Question 696395: Every year there is at least one Friday the thirteenth, but no year has more than 3. On 1998 there are exactly 3: in Feb, Mar, Nov. When will the next year be that contains exactly 3 friday the thirteen's?
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! A leap year (like 2000, 2004, and 2008) has  days,
52 weeks plus  extra weekdays.
All other years (including 1998) have  days,
52 weeks plus  extra weekday.
Each extra weekday causes the days of the week to shift by day.
If the thirteenth of a month was a Friday,
a year later the same date will be a Saturday if no leap day is involved,
but it will be a Sunday if there was a February 29 in between.
The only way to have 3 Friday the thirteenths on a year that is not a leap year is to have the pattern of 1998,
with Feb-13, Mar-13 and Nov-13 being Fridays.
The pattern from 1998 will repeat if there is a shift of a whole number of weeks (7 days, or a multiple of 7 days).
Five years after Feb-13-1998, it will be Feb-13-2003, and having had Feb-29-2000 in between the weekdays will have shifted by 6 days, so it will be Thursday.
A year later, Feb-13-2004 will be Friday,
but with Feb-29-2004 coming up, the 1998 pattern will not repeat.
Leap years have 3 Friday the thirteenths only if January 13 is a Friday.
Four years after Feb-13-2004, it will be Feb-13-2008.
With  years and leap day in between (Feb-29-2004 and Feb-29-2008),
the weekdays will have shifted  days, and it will be Wednesday.
January 13,  days earlier, will have been a Sunday, and 2008 will not have 3 Friday the thirteenths.
Five years after Feb-13-2004, it will be Feb-13-2009.
With two  years and leap days in between (Feb-29-2004 and Feb-29-2008),
the weekdays will have shifted  days, and it will be Friday again.
And because 2009 is not a leap year, the pattern of 1998 will repeat.
After 1998, the next year that contains exactly 3 Friday the thirteens is 
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