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  1. Find a year that has three Friday the thirteenths in it, either past or future. (That is, find a year where the thirteenth of the month falls on a Friday in three different months.)

  2. Find a year with 'a month of Sundays'. That is, find a year that has a Sunday on every day of a month at least once (i.e. a Sunday the 1st, a Sunday the 2nd, a Sunday the 3rd, ..., a Sunday the 31st).

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    $\begingroup$ Please clarify what you mean with "assemble 'a month of Sundays'". $\endgroup$ Dec 2, 2015 at 11:11
  • $\begingroup$ @TheDarkTruth; is that better? $\endgroup$
    – JMP
    Dec 2, 2015 at 11:18
  • $\begingroup$ Check out timeanddate.com/calendar/weekday-friday-13 $\endgroup$
    – Gamow
    Dec 2, 2015 at 11:50

1 Answer 1

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A solution for both problems is

This year (2015)


First we look for the variances between the days for each month.

Normal year

January    0
February   +3   (starts on the weekday that is 3 days after the weekday of January 1st)
March      +3
April      +6   (starts on the weekday that is 6 days after the weekday of January 1st)
May        +1   etc...
June       +4
July       +6
August     +2
September  +5
October    0
November   +3
December   +5

Leap year

January    0
February   +3
March      +4
April      0
May        +2
June       +5
July       0
August     +3
September  +6
October    +1
November   +4
December   +6

First, we need 3 different months with the same variance.

For a normal year, those are February, March and November.
For a leap year, those are January, April and July.

To find a year with 3 Fridays the 13th, we need the three months mentioned above to have their 13th on a Friday, which means those months start on a Sunday.

For a normal year, this means the year starts on a Thursday (those months have a variance of 3, so if they start on a Sunday, January (and thus the year) begins 3 days earlier, on a Thursday).

For a leap year, this means the year starts on a Sunday (one of the three months is January, and it must start on a Sunday, so that means the year starts on a Sunday).

Our current year (2015) began on a Thursday, therefore 2015 had 3 Fridays the 13th.


As for a 'month of Sundays':

Both a normal year and a leap year have all 7 possible variances in their months, so we only need to find a month with 31 days where the 31st is a Sunday. This will mean the whole year has a 'month of Sundays'.

And surprise, this year's May did indeed have a Sunday the 31st.

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  • $\begingroup$ i can't work your variances out 1jan == 5feb, so var should be +4? $\endgroup$
    – JMP
    Dec 2, 2015 at 12:46
  • $\begingroup$ @ Jon Mark Perry: The variances i state are between weekdays. First I say that the first of january has a variance of 0. Now january has 31 days so the first of february is 31 days later or in other words 4 complete weeks plus 3 days therefore having a variance of +3 weekdays. $\endgroup$ Dec 2, 2015 at 13:06
  • $\begingroup$ interestingly enough you can roll the years on, december means var(jan|[year+1])=1 etc..., so we get a pattern 111211121112..., except for years divisible by 400 $\endgroup$
    – JMP
    Dec 2, 2015 at 13:24
  • $\begingroup$ Actually it was May this year that had Sunday the 31st, not March $\endgroup$
    – orp
    Dec 2, 2015 at 15:25
  • $\begingroup$ @orp My bad. Thanks for the pointer. Fixed it now. $\endgroup$ Dec 2, 2015 at 15:46

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