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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
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
This table should answer the second part of the question: – 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
up vote 2 down vote accepted

The table at 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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The link is no longer valid. – Alexander Belopolsky Oct 6 '15 at 20:14

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