If first March of a year is Sunday, which day will be the first February of the next year?

Correct — D, Monday. Count the days from 1 March of the given year to 1 February of the next: March 31, April 30, May 31, June 30, July 31, August 31, September 30, October 31, November 30, December 31 and January 31, which total 337. Divide by seven — 7 × 48 = 336 — and one day is left over. One odd day past Sunday is Monday. The leap-year question never arises: February of the starting year is already behind us, and the 29th of the following February would fall after the date asked about, so neither can add a day to the count.

• (a)Friday — Three days behind the answer. It comes from counting backwards from Sunday instead of forwards, or from losing a 31-day month somewhere in the tally.
• (b)Tuesday — Two odd days, which is what you get by adding a leap day the calendar does not supply here. February 29 of the next year falls after 1 February, so it cannot affect this count.
• (c)Saturday — Six odd days, or equivalently one day short of Sunday. This is the value produced by counting 336 days as one full week too many.

Calendar problems are arithmetic modulo seven. Every seven days returns the same weekday, so only the remainder matters — the odd days. An ordinary year has 365 days and so one odd day; a leap year has two. A month contributes 3 odd days if it has 31 days, 2 if it has 30, and 0 for an ordinary February.

There is a second route that acts as a check. From 1 March of one year to 1 March of the next is 365 days in an ordinary year and 366 if that February has 29 days, so 1 March of the next year is a Monday or a Tuesday accordingly. Step back from 1 March to 1 February — 28 days in an ordinary year, 29 in a leap year. In the ordinary case Monday minus 28 days is Monday; in the leap case Tuesday minus 29 days is again Monday. Both branches converge, which is exactly why the question can be asked without telling you whether a leap year is involved.

• Only the remainder on division by seven matters; these are the odd days.
• 1 March to 1 February of the next year spans 337 days, and 337 = 7 × 48 + 1.
• One odd day after Sunday gives Monday.
• An ordinary year contributes 1 odd day, a leap year 2.
• The 29th of February can only affect a count that straddles it; here it falls after the target date.

• Adding a leap day that falls outside the interval being counted.
• Counting the starting date twice, which shifts the answer by one day.
• Working forwards from March but subtracting as though moving backwards.

A pure odd-days item. The four options are the answer plus the three nearest common miscounts, so the count has to be exact.

• practice — not a real PYQ

  If 1 January of a non-leap year is a Wednesday, what day is 31 December of the same year?

  • (a)Tuesday
  • (b)Wednesday
  • (c)Thursday
  • (d)Friday

  Answer(b) Wednesday — a 365-day year carries one odd day, so it ends on the weekday it began.

• practice — not a real PYQ

  How many odd days are there in 100 years?

  • (a)0
  • (b)3
  • (c)5
  • (d)6

  Answer(c) 5 — 76 ordinary years and 24 leap years give 124 odd days, and 124 leaves 5 on division by 7.