GeekDad Puzzle of the Week Solution – Friggatriskaidekaphobia

Here is the puzzle as previously posted:

The Dark Ages, in Europe, were a time of fear. There was the fear of invading armies, the plague, dragons — there was a lot the be afraid of. Some people were superstitious — afraid of things based upon omens or signs that weren’t necessarily related to bad luck.

One specific fear that people still encounter today is Friggatriskaidekaphobia – the fear of a Friday falling on the 13th day of the month.

If we consider the Dark Ages as starting on July 31, 476 AD, and ending January 1,1000 AD, how many “Friday the 13th” dates were encountered during the Dark Ages? For purposes of this puzzle we will use the standard, current month names and day counts, the current rules for “Leap Day” calculations, and will assume that the Julian calendar was in use the entire time. Per most sources, the last day that the Julian calendar was in use was Wednesday, 02 September 1752 AD — please use that date as an anchor for any and all calendar calculations.

Solutions to week’s puzzle were interesting; each had a logical flow and was internally consistent (i.e., no obvious errors), but none of them matched. Therefore, all answers sent in were entered into the random drawing. Congratulations to Paul Gajewski, selected as the winner of this week’s $50 ThinkGeek gift certificate.

Here’s how I arrived at my answer of XXX “Friday the 13th”s from July 31, 476 AD to January 1, 1000 AD:

Each year can begin with one of seven says of the week, and can either be a leap year or a standard (non-leap) year. Therefore, there are only fourteen different calendars that can possibly happen, and each has its own profile for Fridays the 13th.

This year, 2013, started on a Tuesday and is not a leap year. Looking at my calendar, I can see that 13 September and 13 December will both be Fridays.

If September 02, 1752 AD (a leap year) was a Wednesday, simple math gets to it beginning on a Wednesday. As a leap year beginning on a Wednesday, given no changes to the calender it would have March and November 13th falling on Fridays.

Performing a simple “mod 7” on the number of days from January 1, 1752 to January 1, 1751, we can see that 1751 began with a Tuesday, and (as a non-leap year) would have Fridays the 13th in September and December. Using the same method to back up to 1750, 1749, 1748, … 477, and finally 476 AD, it is straightforward to determine which calendars each used, and therefore which months had Fridays the 13th.

Not counting Friday, February 13th, 476, I got a total of 861 days that friggatriskaidekaphobics should avoid.

Many thanks to everyone that posted a solution, and keep a sharp eye out for this week’s upcoming puzzle.

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