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Topic: Calendar (Read 7159 times) |
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Benny
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2011 calendar
http://www.thfire.com/wp-content/uploads/2010/12/2011_calendar.jpg
July has 5 Fridays, 5 Saturdays and 5 Sundays.
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How often can this combination occur, and will occur?
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| « Last Edit: May 15th, 2011, 12:59pm by Benny » |
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towr
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Re: Calendar
« Reply #1 on: May 15th, 2011, 11:41am » |
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Assuming the world started on Sunday, October 23, 4004 BC* and ends on Friday December 21, 2012**, then*** the combination occurs 872 times. That is, interpreting the question to include the past. Under the given assumptions it will occur just once more in the future.
It breaks down as follows,
Since 4004 BC starts in October, we can skip that year. So there are 4003 years BC to consider, and since we have to count in blocks of 400 (using the Gregorian calendar, to take into account leap years and non-leap centuries and leap 4-centuries), take the first 1997 years CE as well, this gives 6000 years for 4003-1997. In each 400 year period we have 58 Julies starting on a Friday (which is the necessary and sufficient condition for there to be 5 Fridays, Saturdays and Sundays). So this gives 15*58=870
Then to complete the past years, in 1998-2010 we have 1, and for the coming years (this one included), 2011-2012 gives 1 more. So this adds up to 872.
*)Which it didn't.
**)Which it won't.
***)Actually, starting with false premises, we can deduce anything now.
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| « Last Edit: May 15th, 2011, 11:47am by towr » |
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towr
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Re: Calendar
« Reply #3 on: May 15th, 2011, 1:13pm » |
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Did you skip 2016 for any particular reason?
For the next 400 years:
Python Code:from datetime import *
for i in range(2011, 2411):
if (date(i, 7, 1).weekday() == 4):
print i;
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2011
2016
2022
2033
2039
2044
2050
2061
2067
2072
2078
2089
2095
2101
2107 |
2112
2118
2129
2135
2140
2146
2157
2163
2168
2174
2185
2191
2196
2203
2208 |
2214
2225
2231
2236
2242
2253
2259
2264
2270
2281
2287
2292
2298
2304
2310 |
2321
2327
2332
2338
2349
2355
2360
2366
2377
2383
2388
2394
2405
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| « Last Edit: May 15th, 2011, 1:21pm by towr » |
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Benny
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Re: Calendar
« Reply #4 on: May 15th, 2011, 1:15pm » |
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I've just added 2005 and 2016
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SWF
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Re: Calendar
« Reply #5 on: May 15th, 2011, 7:02pm » |
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If you believe an email of facts somebody sent me, this occurs once every 823 years, and is called "Moneybags".
The 823 years is a bit off, but maybe such months do bring enormous wealth.
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Benny
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Re: Calendar
« Reply #6 on: May 16th, 2011, 4:57pm » |
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Towr,
Thanks for taking the time to provide a computer solution. Is a math solution possible?
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technocian
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Pritam
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Re: Calendar
« Reply #7 on: Dec 13th, 2011, 9:25pm » |
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There is no formula to bring the years for 5, Friday, Sat and Sunday in July. It was last occurred in 2005 and 2011 but next it will come in 2016, 22, 33, 39, 44, 50. You see most of the time there is a difference of 6 years but between 2022 and 2033, there was a difference of 11 years. Even after 2050, there is a difference of 11 years and next it will come in 2061.
Simple you can derive a formula from this.
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Grimbal
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Re: Calendar
« Reply #8 on: Dec 14th, 2011, 1:34am » |
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Normally the pattern repeats every 28 years. But each turn of the century it messes up the cycle.
for this century, you can check
year%28 in {0, 6, 17, 23}
If you add century turns, it becomes
((year/100)*16+year)%28 in {1, 7, 12, 18}
where (year/100) is an integer division
Even that works only 400 years. After that you have to add another adjustment. And after 2800 the adjustment depends on the country.
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towr
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Re: Calendar
« Reply #9 on: Dec 14th, 2011, 8:52am » |
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on Dec 14th, 2011, 1:34am, Grimbal wrote:| And after 2800 the adjustment depends on the country. |
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??
Sounds rather inconvenient to diverge in calender system after it took so long to converge on one.
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Grimbal
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Re: Calendar
« Reply #10 on: Dec 14th, 2011, 3:31pm » |
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Hm... I once heard Russia adopted the Gregorian calender quite late, during the revolution, so they had a better knowledge of the actual length of a year, and they adopted a more accurate rule for leap centuries (2 in 900 years instead of the standard 2 in 800).
But I haven't found references of the fact, except for religious calendars, such as
http://orthodoxwiki.org/Revised_Julian_Calendar
So I am not sure Russia as a country actually adopted that alternate calendar system.
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