From a calendar standpoint.
(What follows is for a non-leap year.)
That is Sept. and Dec. start on the same day. Other than the extra day in Dec., the calendar features match up. For example, if one has 5 Sundays, so does the other; if one has Friday the 13th, so does the other.
Of course, as a mathematician, I wondered if this phenomenon occurred other times in the year. It does, April and July (for the same reason—which I’ll leave to the reader to determine).
Another pair of months that match are January and October.
The months that follow, February and November, must match, by definition. In this group add March due to Feb. and March matching (this is for a non-leap year).
Finally, there are 3 months whose start day does not match any other month: May, June, and August.
Summarized in the following graphic. (Another comment below the graphic.)
On a related, but different, note. Some months have 5 Sundays (or 5 Mondays, etc.).
.365 – 12*28 = 365 – 336 = 29
29 = 28 + 1 = 7*4+1.
So, of the 7 days of the week, 6 are repeated an extra 4 times. One is repeated the 5th time. That is, most years there are 4 months with 5 Sundays. There’s a 1 in 7 chance that there will be 5 months with 5 Sundays.
This can also be seen (maybe easier!) by the fact that 365/7 is 52, remainder 1. There are 52 full weeks. The year begins and ends on the same day of the weed.
Finally, the next year starts 1 day later.
OK, I’ve now beat a dead horse.
