Last reviewed 22 Sept 2026 · 6 min read
Odd days
Every calendar question reduces to counting odd days — the remainder when a number of days is divided by 7.
The remainder is then read as a day of the week, counting from a known reference:
| Odd days | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|---|
| Day | Sunday | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday |
Leap years
A year is a leap year if it is divisible by 4, except a century year, which must be divisible by 400.
So 1996, 2004, 2024 and 2000 are leap years; 1900, 2100 and 2200 are not.
An ordinary year has 365 days weeks odd day. A leap year has 366 days weeks odd days.
Odd days in longer periods
| Period | Odd days |
|---|---|
| Ordinary year | 1 |
| Leap year | 2 |
| 100 years | 5 |
| 200 years | 3 |
| 300 years | 1 |
| 400 years | 0 |
Because 400 years contain no odd days, the calendar repeats every 400 years — 1 January 1601, 2001 and 2401 all fall on the same weekday.
Odd days in each month
| Month | Days | Odd days |
|---|---|---|
| January | 31 | 3 |
| February | 28 / 29 | 0 / 1 |
| March | 31 | 3 |
| April | 30 | 2 |
| May | 31 | 3 |
| June | 30 | 2 |
| July | 31 | 3 |
| August | 31 | 3 |
| September | 30 | 2 |
| October | 31 | 3 |
| November | 30 | 2 |
| December | 31 | 3 |
Finding the day of a date
- Count the odd days in the complete centuries before the year.
- Add the odd days of the complete years since: one for each ordinary year, two for each leap year.
- Add the odd days of the complete months of the given year, plus the date itself.
- Take the total modulo 7 and read off the day.
Repeating calendars
Add odd days year by year until the running total is a multiple of 7 — and then check one more thing: the repeating year must have the same leap status as the original, or the two calendars agree only until the end of February. An ordinary year's calendar therefore returns after 6 or 11 years, and a leap year's after 28 (away from the century exceptions).
Worked examples
What day was 15 August 1947?
Solution. 1600 years → 0 odd days. 300 more years (to 1900) → 1 odd day. 46 complete years (1901–1946): leap years are 1904, 1908, … 1944 → 11 leap years, 35 ordinary → odd days → . In 1947 to 15 August: Jan 31 + Feb 28 + Mar 31 + Apr 30 + May 31 + Jun 30 + Jul 31 + 15 = 227 → . Total → Friday.
1 January 2020 was a Wednesday. What day was 1 January 2021?
Solution. 2020 was a leap year → 2 odd days → Wednesday + 2 = Friday.
If 1 January of a non-leap year is a Monday, what day is 31 December of the same year?
Solution. An ordinary year has 1 odd day, so it ends on the same weekday it began → Monday.
How many days are there from 3 March to 17 May in a non-leap year?
Solution. March: remaining; April 30; May 17 → 75 days after 3 March.
Was 1900 a leap year?
Solution. It is a century year and is not whole → no.
If 5 March is a Tuesday, what day is 5 April of the same year?
Solution. March has 31 days → 3 odd days → Tuesday Friday.
If today is Thursday, what day will it be after 100 days?
Solution. → Thursday Saturday.
If 26 January 2024 was a Friday, what day was 26 January 2023?
Solution. From January 2023 to January 2024 spans 365 days (the leap day of 2024 falls later, in February) → 1 odd day → Friday Thursday.
In which year will the calendar of 2026 repeat? (2028, 2032 and 2036 are leap years.)
Solution. Add the odd days year by year from 2026: 1, 1, 2 (2028), 1, 1, 1 — a total of 7 after 2031, so 1 January 2032 falls on the same weekday as 1 January 2026.
But 2032 is a leap year and 2026 is not, so the two calendars part company at the end of February. Keep going: 2032 → 2, 2033 → 1, 2034 → 1, 2035 → 1, 2036 → 2, giving 14 in all after 2036. So the whole calendar of 2026 returns in 2037, an ordinary year — the usual 11-year gap.