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Let $A$ be the set of days in week, $A = \{Mon,Tue,Wed,Thu,Fri,Sat,Sun\}$, and $B = \{1,2,\dots,31\}$. For each date, we have a pair $(x,y)$, $x \in A$ and $y \in B$. Then we will have an infinite sequence of these pairs. For example, starting from today (Friday, April 27, 2012), we have $(Fri,27), (Sat,28), (Sun,29),\dots$ and so on. Now my questions are:

  1. What is the smallest period $T$ after which the sequence repeats? I know that in every 400 years there are 146097 days, which is a multiple of 7, therefore 146097 is a period. But does there exist a smaller period?
  2. During each period $T$, what is (are) the pair (pairs) that appear the most (i.e. highest frequency)? What is the frequency of $(Fri,13)$?

Edit: we are using the Gregorian calendar, with different lengths of months, leap years, etc.

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Are we working on an actual calender, where we have different lengths of months, leap years, etc? Or do you want to know how long it takes to get a repeat if you just consider months of length 31? –  Alex Becker Apr 28 '12 at 1:08
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There is no smaller period than $400$, because of the special rule about leap year in years divisible by $100$. –  André Nicolas Apr 28 '12 at 1:13
    
@AlexBecker: I've just edited the questions to address your comment. –  Truong Apr 28 '12 at 1:14
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This table should answer the second part of the question: http://www.merlyn.demon.co.uk/freq-tbl.txt. –  Doug Chatham Apr 28 '12 at 1:21
    
@DougChatham: why don't you make it an answer and I will accept it? –  Truong Apr 29 '12 at 5:11
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up vote 2 down vote accepted

The table at http://www.merlyn.demon.co.uk/freq-tbl.txt gives the frequencies for each the days of the year over the minimum period, which is 400 years. Several days share the maximum frequency, including Friday the 13th (688/146097).

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