In a calendar year, how many months, maximum, can start on a Sunday? What will be the next (very unlucky?) year when this maximum occurs?

Must every calendar year have at least one month that starts on a Sunday?

Looking at this problem modulo 7 the cumulative values of the various 1st’s of the months are:

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Non-leap year: (Taking January as being day 1)

Feb = 4

Mar = 4

Apr = 7

May = 2

Jun = 5

Jul = 7

Aug = 3

Sep = 6

Oct = 1

Nov = 4

Dec = 6

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Leap year: (Taking January as being day 1)

Feb = 4

Mar = 5

Apr = 1

May = 3

Jun = 6

Jul = 1

Aug = 4

Sep = 7

Oct = 2

Nov = 5

Dec = 7

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Each of the lists contains all of the days from 1 to 7.  So to answer the second question first, yes, either type of year must thus have at least one Sunday.

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As for the 1^st question the most frequent day is 3 in either type of year.  For non-leaps, the maximum is a ‘day 4’, and leaps the most frequent is a ‘day 1’.  Thus maximum Sundays occur, in the case of a non-leap year, if the 1^st January (day 1) is a Thursday, and in a leap-year if the 1^st January is a Sunday.

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Finally for the part asking how long before the next maximum Sunday year, we need to look at the overall days in the year:  Non-leap years have 365 days, which means if January 1^st is day 1, the next year’s 1^st January will be day 2.  Leap years with 366 have the subsequent year starting at day 3.  Maximum frequencies being day 4 for non-leaps, day 1 for leaps, gives the following result, that we have to wait 3 years for the next max: Either,

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If we start with the year immediately following a leap, it has day 4 as a max, the next year, day 5, the next day 6, and the leap following that will start on a day 1 (a leap year’s most frequent day)

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OR

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Starting with a leap year with its most frequent as day 1, the next year will be starting with day 3 (making day 6 the most frequent), the one after starts on day 4 (maximising the number of day 7s) and the 3^rd year will start on day 5 (maximising the day 1s)

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To quantify this, our options are for quickest return to another 3 Sunday year:

Start with a leap-year which starts on a Sunday (the last one was 1984, the next will be 2012), or start in the year immediately after a leap which begins on a Thursday (last occurring in 1981, next occurring in 2009). Either has 3 Sundays, and 3 years after also has 3 Sundays.

Posted by Alec on 2005-04-18 11:25:36