Problem 3 (Friday-the-13th Problem)

This month (November, 1998) the 13th falls on Friday, making it the third Friday-the-13th this year (February and March of this year also had Friday-the-13ths). When will Friday-the-13th fall during the next two years in 1999 and 2000 ? Keep in mind that the year 2000 is a leap year.

Solution

The number of days in a month is 30 or 31, with the exception of February, which has 28 days in a non leap year, and 29 in a leap year. Leap years occur every 4th year (years divisible by 4) with the exception of the first year in every century (century years), which are not leap years, unless they are divisible by 400. In other words, 1700, 1800, and 1900 were not leap years, but 2000 will be a leap year, 2100, 2200, 2300 will not be leap years, but 2400 will be a leap year. Our Gregorian calendar repeats itself every 400 years, and so using this strategy for leap years, we have 97 leap years every 400 years, which means we are saying the number of days in a year averages out to 365 = 365.2425, which is close to the real number of days in a year as determined by astronomers.

But we digress, to find the next Friday-the-13th after the present November Friday-the-13th, we know that a 30-day month is 4 weeks and 2 days, and a 31 day month is 4 weeks and 3 days. Hence, in a 30-day month, the day of the week the 13th occurs on the next month is two days later (like from a Friday to a Sunday), and in a 31 day month, the day of the week the 13th occurs on the next month is three days later (like from a Friday to a Monday). So, we make the following table to illustrate the total number of day's lag following the Friday-the-13th in November, 1998 and the day in every month on which the 13th falls.

1998	DAYS	LAG	TOTAL	1999	DAYS	LAG	TOTAL	2000	DAYS	LAG	TOTAL
Jan	31	3	...	Jan	31	3	5-Wed	Jan	31	3	13-Th
Feb	28	0	...	Feb	28	0	8-Sat	Feb	29	1	16-Sun
March	31	3	...	March	31	3	8-Sat	Mar	31	3	17-Mon
April	30	2	...	April	30	2	11-Tue	April	30	2	20-Th
May	31	3	...	May	31	3	13-Th	May	31	3	22-Sat
June	30	2	...	June	30	2	16-Sun	June	30	2	25-Tue
July	31	3	...	July	31	3	18-Tue	July	31	3	27-Th
Aug	31	3	...	Aug	31	3	21-Fri	Aug	31	3	30-Sun
Sept	30	2	...	Sept	30	2	3-Mon	Sept	31	2	33-Wed
Oct	31	3	...	Oct	31	3	5-Wed	Oct	31	3	35-Fri
Nov	30	2	0-Fri	Nov	30	2	8-Sat	Nov	30	2	3-Mon
Dec	31	3	2-Sun	Dec	31	3	10-Mon	Dec	31	3	5-Wed

We look for the first month following the Friday-the-13th when the total day's lag is a multiple of 7, like a 7, 14, 21, 28, ... . After we hit a multiple of 7, we know that month will be a Friday-the-13th month, and so we start counting the days lag from that point until we get the next multiple of 7. We see that during the years 1999 and 2000, there will be two Friday-the-13ths, one in August, 1999 and the other in October of 2000.

If you continue this analysis, you will discover that in a non leap year, the different possibilities for Friday-the-13ths in one year is one of the following:

• {Jan, Oct}
• {Feb, March, Nov}
• {April, July}
• {May}
• {June}
• {August}
• {Sept, Dec}

In the case of a leap year, the months when a Friday-the-13th occurs will be one of the seven possibilities:

• {Jan, April, July}
• {Feb, August}
• {March, Nov}
• {Sept, Dec}
• {May}
• {June}
• {Oct}

The year 1998 was interesting since it had three Friday-the-13ths, which is the most number of Friday-the-13ths that can occur in a year. You will also discover that every year has at least one Friday-the-13th. If you were really ambitious, you could write a computer program to determine the occurances of Friday-the-13th over the next 400 years.

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Last modified on Tuesday, January 12, 1999