What day of the week were you born?

Tonight's idea

A week is 7 days, so weekdays are clock arithmetic with a 7-hour clock. A normal year is 365 days, and \(365 = 52 \times 7 + 1\). So any date moves forward one weekday per year. If your birthday was a Tuesday this year, it is a Wednesday next year, unless a 29 February sneaks in between, which pushes it two.

Leap years. A year is really 365.2422 days, so every four years we add a day. That overshoots slightly (it gives 365.25), so century years skip the leap day, except every fourth century, which keeps it. The rule gives \(365 + \frac14 - \frac1{100} + \frac1{400} = 365.2425\) days. So 1900 was not a leap year, 2000 was, and 2100 will not be.

Doomsday. John Conway noticed that in any year, one easy date in each month lands on the same weekday, which he called that year's doomsday. Even months from April on are easy (4/4, 6/6 and so on); for odd ones remember "I work 9 to 5 at the 7-Eleven".

Finding the doomsday. For a year in the 2000s, take the last two digits, add the number of whole times 4 goes into them, and take the remainder on dividing by 7. Count that many days on from Tuesday. In symbols, with \(yy\) the last two digits: Tuesday + \((yy + \lfloor yy/4 \rfloor) \bmod 7\), where \(\lfloor\ \rfloor\) means round down and \(\bmod 7\) means the remainder. For 2026: \(26 + 6 = 32\), and 32 leaves remainder 4. Tuesday + 4 = Saturday. Check: 4 April 2026 is a Saturday. So are 6 June, 8 August, 10 October, 12 December, 9 May, 5 September, 11 July, 7 November, and 28 February.

The \(yy/4\) part is the leap days. Each year moves things one day, each leap year one extra. For the 1900s, start from Wednesday instead of Tuesday.

Try the weekday demo to check yourself.