# Proof about Friday the 13th

Few years ago I learned that there are more friday the 13th in the 400 years period of our calendar, than all other days, but I have never found any proof !

Can you help me please ?

• What is your question? More Fridays than what? Apr 1, 2015 at 22:35
• The number of Friday the 13ths is exactly the same as the number of Saturday the 14ths. So (Friday,13) is at least not a strict maximum of anything. Apr 1, 2015 at 22:39
• Ok so Friday 13, sunday 14, monday 9 etc... but there are no proof of that ? Apr 1, 2015 at 22:46
• See related question: math.stackexchange.com/questions/27083/… The proof would be similar, involving actually counting how many of the "$13^{th}$"'s are fridays. Also read scienceworld.wolfram.com/astronomy/FridaytheThirteenth.html Apr 1, 2015 at 22:52

In the Gregorian calendar, the relation between dates and weekdays repeats exactly every 400 years.

Over each 400-year period, the 13ths of the 4800 months fall on the following weekdays:

Mon: 685 times
Tue: 685 times
Wed: 687 times
Thu: 684 times
Fri: 688 times
Sat: 684 times
Sun: 687 times


(which I found simply by letting a computer iterate through all 4800 months, keeping count). So Friday is indeed slightly more common than other days of the week among all 13ths.

• Not something one can really see the effect of in a lifetime, I'd say. Apr 1, 2015 at 23:20
• Yes but is it possible to make this with a paper and pencil ? In less than one hour ^^ ? I think that someone knew this a centuary before computer :) Apr 1, 2015 at 23:26
• @Shadock: There are some shortcuts one could use if one wanted to compute it without computer support -- for example any 28-year period that doesn't contain a not-divisible-by-400 century will have equally many of each day-of-week, and by excluding 13 appropriately chosen such periods we can get down to 36 years that need manual checking. Computing the which day March 1 is in those is fairly quick, and how many 13ths of each kind there is in the subsequent 12 months depends only on that. So only 7 different sample years have to be drawn up completely. Apr 1, 2015 at 23:37
• However, all that is not clearly relevant, given that we're both typing and reading this on computers, so computers obviously exist and are available to us! Apr 1, 2015 at 23:38