Counting when the 13th is a Friday

There is common superstition of Europeans that the number 13 is unlucky and that it is particularly unlucky for the 13th day of a month to fall on a Friday. I have heard it claimed that the 13th of the month is more often a Friday than any other day of the week. Let's see how often it really happens.

Start on the 13th of March (since it's February whose length isn't the same every year) and number its day of the week 0; number successive days of the week 1, 2, … but count modulo seven, so that two dates falling on the same day of the week get the same number. We can then determine the numbers for days of the week on which successive 13ths of the month fall: March is 31 days long, which is equal to 3 modulo 7, so April 13th falls on day 3; April is 30 days long, so May 13th falls on day 5; and so on. The following list shows each month against the number for the day on which its 13th day falls:

March, November
June, February, Next March
September, December, Leap March
April, July
May, January

Note that this is true in any given year: November 13th falls on the same day of the week as March 13th, and so on – the same may likewise be said of the 1st or 27th days of the months, though not of the last. Thus days 0, 1, 2, 3 and 5 get more months' 13ths in the given year; but the pattern shifts by a day or two for the following year. Here Next March and Leap March indicate the days on which the following year's March 13th shall fall if the given following year is a normal year or a leap year, respectively. Since 364 is a multiple of seven, normal years advance by one day and leap years advance by two.

If our initial March 13th fell in a leap year, successive years' March 13ths would fall on days 0, 1, 2, 3, 5, 6, 0, 1, 3, 4, 5, 6, 1, 2, 3, 4, 6, 0, 1, 2, 4, 5, 6, 0, 2, 3, 4 and 5, after which it would repeat itself in an endless 28-year cycle until the next century that isn't a multiple of 400 (and hence skips being a leap year). In this cycle, the digits 0 through 6 appear equally often – four times each. This implies that, in a full 28-year cycle, each day gets to be March and November 13th four times, June and February 13th four times, and so on. Thus each month's 13th day falls four times on each day of the week in any given 28-year cycle: every day of the week is the 13th of the month in 48 months of the cycle. Thus no day of the week is the 13th of the month any more often than any other day.

Thus the 13th of the month is no more apt to be a Friday than any other day of the week, when averaged across 28 years (as long as no non-leap century falls in that interval – and we haven't had one of those since 1900, nor shall we have another until 2100).

Written by Eddy.