How Often Are There 5 Sundays in February?Date: 02/27/2004 at 09:51:54 From: Don Subject: How often are there 5 Sundays in February How often are there 5 Sundays during the month of February? Date: 02/27/2004 at 15:22:56 From: Doctor Rob Subject: Re: How often are there 5 Sundays in February Thanks for writing to Ask Dr. Math, Don! In order for there to be 5 Sundays in February, two things must happen. First, it must be a leap year. Second, February 1st must fall on a Sunday, so that the remaining four Sundays will fall on the 8th, 15th, 22nd and 29th. This just happened in 2004, which I assume prompted your question! Now, if February 1st is a Sunday, then January 1st must have been on a Thursday (you can confirm that with any calendar). So an equivalent question is asking how often January 1 occurs on a Thursday in a leap year. At first glance one would think that, since leap years are 1/4 of the years, and Thursdays are 1/7 of the days, that the answer should be 1/28 of the years have that property. In the Julian calendar, this is the correct answer, and those years come exactly 28 years apart. However, things are slightly more complicated in the Gregorian calendar, because not every year divisible by 4 is a leap year. For example, 1800 and 1900 were not leap years, and 2100 will not be, either. The Gregorian calendar pattern of years repeats with a cycle of 400 years. This is equal to 146,097 days, or 20,871 weeks. Within those 400 years there are 97 leap years. Since 97 is not divisible by 7, the day of the week for 1 January cannot be the same for all the days of the week. If we look at the 97 leap years in a 400-year cycle, and count the number with January 1 on a Thursday, we find just the following years: 2004, 2032, 2060, 2088, 2128, 2156, 2184, 2224, 2252, 2280, 2320, 2348, 2376. Notice that between century years, there are gaps of 28 years, but across a century year, there is a gap of 40 years, and 10*28 + 3*40 = 400, and 10 + 3 = 13. Thus there are 13 leap years containing five Sundays in every 400 years. The situation for other days of the week that January 1 falls on in leap years in any cycle of 400 years is: Sunday 15 Monday 13 Tuesday 14 Wednesday 14 Thursday 13 Friday 15 Saturday 13 ---- 97 Now between century years there are gaps of 28 years, but across a century year, there are gaps of either 12 or 40 years. There can be 0, 1, or 2 12-year gaps. 10*28 + 3*40 + 0*12 = 400, 10 + 3 + 0 = 13, 11*28 + 2*40 + 1*12 = 400, 11 + 2 + 1 = 14, 12*28 + 1*40 + 2*12 = 400, 12 + 1 + 2 = 15. Finally, the answer to your question is, 13 times every 400 years, or, on average, about once every 30.77 years. The gaps between can be 28 or 40 years. Feel free to reply if I can help further with this question. - Doctor Rob, The Math Forum http://mathforum.org/dr.math/ |
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