20190305, 23:54  #1 
"William"
May 2003
New Haven
23·103 Posts 
Monday to Tuesday
This is really too simple for the puzzles section, but I think others might enjoy it.
My local gym has a monthly offering of "Pizza on the first Monday" and "Coffee and Bagels on the second Tuesday," These were consecutive days in January. So how long is it, on average, from the first Monday to second Tuesday? How about the second Wednesday, Thursday, etc.? 
20190306, 03:34  #2 
Einyen
Dec 2003
Denmark
2·3·13·41 Posts 
1st monday to 2nd tuedays is 8 days except when the month starts with tuesday, then it is 1 day.
In the 28 years cycle when weekdays match the same dates compared to leap years (not counting the year%100 and year%400 issues) there are actually exactly 1/7th of the months that starts with tuesdays. So on average (6/7)*8 + (1/7)*1 = 7 days. Why is it in the 28 years (336 months) cycle the 1st of the months falls on a monday exactly 48 times, tuesday 48 times, ..., and sunday 48 times, even though the length of the months switches in the weird pattern between 28,29,30 and 31 days long ? Last fiddled with by ATH on 20190306 at 14:32 
20190306, 13:17  #3  
Feb 2017
Nowhere
2×3^{3}×5×19 Posts 
This reminds me of the cumbersome phrasing specifying Election Day here in the good ol' USA, which <google google> dates back to 1845, when a law was enacted mandating (my emphasis)
Quote:


20190306, 14:42  #4 
6809 > 6502
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Aug 2003
101×103 Posts
67·151 Posts 

20190306, 15:02  #5  
Feb 2017
Nowhere
5130_{10} Posts 
Quote:
In fairness to lawyers, though, the insistence on what would seem to most people to be an unnecessary degree of specificity, almost certainly means that it actually is necessary. Last fiddled with by Dr Sardonicus on 20190306 at 15:03 Reason: Brevity 

20190307, 22:12  #6 
"William"
May 2003
New Haven
23×103 Posts 
I agree with ATH's calculation of average time from First Monday to Second Tuesday. How much longer is it to Second Wednesday?

20190330, 06:24  #7  
"Jeppe"
Jan 2016
Denmark
10101111_{2} Posts 
Quote:
So you do each of the 7 possible starts of the 1,461day cycle. So any particular date (like the 1st of January, the 13rd of February, or anything) falls equally frequently on any day of week. Let us see what happens in the Gregorian calendar. If we ignore the weekdays at first, the Gregorian calendar repeats after 400 years. 400 years is 146,097 (the 97part is the 97 leap days in 400 years) in that system. Now, you would presuppose that you needed 7 times that period, i.e. 2,800 years, before both leap year and dayofweek pattern repeated, but that is wrong. Because, by accident, 7 divides 146,097 (= 400*365 + 97). In the Gregorian system, 400 years is exactly 20,871 weeks. Because of this coincidence, we do not get every possible date/dayofweek combination in the "expected" frequency. 2001Jan01 was a Monday. 2401Jan01 will be a Monday. 2801Jan01, 3201Jan01, etc. are all Mondays. /JeppeSN 

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