This Week's Fiddler: Feb 27, 2026

Posted on Feb 27, 2026

Problem

In 2026, every day of the week appears as the first day of at least one month.

Is 2026 special in this regard? If so, when is the next year where at least one weekday does not appear among month-firsts?

Main Solution

2026 is not special. In fact, this happens in every Gregorian year (common or leap): all seven weekdays appear as firsts of the month.

Here is the general calculation.

Measure each month start as an offset (mod 7) from January 1. For a common year, month lengths mod 7 are 3, 0, 3, 2, 3, 2, 3, 3, 2, 3, 2, so the cumulative month-start offsets are:

0, 3, 3, 6, 1, 4, 6, 2, 5, 0, 3, 5.

For a leap year, only February changes (29 days), giving offsets:

0, 3, 4, 0, 2, 5, 0, 3, 6, 1, 4, 6.

Each list contains all residues 0 through 6, so every weekday appears at least once as a month-first.

Symmetry point: the actual weekday names depend only on the weekday of January 1. Changing January 1 from (say) Thursday to Monday just adds a constant shift mod 7 to the whole list, which permutes weekday labels but does not remove any weekday.

So there is no next Gregorian year where a weekday is missing among month-firsts.

For the concrete 2026 calendar, month-first weekdays are:

• Jan 1: Thursday
• Feb 1: Sunday
• Mar 1: Sunday
• Apr 1: Wednesday
• May 1: Friday
• Jun 1: Monday
• Jul 1: Wednesday
• Aug 1: Saturday
• Sep 1: Tuesday
• Oct 1: Thursday
• Nov 1: Sunday
• Dec 1: Tuesday

All seven weekdays are represented, exactly as the general argument predicts.

Extra Credit

Suppose month lengths can be reassigned, with each month between 28 and 31 days, and total year length still 365 or 366 days.

The fewest possible weekdays that can appear as firsts of the month is 3.

One valid 365-day example calendar is:

Jan 30, Feb 31, Mar 30, Apr 30, May 31, Jun 30, Jul 30, Aug 31, Sep 30, Oct 30, Nov 31, Dec 31.

Total = 365, and each month is in the allowed range 28 to 31.

If Jan 1 is Thursday in this custom calendar, month-first weekdays cycle through only three values. Here is the full year grid:

So only Tuesday, Thursday, and Saturday appear.

For fun: there are 4 such optimal calendars (ordered 12-month length patterns) achieving this minimum.

View the full gallery of all 4 optimal calendars

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