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3 months, maximum, can start on a Sunday. (The months will always be February, March and November for a non-leap year, and January, April and July for a leap year.) The next year when 3 months will all start on Sunday is the non-leap year 2009. There is always at least one month that starts on a Sunday.
Mod 7, the starting day of the next month advances by M%7 where M is the number of days of the current month and M%7 is its remainder when divided by 7. If the 1st of January falls on day d, then the first of February always falls on day d+3, and the first of March falls on day d+3 in non-leap years but on day d+4 in leap years, etc. For non-leap years, if January falls on day d, then for the 12 months in order starting with January, the 1st of the month will fall on day d+k (mod 7), where the sequence of k's is 0 3 3 6 1 4 6 2 5 0 3 5. For leap years, the similar sequence is 0 3 4 0 2 5 0 3 6 1 4 6. We see that the high runner for k is 3 in non-leap years, and 0 in leap years, each occurring 3 times. To find the next year where three months start on a Sunday is not very difficult -- a simple way is to just look at calendars produced by some standard program such as the Unix program cal.
Notice that the two sequences of k's each contain every number from 0 through 6 at least once. This means that given any day of the week, at least one month will start on that day. |
