## Five Fridays, Five Saturdays and Five Sundays

I received a message at the beginning of October: “This month has 5 Fridays, 5 Saturdays and 5 Sundays; this only happens every 823 years.”

Wait a minute. The Gregorian calendar cycles every 400 years. Where is the figure of 823 coming from?

Wait another minute. Within a century the calendar repeats itself every 28 years. So we are guaranteed that October 2038 will be the same as October 2010.

Wait one more minute. To have a month with five Fridays, Saturdays and Sundays, we need a month that has 31 days and starts on a Friday. There are seven months a year with 31 days, so on average we would expect to have such a month once a year.

If you study the calendar you can see that the seven long months start on six different days. This means that two of the months start on the same day and one of the days is skipped altogether. We see this in both leap years and non-leap years.

Ironically, 2010 is the year with two long months starting on Friday — October and January. Despite the claims of the email about this only happening every 823 years, in fact the same phenomenon occurred twice this year. The next time this will happen is in July 2011.

For those people who get all excited when a month has five Fridays, five Saturdays and five Sundays, I have good news for you. The month following each of these months has to start on Monday. And unless it is a February of a non-leap year, it will have five Mondays.

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1. #### Jonathan:

Perhaps the e-mail was making its way around the internet, and the interesting bit about the full moon was lost?

Jonathan,

3. #### Jonathan:

823/28 ≈ 29.4, which is pretty close to a lunar month (29.53). So the original post was probably referring to something relating phases of the moon to days of the week.

Lame, actually, because, omitting leap year Februarys, they all occur about once every 820 years. Next time the full moon is on the 5 Wednesday of July? I don’t know, but it will happen again 823 years (this has to be close to correct).

4. #### Vishal:

Can you please explain the difference between: “The Gregorian calendar cycles every 400 years” and “Within a century the [Gregorian] calendar repeats itself every 28 years”? Thanks!

5. #### Tanya Khovanova:

Vishal,

Every fourth year is a leap year, except at the turn of a century.

6. #### Pavel Litvinov:

Tanya, I believe, that a turn of the century year is a leap year if it is divisible over 400. Then 2000 is a leap year, but 1700, 1800 and 1900 are not.

7. #### Pavel Litvinov:

Tanya, turn of a century year divisible over 400 is a leap year. Thatis the reason of 400 years periodicity.

8. #### Barry:

the five fri, sat, sun thing every 823 years is correct when used in conjunction with the first day of the month being a new moon–july 2011