AMC 10 · 2015 · #6
Easy mode Grade 2Mahdi plays exactly one sport each day of the week. He runs on three days, but never on two days in a row. On Monday he plays basketball, and two days later he plays golf. He also swims on one day and plays tennis on one day. He never plays tennis on the day right after a day he runs or swims. Which day of the week does Mahdi swim?
Pick an answer.
AMC 10 2015 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: Mahdi does exactly one sport on each of the seven days of the week. Some days and rules are fixed, and the rest must be deduced. Find the day he swims.
Givens: He runs on three days, and never on two days in a row.; Monday is basketball, and two days later (Wednesday) is golf.; He also swims one day and plays tennis one day.; He never plays tennis on the day right after a running day or a swimming day.
Unknowns: Which day of the week Mahdi swims.
Understand
Restated: Mahdi does exactly one sport on each of the seven days of the week. Some days and rules are fixed, and the rest must be deduced. Find the day he swims.
Givens: He runs on three days, and never on two days in a row.; Monday is basketball, and two days later (Wednesday) is golf.; He also swims one day and plays tennis one day.; He never plays tennis on the day right after a running day or a swimming day.
Plan
Primary tool: #3 Eliminate Possibilities
Secondary: #1 Draw a Diagram, #2 Make a Systematic List
Two days are already fixed, leaving only five open days for the three runs plus swimming and tennis. List the few legal ways to place the runs, then knock out the ones that force an illegal tennis day. One arrangement survives, and it pins down the swim day.
Execute — Answer: E
1.OA.A.1 Step 1 Fill in the fixed days
- Monday is basketball.
- Two days after Monday is Wednesday, so Wednesday is golf.
- That leaves Tuesday, Thursday, Friday, Saturday, and Sunday open for the three runs, the swim, and the tennis.
💡 Nail down what you already know before guessing anything.
1.OA.A.1 Step 2 Tuesday has to be a run
- The open days are Tuesday, then the block Thursday-Friday-Saturday-Sunday.
- Inside four days in a row you can fit at most two runs without two touching (like Thursday and Saturday).
- Three runs need a day outside that block, and the only one is Tuesday.
- So Tuesday is a running day.
💡 Four days in a row can only hold two spaced-out runs, so the third run is forced outside.
2.OA.A.1 Step 3 List the two remaining runs
- Tuesday is one run; the other two runs come from Thursday, Friday, Saturday, Sunday with none touching.
- The only non-touching pairs are Thursday and Saturday, Thursday and Sunday, or Friday and Sunday.
💡 Only three spaced-out pairs fit in those four days, so just check all three.
2.OA.A.1 Step 4 Knock out the bad run plans
- In each plan the two leftover days take swim and tennis, and tennis cannot follow a run or a swim.
- Runs on Thursday and Saturday leave Friday and Sunday, but both sit right after a run, so neither can be tennis - dead.
- Runs on Thursday and Sunday leave Friday and Saturday: Friday follows a run, so tennis would be Saturday and swim Friday, but Saturday then follows the swim - dead.
- Runs on Friday and Sunday leave Thursday and Saturday: Saturday follows a run, so tennis is Thursday (after Wednesday golf, allowed) and swim is Saturday - this one works.
💡 The tennis rule blocks every leftover day except when the swim lands on Saturday.
1.OA.A.1 Step 5 Read off the swim day
- The only legal week is basketball Monday, run Tuesday, golf Wednesday, tennis Thursday, run Friday, swim Saturday, run Sunday.
- Mahdi swims on Saturday, which is choice (E).
💡 One arrangement survives every rule, and it puts the swim on Saturday.
1.OA.A.1 Monday is basketball. Two days after Monday is Wednesday, so Wednesday is golf. 1.OA.A.1 The open days are Tuesday, then the block Thursday-Friday-Saturday-Sunday. Insid 2.OA.A.1 Tuesday is one run; the other two runs come from Thursday, Friday, Saturday, Sun 2.OA.A.1 In each plan the two leftover days take swim and tennis, and tennis cannot follo 1.OA.A.1 The only legal week is basketball Monday, run Tuesday, golf Wednesday, tennis Th Review
Reasonableness: Check the surviving week against every rule: runs are Tuesday, Friday, Sunday - three runs, none next to each other. Basketball is Monday, golf is two days later on Wednesday. Tennis is Thursday, the day after golf, not after a run or swim. The day after each run is Wednesday (golf), Saturday (swim), Monday (basketball) - never tennis, and the day after the swim is Sunday (run) - not tennis. Every condition holds, so the swim on Saturday is consistent.
Alternative: Instead of starting from the runs, start from tennis. Tennis cannot be the day after Tuesday's forced run, after a swim, and the leftover days are tight - testing each candidate tennis day against the rules also leaves only Thursday for tennis and forces the swim to Saturday.
CCSS standards used (min grade 2)
1.OA.A.1Solve addition and subtraction word problems within 20 (Counting days forward (two days after Monday) and judging which days sit next to each other.)2.OA.A.1Solve one- and two-step word problems using addition and subtraction within 100 (Combining the run, swim, and tennis rules across several days to test and rule out each arrangement.)
⭐ Lock in what you know, list the few legal options, then cross out the ones that break a rule.
⭐ Lock in what you know, list the few legal options, then cross out the ones that break a rule.
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