Prove that the probability that Christmas in a random year will fall on a Wednesday is not 1/7.

arithmetic/clock/christmas.day New

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Problem

Prove that the probability that Christmas in a random year will fall
on a Wednesday is not 1/7.

This cute problem is from a 1950's Putnam competition, quoted in Prof.
Z.A. Melzak's (unpublished) notes on history of Mathematics.

-- Alexander Pruss <pruss@math.ubc.ca>

Show solution

The leap-year cycle is periodic with a period of 400 years.  Every 4th
year is a leap year, except that every 100th year is not, except that
every 400th is.

The number of days in 400 years is

        400 * 365 + 100 - 4 + 1 = 146097 = 7 * 20871

so the entire calendar is periodic with a period of 400 years.  I.e.,
the calendar for Month M of year Y is identical to the calendar for
month M of year Y+400.

There are 400 Christmas days in 400 years, but 400 is not evenly
divisible by 7.  QED.

-- Dave Dodson <dodson@convex.com>