Yes, in fact, they do, and a little "everyday math" can explain why.
With seven days in a week, all months except for non-leap-year February have a few days more than a number of exactly complete weeks. That is, 28 days equal exactly four weeks by the calendar. So, the months with 30 days (April, June, and September) have two days more than exactly four weeks. The months with 31 days (March, May, July, August, and October) have three days more than exactly four weeks. (I'm only including the months from the beginning of March up to the beginning of November.)
All of the "extra" days (3 x 2; 5 x 3) add up to 21 extra days, or exactly three weeks. Since all of the days from the beginning of March up to the beginning of November equal an exact number of weeks, then whatever day of the week was March 1 will be the same day on which November 1 occurs. And so forth.
For the same reason (exact weeks), September and December dates fall on the same days of the week. September and November have 30 days (therefore 4 extra days between the two of them) and October has three extra days; 4+3=7, or exactly one week.
That was an awesome answer Aficionada....You have clearly explained why this happen...Also, I am surprised to know that even September and December dates also fall on the same days of the week... Thanks for the valuable information provided
It's really a fun pattern to observe. There are other pairs, depending on whether it's leap-year or not: in non-leap-years, April+July; leap-year adds January to those 2; in non-leap-years, January+October; in non-leap-years February+March+November.