2015 AMC 10B Problem 6
Marley practices exactly one sport each day of the week. She runs three days a week but never on two consecutive days. On Monday she plays basketball and two days later golf. She swims and plays tennis, but she never plays tennis the day after running or swimming. Which day of the week does Marley swim?
Show solution without answering
Key insight
Tuesday must be a running day, since four days in a row hold at most two runs. The tennis rule then rules out all but one schedule.
Solution
1. Fill in what we know. Monday is basketball. Two days later, Wednesday, is golf. The other five days are Tuesday, Thursday, Friday, Saturday and Sunday. Three of them are running days, one is swimming and one is tennis.
2. Place the runs. No two running days can be next to each other. Thursday, Friday, Saturday, Sunday are four days in a row, and from four days in a row you can pick at most two that are not next to each other. So Tuesday must be a running day, plus two of Thursday to Sunday that are not neighbors. The only choices are:
- Thursday and Saturday,
- Thursday and Sunday,
- Friday and Sunday.
3. Use the tennis rule. Tennis can never be the day after running or the day after swimming. Check each choice:
- Runs on Thu, Sat. Swimming and tennis go on Friday and Sunday. But Friday comes after a run (Thursday) and Sunday comes after a run (Saturday), so tennis has nowhere to go. Impossible.
- Runs on Thu, Sun. Swimming and tennis go on Friday and Saturday. Friday comes after a run, so tennis must be Saturday and swimming Friday. But then tennis is the day after swimming. Impossible.
- Runs on Fri, Sun. Swimming and tennis go on Thursday and Saturday. Saturday comes after a run, so tennis is Thursday (after golf, which is fine) and swimming is Saturday. This works.
Only one schedule works, and in it Marley swims on Saturday.
The answer is .
Why this works
Write down the fixed facts first, then use the strictest rule (three running days, none next to each other) to cut the possibilities down to a few. The last rule then checks each one. Going through every case, instead of guessing one schedule that works, also proves the other answers are wrong.
Concise solution
A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.
Monday is basketball and Wednesday is golf. The remaining days are Tuesday, Thursday, Friday, Saturday, Sunday; three of them are running days (no two consecutive) and the other two are swimming and tennis.
Among Thu, Fri, Sat, Sun, at most two can be non-consecutive, so Tuesday must be a running day, joined by one of the pairs Thu, Sat, Thu, Sun, Fri, Sun.
- Runs Thu, Sun: swim/tennis fill Fri and Sat. Fri follows a run, so tennis is Sat; but then swim is Fri and tennis follows swimming. Not allowed.
- Runs Fri, Sun: swim/tennis fill Thu and Sat. Sat follows a run, so tennis is Thu (after golf, fine) and swim is Sat. Allowed.
- Runs Thu, Sat: swim/tennis fill Fri and Sun. Both follow a run, so neither can be tennis. Not allowed.
Only the middle case survives: Marley swims on Saturday.
The answer is .
Review in the AoPS books
- Intermediate Counting & Probability, Chapter 2: Sets and Logic (§2.4)
- Introduction to Counting & Probability, Chapter 2: Basic Counting Techniques (§2.2)
- Prealgebra, Chapter 14: Counting (§14.3)
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Problem © Mathematical Association of America (MAA), American Mathematics Competitions. Reproduced for non-commercial educational use. Solution and commentary are original to this site.