Friday 13th

One or two people have asked me to explain why the 13th of the month is more likely to fall on a Friday than on any other day.  There's no special reason why it happens to be a Friday, but it arises from two pieces of luck, one of which explains why there is any variation at all between the days of the week, and the other leading to Friday being top of the list.

The first piece of luck is to do with the length of our calendar cycle.  We use the Gregorian calendar, which has a 400-year cycle of year lengths.  Why 400?  Because the rule is that we have a leap year every four years, except for the century years which don't, except every fourth century when we do.  So 2100, 2200 and 2300 will not be leap years, but 2000 was - a double exception which meant that nobody noticed how special it was (apart from the fear of the millennium bug of course, but that's another story).  Now if you work out the number of days in that 400-year cycle, it comes to 400 x 365 + 97 (because of 100 leap years minus the 3 special non-leap years), which is 146,097.  The coincidence is that that huge number of days happens to be a whole number of weeks (20,871 to be precise).  If it weren't a whole number of weeks, then the length of the calendar cycle taking days of the week into account would be 2,800 years (because it would take seven 400-year cycles for everything to come back to start on the same day of the week).  And the 13th (or any other day of the month) would over that 2,800 years occur an equal number of times on every day of the week.

So with the fact that the 400-year cycle is a whole number of weeks, we are left with counting up how many times the 13th of the month falls on each day of the week.  We can know straight away that there will not be an even spread, because there are 4,800 months in the cycle, and 4,800 is not divisible by 7.  When you count it up, this is what you get for each day:

Sunday 687
Monday 685
Tuesday 685
Wednesday 687
Thursday 684
Friday 688
Saturday 684

So Friday wins!  Not by much... but what surprises me a bit is that the variation is more than just one day.  It all comes down to the effect of the three very specific hiccups in the cycle introduced by the century days.

As for the 13th of January, this will be on a Friday, Sunday or Wednesday 58 times each, a Monday or Tuesday 57 times each and a Thursday or Saturday 56 times each.

So now you know.  Not that it will have any effect on anyone of course.  Pure geekery!