Which one of the following years will have identical calendar with the year 2025?

Correct — D, 2031. Two conditions must both hold for a calendar to repeat: the year must start on the same weekday, and it must be of the same type, common or leap. 2025 is a common year starting on a Wednesday. A common year is 365 days, which is 52 weeks and one day, so the next year starts one weekday later; a leap year is 366 days and pushes the start two days on. Counting: 2026 starts Thursday, 2027 Friday, 2028 Saturday. 2028 is a leap year, so it advances two, and 2029 starts on a Monday; 2030 on a Tuesday; 2031 on a Wednesday. 2031 is a common year like 2025, so both conditions are met and its calendar is identical.

• (a)2028 — Fails both tests. 2028 begins on a Saturday and is a leap year, so it has an extra day in February and its dates fall on different weekdays throughout.
• (b)2029 — Begins on a Monday. The leap day of 2028 has pushed the start forward by two, so 2029 is two weekdays off from 2025 rather than back in step.
• (c)2030 — Begins on a Tuesday — one day short. It is a common year, so the type matches, but the starting weekday does not, and every date in the year would fall a day early.

Every ordinary year carries one odd day and every leap year two, an odd day being the remainder when the year's length is divided by seven. A calendar repeats when the odd days since the reference year add up to a multiple of seven and the year type matches. For a common year that follows a leap year in the pattern the repeat gap is usually six or eleven years; for a common year positioned as 2025 is, it is six.

The pitfall is stopping at the first year whose weekday looks right without checking whether it is leap, or counting the leap year as one day instead of two. Set the count out in a line — Wed, Thu, Fri, Sat, then jump two to Mon, then Tue, then Wed — and the leap year's double step is visible. It is also worth remembering that the answer year here is six years later, one of the two standard gaps, while a leap year's calendar repeats only after twenty-eight years in the ordinary case.

• A common year has 365 days, that is 52 weeks and 1 odd day; a leap year has 366 days and 2 odd days.
• 1 January 2025 fell on a Wednesday and 2025 is a common year.
• The starts run Wednesday 2025, Thursday 2026, Friday 2027, Saturday 2028, then Monday 2029 after the leap day, Tuesday 2030, Wednesday 2031.
• For the calendar to repeat, the starting weekday and the leap or common type must both match.
• 2031 is a common year starting on a Wednesday, so it repeats 2025 exactly.

• Advancing a leap year by one day instead of two.
• Matching the weekday but ignoring whether the year is leap.
• Assuming calendars always repeat after a fixed number of years.

A repeating-calendar item; a seven-line day count settles it faster than any remembered rule.

• practice — not a real PYQ

  How many odd days does a leap year contain?

  • (a)0
  • (b)1
  • (c)2
  • (d)3

  Answer(c) 2 — 366 days is 52 weeks and 2 days.

• practice — not a real PYQ

  If 1 January of a common year falls on a Friday, on which day will 1 January of the next year fall?

  • (a)Friday
  • (b)Saturday
  • (c)Sunday
  • (d)Thursday

  Answer(b) Saturday — a common year carries one odd day.