Calendars
CSAT · Logical Reasoning
Introduction
Calendar questions ask for the day of the week for a date, or which year repeats a calendar. The idea of 'odd days' – days left over after complete weeks – makes them mechanical.
Key concepts
Odd days
An ordinary year has 365 days = 52 weeks + 1 odd day; a leap year has 366 = 52 weeks + 2 odd days.
Leap years
A year divisible by 4 is a leap year, except century years, which must be divisible by 400. So 2000 was a leap year, 1900 was not.
Odd days in centuries
100 years → 5 odd days; 200 → 3; 300 → 1; 400 → 0. So any date repeats its weekday pattern every 400 years.
Month codes (odd days by month, ordinary year)
| Month | Days | Odd days |
|---|---|---|
| Jan | 31 | 3 |
| Feb | 28/29 | 0/1 |
| Mar | 31 | 3 |
| Apr | 30 | 2 |
| May | 31 | 3 |
| Jun | 30 | 2 |
| Jul | 31 | 3 |
| Aug | 31 | 3 |
| Sep | 30 | 2 |
| Oct | 31 | 3 |
| Nov | 30 | 2 |
| Dec | 31 | 3 |
Day of the week
Count total odd days from a known reference (1 January of year 1 is counted so that 0 odd days = Sunday, 1 = Monday … 6 = Saturday), or simply count forward from a date whose weekday you know.
Repeating calendars
Two years have identical calendars if they start on the same weekday and are both leap or both ordinary. An ordinary year usually repeats after 6 or 11 years; a leap year after 28.
- Ordinary year = 1 odd day; leap = 2
- 100 years = 5 odd days; 400 years = 0
- 0 Sun, 1 Mon, 2 Tue, 3 Wed, 4 Thu, 5 Fri, 6 Sat
Solved examples
Example 1
The calendar of the year 2030 will be repeated next in which year?
(a) 2041 (b) 2042 (c) 2040 (d) 2037
- A year repeats when the total odd days in between is a multiple of 7 and both years are of the same type (leap/ordinary).
- Counting odd days (1 per ordinary year, 2 per leap year) from 2030, the first match is 2041.
Answer: (a) 2041
Example 2
The calendar of the year 2007 will be repeated next in which year?
(a) 2019 (b) 2014 (c) 2017 (d) 2018
- A year repeats when the total odd days in between is a multiple of 7 and both years are of the same type (leap/ordinary).
- Counting odd days (1 per ordinary year, 2 per leap year) from 2007, the first match is 2018.
Answer: (d) 2018
Example 3
If 21 February 2028 is a Monday, what day of the week is 2 July 2028?
(a) Saturday (b) Sunday (c) Wednesday (d) Monday
- Days from 21 Feb to 2 Jul = 132.
- 132 ÷ 7 leaves 6 odd days.
- Monday + 6 = Sunday.
Answer: (b) Sunday
Example 4
If 16 June 2007 is a Saturday, what day of the week is 7 August 2007?
(a) Sunday (b) Tuesday (c) Wednesday (d) Saturday
- Days from 16 Jun to 7 Aug = 52.
- 52 ÷ 7 leaves 3 odd days.
- Saturday + 3 = Tuesday.
Answer: (b) Tuesday
Example 5
The calendar of the year 2027 will be repeated next in which year?
(a) 2034 (b) 2039 (c) 2038 (d) 2037
- A year repeats when the total odd days in between is a multiple of 7 and both years are of the same type (leap/ordinary).
- Counting odd days (1 per ordinary year, 2 per leap year) from 2027, the first match is 2038.
Answer: (c) 2038
Example 6
If 13 June 2034 is a Tuesday, what day of the week is 16 August 2034?
(a) Thursday (b) Tuesday (c) Friday (d) Wednesday
- Days from 13 Jun to 16 Aug = 64.
- 64 ÷ 7 leaves 1 odd day.
- Tuesday + 1 = Wednesday.
Answer: (d) Wednesday
Example 7
The calendar of the year 2015 will be repeated next in which year?
(a) 2026 (b) 2027 (c) 2022 (d) 2025
- A year repeats when the total odd days in between is a multiple of 7 and both years are of the same type (leap/ordinary).
- Counting odd days (1 per ordinary year, 2 per leap year) from 2015, the first match is 2026.
Answer: (a) 2026
Example 8
The calendar of the year 2016 will be repeated next in which year?
(a) 2043 (b) 2044 (c) 2045 (d) 2023
- A year repeats when the total odd days in between is a multiple of 7 and both years are of the same type (leap/ordinary).
- Counting odd days (1 per ordinary year, 2 per leap year) from 2016, the first match is 2044.
Answer: (b) 2044
- From a known date, count days to the target and take the remainder on division by 7.
- The same date moves forward 1 weekday per ordinary year and 2 after 29 February.
- Remember an anchor: 15 August 1947 was a Friday; 26 January 1950 was a Thursday.
- Treating 1900 as a leap year.
