match, solve for the unknown.
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Approach: total wins = total losses
- Each game makes one winner and one loser, so summed over all three players, wins = losses. This balance lets you skip working out who played whom.
- Total losses: 2 + 3 + 3 = 8. So total wins must also be 8: 4 + 3 + Kyler = 8.
- Kyler's wins = 8 − 7 = 1.
- Why this transfers: any tournament/handshake/edge-counting setup where each event contributes equally to two tallies lets you equate the tallies instead of reconstructing the schedule.
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