Can every day be the first?

In 2026, every day of the week is the first day of the month at least once:

• Monday is June 1.
• Tuesday is September 1 and December 1.
• Wednesday is April 1 and July 1.
• Thursday is January 1 and October 1.
• Friday is May 1.
• Saturday is August 1.
• Sunday is February 1, March 1, and November 1.

Is 2026 special in this regard? If so, when is the next year when one of the days of the week is not represented among the firsts of the month? Otherwise, if 2026 is not special in this regard, then why not?

First let's note that if a month has $28$ days, looking at you February, then the next month will start on the same day of the week that it starts. If a month has $29$ days, again looking less frequently at you February, then then next month will start on the next day of the week. If a month has $30$ days then the next month will start two days after it, and finally for all of those $31$ day months, the following month will start three days of the week after it.

Let's further represent the days of the week as numerals, $\mathcal{D} = \mathbb{Z} / 7\mathbb{Z}.$ If January 1 occurs on some $x \in \mathcal{D}$ then February 1 occurs on $x+3.$ If the year is not a leap year then we have March 1 also on $x+3,$ followed by April 1 on $x+6,$ May 1 on $x+8 \equiv x+1,$ June 1 on $x+4$, July 1 on $x+6,$ August 1 on $x+9 \equiv x+2$, September 1 on $x+5,$ October 1 on $x+7 \equiv x,$ November 1 on $x+3$ and finally December 1 on $x+5.$ Gropuing these all together we see that we have all the days of the week covered $\{ x, x+1, x+2, x+3, x+4, x+5, x+6\} = \mathcal{D}.$

If on the other hand, this is a leap year, then we have January 1 and February 1 on $x$ and $x+3$ as before, but then March 1 on $x+4,$ April 1 on $x+7 \equiv x,$ May 1 on $x+2,$ June 1 on $x+5,$ July 1 on $x+7 \equiv x$, August 1 on $x+3,$ September 1 on $x+6,$ October 1 on $x+8 \equiv x+1,$ November 1 on $x+4,$ and finally December 1 on $x+6.$ Again, grouping these all togehter, we see that we have all the days of the week covered $\{ x, x+1, x+2, x+3, x+4, x+5, x+6 \} = \mathcal{D}.$

Therefore, there is nothing special about 2026. Each and every year under the Gregorian system has this property where each day of the week is represented among the firsts of the month.