An expert question on cycles for P5–P6, with a step-by-step solution, diagrams and the answer. Cycle questions appear in the Hua Xia Cup from the P1 first round to the P5 papers: which comes in a given place, which day of the week, which last digit.
The question
In a certain year, May has 31 days.
The dates of all the Tuesdays in that month add up to 75.
What day of the week is 14 August of the same year? (Answer 1 for Monday, 2 for Tuesday … 7 for Sunday.)
Show the hints (3)
💡Hint 1: How far apart are the dates of the Tuesdays? How many Tuesdays can a month have?
💡Hint 2: Say the first Tuesday is the a-th (a from 1 to 7). Try both "five of them" and "only four", and see which gives a sensible a.
💡Hint 3: The first Tuesday is the 1st. Now count the days from 1 May to 14 August.
Worked solution
Have a go first, then read the solution step by step.
1.Tuesdays come every 7 days. Say the first is the a-th, with a from 1 to 7; a month has five Tuesdays at most, four at least.
2.If there are 5 Tuesdays: a + (a + 7) + (a + 14) + (a + 21) + (a + 28) = 5 × a + 70 = 75, so a = (75 − 70) ÷ 5 = 1; the 29th is within the 31st: it fits.
3.If there are 4 Tuesdays: a + (a + 7) + (a + 14) + (a + 21) = 4 × a + 42 = 75, and a = (75 − 42) ÷ 4 is not a whole number: impossible.
4.So the month has 5 Tuesdays, the first on the 1st (1 May is a Tuesday).
5.From 1 May to 14 August: 31 − 1 = 30 days left in May, then June 30, July 31, then 14 days into August: 30 + 30 + 31 + 14 = 105 days.
6.105 ÷ 7 = 15 exactly, so it is still Tuesday: 2.
⚠️Common mistake: trying only "five Tuesdays" (or only four) without checking that a makes sense; or counting 1 May itself in the gap.
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