If 29th February 2004 was a Sunday, which of the following months starts with…

To find the day of the week for a later date from a known reference date, count the total number of days between them and take that count modulo 7 — this is the 'odd days' method. A remainder of 0 means the same weekday repeats; a remainder of r means the weekday is exactly r days after the reference day (with Sunday as day 0, Monday as day 1, and so on through Saturday as day 6).

1. Reference point: 29th February 2004 is a Sunday.

2. 1st March is 1 day after 29th February, so 1st March = Sunday + 1 = Monday.

3. March has 31 days, so 31 mod 7 = 3 odd days: 1st April = Monday + 3 = Thursday.

4. April has 30 days, so 30 mod 7 = 2 odd days: 1st May = Thursday + 2 = Saturday.

5. May has 31 days, so 31 mod 7 = 3 odd days: 1st June = Saturday + 3 = Tuesday.

6. June has 30 days, so 30 mod 7 = 2 odd days: 1st July = Tuesday + 2 = Thursday.

7. July has 31 days, so 31 mod 7 = 3 odd days: 1st August = Thursday + 3 = Sunday.

So 1st August 2004 falls on a Sunday.

Independent check: the total elapsed days from 29th February to 1st August is 1 (to 1st March) + 31 (March) + 30 (April) + 31 (May) + 30 (June) + 31 (July) = 154 days. Since 154 = 22 x 7 is an exact multiple of 7, 1st August must fall on exactly the same weekday as 29th February — Sunday — confirming the step-by-step result.