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.)
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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1$\begingroup$ Please clarify what you mean with "assemble 'a month of Sundays'". $\endgroup$– The Dark TruthCommented Dec 2, 2015 at 11:11
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$\begingroup$ @TheDarkTruth; is that better? $\endgroup$– JMPCommented Dec 2, 2015 at 11:18
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$\begingroup$ Check out timeanddate.com/calendar/weekday-friday-13 $\endgroup$– GamowCommented Dec 2, 2015 at 11:50
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1 Answer
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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.
answered Dec 2, 2015 at 12:17
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$\begingroup$ i can't work your variances out 1jan == 5feb, so var should be +4? $\endgroup$– JMPCommented Dec 2, 2015 at 12:46
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$\begingroup$ @ 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. $\endgroup$ Commented Dec 2, 2015 at 13:06
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$\begingroup$ 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 $\endgroup$– JMPCommented Dec 2, 2015 at 13:24
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$\begingroup$ Actually it was May this year that had Sunday the 31st, not March $\endgroup$– orpCommented Dec 2, 2015 at 15:25
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$\begingroup$ @orp My bad. Thanks for the pointer. Fixed it now. $\endgroup$ Commented Dec 2, 2015 at 15:46