Calendar — PPSC previous year questions

Two months have the same calendar when they begin on the same weekday. So the days from the 1st of one to the 1st of the other must be a multiple of 7. > April → July: 30 + 31 + 30 = 91 = 13 × 7 ✓ > June → October: 30 + 31 + 31 + 30 = 122 (not a multiple of 7) > October → December: 31 + 30 = 61; April → November: 214 (neither divisible by 7) So April and July always share a calendar. Other standard pairs are January–October (non-leap year), February–March–November (non-leap year) and September–December.

If the third Wednesday is the 15th, the first Wednesday is the 1st (15 − 14). > Wednesdays: 1, 8, 15, 22, 29 > 30th = Thursday, 31st = Friday > Fridays: 3, 10, 17, 24, 31 So the 31st is the fifth Friday of the month. The month has only four Sundays (7, 14, 21, 28). The fifth Wednesday is the 29th, not the last day.

Count the odd days from 31 December 2005 to 31 January 2010. > 2006, 2007, 2009 = 1 odd day each; 2008 (leap) = 2 → 5 odd days to 31 Dec 2009 > Saturday + 5 = Thursday (31 Dec 2009) > January 2010 = 31 days = 3 odd days → Thursday + 3 = Sunday So 31 January 2010 was a Sunday.

Count the days from 16 January to 4 March 2020. Remember that 2020 is a leap year, so February has 29 days. > Rest of January: 31 − 16 = 15 days > February: 29 days; March: 4 days → total 15 + 29 + 4 = 48 days > 48 mod 7 = 6 odd days Friday + 6 days = Thursday.

The fewest working days come in the shortest month. > A February of 28 days is exactly 4 weeks > Holidays: 4 Sundays plus the 2nd and 4th Saturdays = 6 days > 28 − 6 The minimum is 22 working days.

The 3rd is Monday. The 21st is 18 days later, and 18 = 2 × 7 + 4, so the 21st is Monday + 4 = Friday. The fifth day from the 21st is the 26th, which is Friday + 5 = Wednesday. > 21 − 3 = 18 → 18 mod 7 = 4 → 21st = Friday > 26th = Friday + 5 = Wednesday

2008 is a leap year, and the interval 3 January 2008 to 3 January 2009 contains 29 February 2008, so it spans 366 days. > 366 = 52 weeks + 2 odd days > Thursday + 2 = Saturday So 3 January 2009 was a Saturday. A 365-day interval would have shifted the day by only one.

The last day depends on how many days the month has. If the 1st is Monday, a 28-day month ends on Sunday, a 29-day month on Monday, a 30-day month on Tuesday and a 31-day month on Wednesday. Since the month is not named, the answer cannot be determined.

A year repeats its calendar when the odd days since it add up to 0 (mod 7). The new year must also be of the same type. 1896 is a leap year. Count odd days year by year (leap = 2, ordinary = 1; 1900 is NOT a leap year): > 1896–1903: 2+1+1+1+1+1+1+1 = 9 → 2 odd days > add 1904–1907: 2+1+1+1 = 5 → 14 → 0 odd days So 1908 starts on the same weekday as 1896, and is also a leap year. 1902 gives 0 odd days too, but it is an ordinary year. 1904 has 2 odd days. The usual '+28 years' shortcut fails because the non-leap century year 1900 comes in between.

Count the days from 6 December 2007 to 9 March 2008. Remember that 2008 is a leap year, so February has 29 days. > Dec 7–31 = 25, Jan = 31, Feb = 29, Mar 1–9 = 9 → total 94 days > 94 ÷ 7 = 13 weeks, remainder 3 odd days Thursday + 3 = Sunday.