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Why 4/4, 6/6, 8/8, 10/10 and 12/12 share a weekday

calendardoomsday-rulemodular-arithmetic

Here is a calendar fact that holds in every year: the dates 4 April, 6 June, 8 August, 10 October and 12 December all fall on the same day of the week.

Prove it. How many days are there between each consecutive pair, and why does that settle it?

(This is the backbone of the "Doomsday rule" for computing weekdays in your head.)

Show a hint

Count the days from 4 April to 6 June using the month lengths: the rest of April, all of May, and six days of June. Then look for a pattern with the other pairs.

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