Fewest Firsts

As we just noted, in 2026, all seven days of the week appear as the first of the month at least once. But you know, I decided that I don’t like that at all. Instead, I want as few days of the week as possible to appear as the first of the month in a given year.

To accomplish this, I have been granted the authority to change the number of days in each of that year’s 12 months, provided that there are still 365 or 366 days in the year and each month has at least 28 days and at most 31 days.

What are the fewest days of the week that can appear as the first of the month in such a calendar year? (And for fun, rather than for credit: How many such calendars can you design with this property?)

Let $N^*$ be the minimal number of days of the week that can appear as the first of the month of any calendar year. First from the Classic problem we see that the furthest away that two consecutive months can start is 3 days of the week apart, so this means that the smallest that we could possibly hope for is to have a calendar with only three possible days of the week for firsts of the month. That is, $N^* \gt 2,$ or equivalently that $N^* \geq 3.$

Therefore, all we have to do is provide some calendar year where $N=3$ is attained. For instance, take the following setup where March, June, and September have 29 days and all other months have 31 days. This would give $3 \cdot 29 + 9 \cdot 31 = 87 + 279 = 366$ days in the year. Similarly, if January 1 occurs on $x \in \mathcal{D},$ then February 1 occurs on $x+3,$ March 1 occurs on $x+6,$ April 1 occurs on $x+7 \equiv x,$ May 1 occurs on $x+3,$ June 1 occurs on $x+6,$ July 1 occurs on $x+7\equiv x,$ August 1 occurs on $x+3,$ September 1 occurs on $x+6,$ October 1 occurs on $x+7 \equiv x,$ November 1 occurs on $x + 3$ and finally, December 1 occurs on $x+6.$ Putting these altogther gives a subset $\mathcal{N} = \{ x, x + 3, x + 6 \} \subsetneq \mathcal{D},$ with $N= |\mathcal{N}| = 3.$ Therefore, we have that $N^* \leq N = 3$ and hence the fewest days of the week that can appear as the first of the month is $N^*=3.$