Friday the 13th
When's the next year when there won't be a Friday the 13th?
Answer
Never.
The 13th of each month will fall on different days, according to a complex pattern which takes several years to repeat, because of Leap Years. However, each day of the week comes up as the 13th within the first 9 or 10 months every year.
Assign A, B, C, D, E, F, & G randomly to Sun, Mon ... Sat.
If we assume, in a non leap year, that:
* 13 Jan is Day A, then
* 13 Feb is Day D,
* 13 Mar is Day D,
* 13 Apr is Day G,
* 13 May is Day B,
* 13 Jun is Day E,
* 13 Jly is Day G,
* 13 Aug is Day C,
* 13 Sep is Day F
Thus all seven possible days of the week have been used, within a 9month period.
Similarly, in a leap year:
* 13 Jan is Day A, then
* 13 Feb is Day D,
* 13 Mar is Day E,
* 13 Apr is Day A,
* 13 May is Day C,
* 13 Jun is Day F,
* 13 Jly is Day A,
* 13 Aug is Day D,
* 13 Sep is Day G,
* 13 Oct is Day B,
Thus all seven possible days of the week have been used, within a 10month period.
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Comments
randomgirl
Jan 15, 2012
 I did not get that! 
babyjuice
Jan 15, 2012
 Really, thats the best you could think of to say, I thought it was OK for a first teaser.
Attention everyone else, she is my friend and is really just trying to annoy me. Please post if you like or hate it 
randomgirl
Jan 15, 2012
 This is kind of cool! 
zanner2001
Jan 15, 2012
 u 2 r crazy i like this one its ok 
dalfamnest
Jan 15, 2012
 A good teaser, but it requires a lot more thought than you suggest! The answer is right, but your use of probability is not correct  I have submitted a suggestion for a better explanation.
Keep up the good work! 
Mathgeek007
Jan 16, 2012
 Assume it's not a leap year.
When a month starts on Sunday, it will have a Friday the 13th.
Monday, Thursday, Thursday, Sunday, Tuesday, Saturday, Monday, Tursday, Saturday, Tuesday, Thursday, Sunday.
As you see, no Wednesday.
2 days before Friday.
2 days before Monday, the day I started with. There will be no Friday the 13th.
The next nonleap year that starts on Saturday will not have a Friday the 13th. 
jhosek
Jan 17, 2012
 @Mathgeek007  In a leap year that starts on a Saturday, October will have a Friday the 13th 
jhosek
Jan 17, 2012
 Oops. Sorry i misread your comment "Nonleap year" 
jhosek
Jan 17, 2012
 in a nonleap year that starts on a Saturday, May will have a Friday the 13th 
jhosek
Jan 17, 2012
 When a nonleap year starts on a Monday the first day of each month (JanDec) fall on
Mon,Thurs,, Thurs, Sun, Tues, Fri,
Sun, Wed, Sat, Mon, Thur, Sat
In a leap year starting on a Monday the 1st day of the months fall on
Mon, Thur, Thur, Mon, Wed, Sat
Mon, Thur, Sun, Tues, Fri, Sun
In both cases there is a least 1 month starting on each day of the week. So, if the starting day of the year is shifted there will still be at least one month starting on a Sunday meaning there will be at least one Friday the 13th in any year. 
babyjuice
Jan 17, 2012
 You beat me to post! Also, the year is 2022 
wolfpackw2
Feb 02, 2012
 There are only 14 possible distinctive calendars. One for each day Jan. 1 can start on for a nonleap year (7), and another set for leap year (7). If you look at each of these 14 possibilities, there is ALWAYS a year that has at least 1 month that will start on Sunday (resulting in Friday the 13th). Therefore, there will ALWAYS be a Friday the 13th in every year. It has nothing to do with probability. 
dalfamnest
Feb 03, 2012
 Sorry, guys, every year MUST have a Friday 13th. Mathgeek  you need to check your logic, and others do, too. May 13, 2022 will be a Friday.
The answer is correct, but the reasoning is totally wrong, as this has nothing to do with probability. The theory of probability relies on events being random ... and this is not random. 
RGW4
Apr 05, 2012
 I got it right only because there has always been to be at least one Friday the 13th in my 56+ years of living. Some years more than others. Nice work! 
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