Every year has a Friday the 13th

Question

Superstition aside, "Friday the 13th" is not rare, it turns out every single year is guaranteed to have at least one.

Prove that in any common (non-leap) year, at least one month's 13th falls on a Friday, no matter which weekday the year starts on.

Accepted Answer

The trick is to show that across a year's 12 months, the 13th of the month lands on every one of the seven weekdays, so whatever day "Friday" is, some month's 13th hits it. Find each month's starting offset. Weekdays cycle every 7 days, so what matters is the number of days before each month, taken modulo 7. Using the common-year month lengths (Jan 31, Feb 28, Mar 31, ...), the cumulative days before the 1st of each month, reduced mod 7, are: Month Days before (mod 7) --------------------------- Jan 0 Feb 3 Mar 3 Apr 6 May 1 Jun 4 Jul 6 Aug 2 Sep 5 Oct 0 Nov 3 Dec 5 Check which weekdays are c…

Every year has a Friday the 13th

Your answer

More Brainteasers questions