Solution to Problem #6

The first correct solution was given by Harry J. Smith of Saratoga, CA:

In the Gregorian calendar there are exactly 146097 days every 400 years. [Recall that in the Gregorian calendar there are leap years in years that are divisible by 4 but not 100 and in years divisible by 400.] This is calculated by counting 400 * 356 + 100 - 3. It is nice that this is divisible by 7 because this makes it exactly 20871 weeks. The calendar repeats itself every 400 years.

In 400 years there are 4800 months. Of these, 688 start on a Sunday, 687 start on a Tuesday, 687 start on a Friday, 685 start on a Wednesday, 685 start on a Thursday, 684 start on a Monday, and 684 start on a Saturday.

Therefore a month has a better chance of starting on a Sunday than starting on a Saturday, or any other day.

But the question is only about Januarys. There are 400 Januarys in 400 years. So I had to write a program to tally up the answer:

In 400 years, January 1 is 58 times on a Sunday 56 times on a Monday 58 times on a Tuesday 57 times on a Wednesday 57 times on a Thursday 58 times on a Friday 56 times on a Saturday

January 1 is more likely to fall on a Sunday than a Saturday!

Also solved by Dean G. Huffman.

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