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Re: SYMMETRICAL DISTRIBUTION OF MONTHS OVER 4 YEARS:

by Walter Ziobro :: Rate this Message:

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Thank you, Karl, for your observations.

A thought occurred to me after reading your response. Another way that the 31 day months can be distributed is over a period of 33 tropical years. There would be 173 31-day months, and 223 30-day months, for a total of 12053 days, but I haven't been able to think how they could be distributed symmetrically.

-Walter Ziobro

From: "karl.palmen@..." <karl.palmen@...>
To: petersonwalter@...; CALNDR-L@...
Sent: Wednesday, April 4, 2012 8:03 AM
Subject: RE: SYMMETRICAL DISTRIBUTION OF MONTHS OVER 4 YEARS:

Dear Walter and Calendar People

I show Walter’s suggestion but highlighted in red the even-numbered months of 30 days and the odd-numbered months of 31 days .

Year/
Olympiad:    1   2      3 4

January      30    31    31    31
February    31    30    30    30
March        30    30    31    31
April          31    31    30    30
May          30    30    30    31
June          31    31    31    30
July           30    30    30    30
August      30    31    31    31
September31    30    30    30
October    30    30    31    31
November 31    31    30    30
December 30    30    30    31

The black and red periods are all 7 months except one period of 13 months. So the Olympiad is symmetrical about the middle month of the interval of 13 months which is January of year 1.

So the symmetry like this:

Feb year 1 same length as Dec year 4 with 31 days

Mar year 1 same length as Nov year 4 with 30 days

Apr year 1 same length as Oct year 4 with 31 days

etc.

The 30-day and 31-day months are not spread as uniformly as possible. This would be achieved by changing the pairs of consecutive 30-day months as follows while preserving the symmetry

Jul/Aug year 1 -> May/Jun year 1 (2 months earlier)

Feb/Mar year 2 -> Dec year 1/Jan year 2 (2 months earlier)

Sep/Oct year 2 -> Sep/Oct year 2 (no change)

Apr/May year 3 -> Apr/May year 3 (no change)

Nov/Dec year 3 -> Jan/Feb year 4 (2 months later)

Jun/Jul year 4 -> Aug/Sep year 4 (2 months later)

The black and red periods are then 7 or 9 months spread as uniformly as possible. Indeed, the back periods have 9 months and the red periods have 7 months.

Year/
Olympiad:    1   2      3 4

January      30    30    31    30
February    31    31    30    30
March        30    30    31    31
April          31    31    30    30
May          30    30    30    31
June          30    31    31    30
July           31    30    30    31
August      30    31    31    30
September 31    30    30    30
October    30    30    31    31
November 31    31    30    30
December 30    30    31    31

However I do notice that year 3 rather than year 4 is the 366-day year of the Olympiad. Such a distribution (uniform as possible) cannot be made symmetrical about January of the year after a 366-day year, but can be made symmetrical about December of a 366-day year. I leave as a puzzle for calendar people to work this out. A simple manipulation will lead to the answer.

Spreading the months as uniformly as possible across an entire Gregorian 400-year cycle would be much more complicated. 400 years would have 606 black and red periods (rather than 600).

If one does not care about the months begin distributed as uniformly as possible, one can simplify. Make all even number months have 30 days and all odd-numbered months have 31 days with three exceptions and distribute the exceptions symmetrically. The exceptions can be Jan year 1, July years 2 and 3. Also year 4 has no exception and so has 366 days.

Year/
Olympiad:    1   2      3 4

January      30    31    31    31
February    30    30    30    30
March        31    31    31    31
April          30    30    30    30
May          31    31    31    31
June          30    30    30    30
July           31   30    30    31
August      30    30    30    30
September 31    31    31    31
October    30    30    30    30
November 31    31    31    31
December 30    30    30    30

There is an even simpler scheme that is symmetrical about December in a leap year. I’ll leave it to other calendar people to work it out.

Karl

12(10(13

From: East Carolina University Calendar discussion List [mailto:CALNDR-L@...] On Behalf Of Walter Ziobro
Sent: 04 April 2012 01:19
To: CALNDR-L@...
Subject: SYMMETRICAL DISTRIBUTION OF MONTHS OVER 4 YEARS:

SYMMETRICAL DISTRIBUTION OF MONTHS OVER 4 YEARS:

Here's a novel idea I got for distributing the lengths of the months: instead of fixing the lengths of the months in each year, why not distribute the 31 day months symmetrically over a 4 year period, with the leap month at the end? My Olympiad calendar would look like this:

Year/
Olympiad:    1   2      3 4

January      30    31    31    31
February    31    30    30    30
March        30    30    31    31
April          31    31    30    30
May          30    30    30    31
June          31    31    31    30
July           30    30    30    30
August      30    31    31    31
September31    30    30    30
October    30    30    31    31
November 31    31    30    30
December 30    30    30    31

The basic 7 month pattern is 30-31-30-31-30-31-30, which repeats 6 times, with 30-31-30-31-30-31 at the end.

Walter Ziobro

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