Lucky Friday the 13th

It's a spooky time. Halloween has just passed, and this week we have Friday the 13th, the third such Friday this year. How unlucky is that! Or, how lucky, if you're a black cat or a demonised Maths Master.

Judged by this criterion, 2009 has been unusually spooky. By comparison, there was only one Friday the 13th in 2008, and there will be only one next year, and one again in 2011.

How many do we expect in a given year? There are seven days of the week and twelve months of the year, and 7 into 12 won't go. However, in the long run it's a fair guess that it will all even out, with a 1/7 chance of the 13th of a given month being a Friday.

Surprisingly, this is not the case. In fact, vampires rejoice, it turns out that the 13th is most likely to fall on a Friday!

To see how this works, we have to review our calendar system. Recall that Australia, along with most countries, uses the Gregorian calendar. We wrote of this calendar earlier in the year, when discussing the dates of Easter.

The Gregorian calendar was introduced in 1582 and is based on a 400 year cycle. The cycle refers to the choosing of leap years, but it turns out that a 400-year period contains 146,097 days, which comes to exactly 20,871 weeks. This tells us that the days of the week have the same 400-year cycle. For example, today is Monday November 9, and so the cycle tells us that November 9 in the year 2409 will also fall on a Monday.

This means that to determine the frequency of Friday the 13th, we just have to check through the 4800 months in a 400-year period, and count the number of spooky Fridays. Below is the frequency table for the 13th falling on the days of the week for such a 400-year cycle.

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

As you can see, Friday is the winner with 688 hits. But the vampires shouldn't get overly excited: the chances of the 13th being a Friday is 688/4800 = 14.3%, as compared with 684/4800 = 14.25% for Thursday or Saturday, the least likely days.

Though the calendar takes 400 years to cycle, there are only 14 possible calendars for a given year. Why? There are seven possible days on which January 1 can fall, and it is either a leap year or not: then, 7 x 2 = 14.

Day of the week Jan 1	Non-leap year Fri 13th	Leap year Fri 13th
Monday	April, July	September, December
Tuesday	September, December	June
Wednesday	June	March, November
Thursday	February, March, November	February, August
Friday	August	May
Saturday	May	October
Sunday	January, October	January, April, July

By inspecting all 14 calendars we obtain the complete picture above. We see that every year must have at least one Friday 13th. And, with three such Fridays, this year was indeed the spookiest possible.

Burkard Polster teaches mathematics at Monash and is the university's resident mathemagician, mathematical juggler, origami expert, bubble-master, shoelace charmer, and Count von Count impersonator.

Marty Ross is a mathematical nomad, currently lecturing at the University of Melbourne. His hobby is smashing calculators with a hammer.

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