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2015 AMC 10B Problem 6

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2015 AMC 10B Problem 6AMC 10 Step by Step
Name:                             Date: September 13, 2026
Logic Puzzles · difficulty 2 of 5 · about 3 min · position P1-10

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?

  1. A)

    Sunday\text{Sunday}

  2. B)

    Tuesday\text{Tuesday}

  3. C)

    Thursday\text{Thursday}

  4. D)

    Friday\text{Friday}

  5. E)

    Saturday\text{Saturday}

Answer

E

Key insight

Three non-consecutive running days among Tue, Thu, Fri, Sat, Sun force Tuesday plus two of Thu/Sat, Thu/Sun, Fri/Sun; only the first leaves a legal tennis day.

Solution

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 (E) Saturday\boxed{\textbf{(E)}\ \text{Saturday}}.

Why this works

Fix the given facts, then attack the most restrictive requirement (three pairwise non-consecutive days out of five) to cut the possibilities to three cases. The tennis rule then acts as a filter. Systematic casework beats trial and error because it proves the other placements fail.

The trap

Forgetting that tennis also cannot follow swimming, which wrongly allows swim Friday, tennis Saturday (choice (D)).

Common mistakes

  • Forgetting that tennis also cannot follow swimming, which wrongly allows swim Friday, tennis Saturday (choice (D)).
  • Treating Sunday and Monday as non-adjacent or forgetting Tuesday is available, which changes which running triples are possible.

Techniques

Split into exhaustive cases and handle each