Problem 20 · 1985 AJHSME
Hard
Number Theory
31-day-monthweekday-frequency
In a certain year, January had exactly four Tuesdays and four Saturdays. On what day did January 1 fall that year?
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Answer: C — Wednesday.
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Hint 1 of 3
January has 31 days, and 31 = 4 weeks + 3 leftover days. That means SOME weekdays squeeze in a 5th time. Which ones? The first three days of the month — they're the extras.
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Hint 2 of 3
So exactly three weekdays appear 5 times (the day Jan 1 lands on, plus the next two), and the other four appear only 4 times. The problem wants Tuesday AND Saturday to be among the rare (4-time) ones.
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Hint 3 of 3
For both Tuesday and Saturday to appear only 4 times, neither can be in the 'first three days' trio. Test each possible starting day and see whose trio of three consecutive weekdays misses both Tue and Sat.
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Approach: find which start makes Tue and Sat both rare
- 31 days = 4 full weeks (28 days) + 3 extra days. Every weekday shows up 4 times for the 28, and the 3 extras — Jan 1, 2, 3's weekdays — get a 5th appearance. So three consecutive weekdays appear 5 times; the rest appear 4.
- We need Tuesday and Saturday to both be 4-time days, i.e. NOT in that trio of three-in-a-row starting at Jan 1. List the trios: a Monday start gives Mon-Tue-Wed (hits Tue ✗); a Tuesday start hits Tue ✗; a Wednesday start gives Wed-Thu-Fri — misses both Tue and Sat ✓.
- So January 1 fell on Wednesday.
- Why this transfers: in any month, take the day count mod 7 to find how many 'extra' days there are; those extras (counting from the 1st) are exactly the weekdays that occur one more time than the others.
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