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A year normally has at most three Fridays the 13th. Year 2015 has three such days. What is the next year that has three Fridays the 13th again for the first time?

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If you look at the property of the 13th day of each month, you will see that only Feb, March and November are the months that can have the same weekday occurring on the 13th of those months, and that too, only in a non-leap year. All other 13th's in the rest of the non-leap year (and all 13th's in leap years) represent weekdays whose frequency within that year (of occurring on 13th) is atmost 2.

This is apparent in the below tables, where the last column represents the difference between the weekday occurring on 13th of the 'n'th month vis-a-vis 13th Jan, and the same difference occurring for multiple values of 'n' indicates that the 13th's of those months fall on the same weekday -

  1. Cumulative difference between 13th of any month and 13th January in a NON-LEAP year.

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  1. Cumulative difference between 13th of any month and 13th January in a LEAP year.

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Once, this is clear, the problem reduces to finding the next NON-LEAP year when 13th Feb falls on a Friday. Rest should be easy.

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A year with $365$ days may have three Fridays the $13$th only on February, March and November.

A year with $366$ days cannot have three Fridays the $13$th, hence the next year with three Fridays the $13$th is $\color{red}{2026}$.

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  • $\begingroup$ Hi, can you provide the line of thought that could be used to deduce these two statements easily? Thanks. $\endgroup$ Sep 21, 2015 at 15:40

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