Friday 13th


I wouldn't be surprised if it was just my foolishness, but somehow I always used to think that Friday the thirteenths were not such frequent occurrences. A little while ago I thought I'd try to work out just how common they were.

Obviously 13th January may be a Friday, but it may be any other day of the week too. Suppose we number the days 1, 2, ..., 7. Then perhaps later we can work out which weekday might go with each number. There are 31 days in January, that is four weeks, or 28 days precisely, plus 3. So if 13th January is day 1, 13th February will be day 4. But now we see that it would be better to number the days from 0 to 6, for then 28 + 3 brings 13th February to day 3, the same number nice and simply.

Now February may have 28 days precisely, or 29 in a leap year. Thus 13th March may also be day 3, or it may be day 4. March has 31 days, so 13th April will be day 6 or day 6 + 1, that is back to day 0 in a leap year.

Working through the whole year, the days will clearly be as follows:

  Jan    Feb    Mar     Apr    May    Jun     Jul    Aug    Sep     Oct    Nov    Dec  
    0      3      3       6      1      4       6      2      5       0      3      5  
                   4       0      2      5       0      3      6       1      4      6  

The second row shows leap year days. We still don't know what days the numbers stand for, but look at what numbers they are. In a non-leap year, we can find 0, 1, 2, 3, 4, 5 and 6 - every weekday is represented somewhere, indeed some more than once. The same happens in a leap year.

In a non-leap year indeed, if day 3 is Friday, there will be three Friday 13ths, namely in February, March and November. Otherwise a Friday day 0 or day 6 will account for two such Fridays. In a leap year there may also be three Friday 13ths, with Friday day 0. Apart from those, there may be two supposedly unlucky days with Friday day 3 or day 4 or day 6.

So as many as three Friday 13ths in any one year! I must say I was surprised, or maybe I was just ignorant.

Anyway, that calculation might make a good elementary mathematics exercise for school students!

This year, 2013, is not a leap year, and 13th January was a Sunday, so Sunday is day 0, meaning that Friday is day 5. Thus September and December have Friday 13ths. Just two. Last year was a leap year, and 13th January, day 0, was a Friday. That means two more as well as January, in April and July. Next year, 2014, has a Monday 13th January, so Friday is day 4. Hence only one Friday 13th, in June.