what year might that have been?

Given an invitation for an event on “Saturday, April 26th,” for instance, might it (if correct, of course) have been from last year?

Without consulting a perpetual calendar¹⁾, here's one solution that uses a sort of back of the envelope perpetual calendar that is easy to construct.

The idea is to figure out what day of the week January 1 had to fall in order for your date to fall on the right day of the week; for dates after February, two weekdays will be possible, one for leap years and one for other years. Then, figure out which years started that way; that is easy to do from the calendar of the current year (or a good memory).

1. calculate how many days came before your date (including January 1).
2. calculate the remainder modulo 7 to get the weekday difference between January 1 and your date.
3. build the table of January 1 weekdays.
4. look up the possibilities.

The serial number for a date can be computed as the sum of month lengths of prior months in the year plus date minus 1.

What this means is that April 26 falls some number of weeks plus either 3 or 4 days after January 1. Thus, if April 26th were a Saturday, January 1 must have been either a Wednesday (non leap year, 3 days before Saturday) or a Tuesday (leap year, 4 days before Saturday).

To compute “modulo 7” just divide by seven and keep the remainder: 115 = (7 * 16) + 3 .

Now we need to know what weekday January 1 fell (or will fall) in each year! Here's a table with the answers, but it is pretty easy to reconstruct if you don't have it at hand.

• The year after a leap year starts two weekdays later than the leap year did,
• otherwise a year starts one weekday later than the previous year did.
• It repeats after 28 years (7 rows).

So starting with Sunday in 2012, we can easily construct this table:

Done!

When has there been Saturday, April 26? When have there been leap years starting on Tuesday or other years starting on Wednesday?

• Answer: 2008, 2003, 1997,
• Also 28, 56, 84 (etc.) years prior: 1980, 1975, 1969, 1952, 1947, 1941 (etc.)