Brain Teaser: Can you figure out why this method works for finding the day of the week for a calendar date?

This is a method devised by the matematician John Conway. There are two parts, one for even numbered months and one for odd numbered months. The part for odd numbered months is not nearly as good as the even month one, so we will just work with the even numbered months.

The idea is to find a way of determining the day of the week for one day in each month and then use that date to figure out any other day in the month. All that you have to know is that the following dates all fall on the same day of the week: last day of February, April 4 (4/4), 6/6, 8/8, 10/10 and 12/12. The associated day for this year is Saturday.

That the last day of February and 4/4 fall on the same day is a happy coincidence, but the matchup for the rest of the days depends on a regularity in the way months are numbered that has only one exception. If not for the exception, there would be no need to remember “30 days have September, April, June and November.” Take a look at a calendar and check the pattern for which months have 30 days and see if you can do some simple arithmetic to show why the indicated dates all fall on the same day of the week.

As a bonus, when I give the answer I will tell a legend which does a great job of explaining the exception in the month numbering, whose only fault is that it is completely untrue, but it does make for a great story.