From Newsgroup: rec.puzzles

The following puzzle is taken from "The Penguin Book of Puzzles". The paperback book is an excellent collection of puzzles many of which are familiar from other sources. I don't recall seeing this one before.

ISBN 978-0-718-188627

Under the present calendar rules, the first day of a century can never
fall on a Sunday or a Wednesday or a Friday. Can you explain the mystery
in as simple way as possible?

Note that 1901 was the first day of a century: not 1900.
--
David Entwistle
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From Newsgroup: rec.puzzles

David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> wrote or quoted:

Under the present calendar rules, the first day of a century can never
fall on a Sunday or a Wednesday or a Friday. Can you explain the mystery
in as simple way as possible?

Spoilerspace follows




























































(The simplicity of this explanation lies in the fact that one
gets a simple answer by reading just the following paragraph.
One then can stop reading if one is satisfied with the expla-
nation given or read on to learn more details. These details
will make it less simple, but by choosing how far one is reading,
one can adapt the simplicity to the level of one's choosing.)

The first days of centuries are Monday, Saturday, Thursday,
and Tuesday, and this repeats forevery without any Sunday,
Wednesday or Friday ever occuring.

How do we know that this sequence repeats forever?

In the Gregorian calendar, the first four centuries have the
lengths of 36,524, 36,524, 36,524, and 36,525 days, and then
this sequence of lengths repeats forever. To learn the effect
the passing of a number of days has on the day of the week,
it suffices to just take the remainders of the division by
seven into account, which is 5, 5, 5, and 6 for those numbers:

1601-01-01 was a Monday.

Monday + 5 = Saturday
Saturday + 5 = Thursday
Thursday + 5 = Tuesday
Tuesday + 6 = Monday, again

. So these days of the week keep repeating forever!

How do we know 1601-01-01 was a Monday?

The Gregorian calendar started Friday, 15 October 1582. To get the
number of days passed after that Friday and before 1601-01-01, one
can count the days remaining in 1582 (counting the days remaining
in October and then adding the days of November and December)
and then the days in the full years from 1583 to 1600, taking
leap years into account. Adding the remainder of the division
of "that number plus 1" by seven to Friday then gives Monday.

How do we know that the lengths of the centuries are
36,524, 36,524, 36,524, and 36,525 days?

The final year of a century is a leap year if its number is
divisible by 400. So, other centuries have 36,524 days, but
"leap centuries" have 36,525 days. A non-leap century has
76 standard years, giving 76 * 365 days = 27,740 days, and
24 leap years, giving 24 * 366 days = 8,784 days, which gives
these 36,524 days. A leap century has one day more.

How do we know which years are leap-years?

A year is a leap year if it is evenly divisible by four,
but not by 100. But when it is divisible by 400, as a
higher-ranking rule, it /is/ a leap year.


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From Newsgroup: rec.puzzles

In article <10ve1e2$l8ae$2@dont-email.me>,
David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> wrote:

Under the present calendar rules, the first day of a century can never
fall on a Sunday or a Wednesday or a Friday. Can you explain the mystery
in as simple way as possible?

Before the answer, a better-known oddity: on which day of the week
does the 13th of the month most often fall?

Answer to the original question below.

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Short version:

The calendar repeats every 4 centuries because the leap-year rules
repeat after 4 centuries and 4 centuries happen to contain a whole
number of weeks. So the days on which 4 successive centuries start
are repeated in the next 4, and so on.

Long version:

A year contains 365 or 366 days depending on whether it's a leap year,
which is 52 weeks and 1 or 2 days. A century contains 24 or 25 leap
years depending on whether it includes a year that's a multiple of
400. That's 5200 weeks plus 124 or 125 days, which is 5217 weeks and
5 or 6 days.

So the day of the week of the first day of a century will advance by 6
days when the century includes a year that's a multiple of 400, and 5
days otherwise. After 4 centuries it will have advanced 3*5+6 = 21
days - a multiple of 7 - and will be back where it started.

1/1/1901 was a Tuesday and the following century contained the year
2000, so 1/1/2001 was 6 days later, a Monday. 1/1/2101 will be 5
days later, a Saturday. 1/1/2102 will be 5 days later, a Thursday.
And 1/1/2103 will be 5 days later, a Tuesday again.

So the repeating pattern is Tuesday, Monday, Saturday, Thursday.

-- Richard
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