# Friday the 13th

Do you suffer from *paraskevidekatriaphobia* (the fear of Friday 13^{th})? Well may you do so, for the 13^{th} day of a month is (slightly) more likely to be a Friday than any other day of the week. And 2015 has had three Friday the 13^{th}s (the most possible), so it’s no wonder you’re ducking for cover at the slightest hint of trouble.

How might this happen? Surely things average out in the end? Well, no.

The Earth-Moon-Sun system has three basic parameters used by civilisations in the past few thousand years to create their calendar:

- The time for he Earth to rotate once about its axis, like a spinning top. If measured as the average time between the sun reaching its zenith (highest point), then it is very close to 86400 seconds. In fact, in 1895, 1/86400
^{th}of an average day was taken as the definition of the second (and therefore the minute, hour etc). Since then, tidal friction has slowed the earth’s rotation by a couple of milliseconds per day, meaning we need a leap second from time to time. - The time for the Moon to revolve about the Earth is 29.530589 days – not a nice tidy number at all, making lunar calendars rather uneven.
- The time for the Earth to revolve about the Sun, about 365 days 5 hours 48 minutes 46 seconds or 365.2422 days. Again, nothing neat and tidy about this one.

Here I’m (very parochially) looking only at the Western calendars of the last 2000 years. (If you want to be less parochial, you might start with History of Calendars.)

The Julian calendar was instituted by Julius Caesar in 46BC. It had 365 days in each year, except for leap years, which had 366. This assumes a year length of exactly 365.25 days – pretty good, but about 11 minutes and 14 seconds too long. This meant that, by the 16th century, the year (and its seasons and so on) was about 12 days behind what the seasons would say.

So, in 1562, Pope Gregory introduced an adjustment to the Julian calendar, making years divisible by 100 but not by 400 to be non-leap years. This assumes a year length of 365.2425 days. The difference between the assumed and actual year length is then 26 seconds, and so will still accumulate, but only by one day in 3323 years. I think we can fairly leave it to our 54th century descendants to figure out what to do then.

Now, what about Friday the 13^{th}?

In the Gregorian calendar, there are 146097 days, which is exactly 20871 weeks. This means that each 400 years, New Years Day is the same day of the week. For example, in 2015, New Years Day was a Thursday, as will New Years Day be in 2415.

It also means that other days, like the 13^{th} of a month, are similarly unbalanced. In fact, the number of Friday the 13^{th}s per day of the week in each 400 years of the Gregorian calendar is as follows:

Day of week |
13th of month |

Sunday | 687 |

Monday | 685 |

Tuesday | 685 |

Wednesday | 687 |

Thursday | 684 |

Friday | 688 |

Saturday | 684 |

So there you go, more bad things do happen on Friday the 13^{th} – because there are more of them!

Note: My apologies to Spanish or Italian readers, for whom I understand the equivalent culturally unlucky dates are Tuesday 13^{th} and Friday 17^{th} respectively.