- Counting both the start and end dates.
- Forgetting the extra day only applies after 29 February.
Practice set
Attempt all 12 questions (about 90 seconds each), then check the worked solutions below.
Q1. What day of the week was/will be 10 April 1920?
(a) Tuesday (b) Wednesday (c) Friday (d) Saturday
Q2. The calendar of the year 2008 will be repeated next in which year?
(a) 2035 (b) 2036 (c) 2015 (d) 2037
Q3. What day of the week was/will be 22 January 1913?
(a) Monday (b) Friday (c) Wednesday (d) Tuesday
Q4. If 18 May 2012 is a Friday, what day of the week is 8 October 2012?
(a) Wednesday (b) Monday (c) Friday (d) Saturday
Q5. What day of the week was/will be 16 May 1906?
(a) Friday (b) Tuesday (c) Wednesday (d) Sunday
Q6. If 3 January 2029 is a Wednesday, what day of the week is 8 September 2029?
(a) Sunday (b) Tuesday (c) Friday (d) Saturday
Q7. If 2 April 2019 is a Tuesday, what day of the week is 24 November 2019?
(a) Sunday (b) Tuesday (c) Monday (d) Friday
Q8. What day of the week was/will be 12 February 1963?
(a) Tuesday (b) Saturday (c) Friday (d) Thursday
Q9. The calendar of the year 2005 will be repeated next in which year?
(a) 2212 (b) 2012 (c) 2010 (d) 2011
Q10. If 15 March 2002 is a Friday, what day of the week is 1 September 2002?
(a) Thursday (b) Friday (c) Sunday (d) Wednesday
Q11. If 23 April 2028 is a Sunday, what day of the week is 19 December 2028?
(a) Tuesday (b) Thursday (c) Wednesday (d) Monday
Q12. What day of the week was/will be 5 January 1981?
(a) Saturday (b) Monday (c) Tuesday (d) Friday
Q1 – (d) Saturday. Odd days up to 31 Dec 1919: 1919 + 479 leap years − 19 + 4 → remainder 3 (0 = Sunday … but checking with a calendar is fastest). Add the odd days of 1920 up to 10 April: 101 days → 3 odd days. Total gives Saturday (verified with the calendar).
Q2 – (b) 2036. A year repeats when the total odd days in between is a multiple of 7 and both years are of the same type (leap/ordinary). Counting odd days (1 per ordinary year, 2 per leap year) from 2008, the first match is 2036.
Q3 – (c) Wednesday. Odd days up to 31 Dec 1912: 1912 + 478 leap years − 19 + 4 → remainder 2 (0 = Sunday … but checking with a calendar is fastest). Add the odd days of 1913 up to 22 January: 22 days → 1 odd days. Total gives Wednesday (verified with the calendar).
Q4 – (b) Monday. Days from 18 May to 8 Oct = 143. 143 ÷ 7 leaves 3 odd days. Friday + 3 = Monday.
Q5 – (c) Wednesday. Odd days up to 31 Dec 1905: 1905 + 476 leap years − 19 + 4 → remainder 0 (0 = Sunday … but checking with a calendar is fastest). Add the odd days of 1906 up to 16 May: 136 days → 3 odd days. Total gives Wednesday (verified with the calendar).
Q6 – (d) Saturday. Days from 3 Jan to 8 Sep = 248. 248 ÷ 7 leaves 3 odd days. Wednesday + 3 = Saturday.
Q7 – (a) Sunday. Days from 2 Apr to 24 Nov = 236. 236 ÷ 7 leaves 5 odd days. Tuesday + 5 = Sunday.
Q8 – (a) Tuesday. Odd days up to 31 Dec 1962: 1962 + 490 leap years − 19 + 4 → remainder 1 (0 = Sunday … but checking with a calendar is fastest). Add the odd days of 1963 up to 12 February: 43 days → 1 odd days. Total gives Tuesday (verified with the calendar).
Q9 – (d) 2011. A year repeats when the total odd days in between is a multiple of 7 and both years are of the same type (leap/ordinary). Counting odd days (1 per ordinary year, 2 per leap year) from 2005, the first match is 2011.
Q10 – (c) Sunday. Days from 15 Mar to 1 Sep = 170. 170 ÷ 7 leaves 2 odd days. Friday + 2 = Sunday.
Q11 – (a) Tuesday. Days from 23 Apr to 19 Dec = 240. 240 ÷ 7 leaves 2 odd days. Sunday + 2 = Tuesday.
Q12 – (b) Monday. Odd days up to 31 Dec 1980: 1980 + 495 leap years − 19 + 4 → remainder 3 (0 = Sunday … but checking with a calendar is fastest). Add the odd days of 1981 up to 5 January: 5 days → 5 odd days. Total gives Monday (verified with the calendar).
- Ordinary 1, leap 2 odd days
- Century leap only if ÷400
- Count days → remainder mod 7
- Same calendar: same start day & same type
