Now that we’ve just started another
year, we have to confront the complicated
pattern of our calendar. There isn’t
any way to avoid some complications with
the calendar, since our year does not contain
a whole number of days. It also helps to
arrange our lives around a seven day week,
but the year does not contain a whole
number of weeks. Through most of human
history (before electric lights were
available) it was important to keep
track of the phase of the moon because
that determined how much light there
would be at night. Even now a month
is a convenient unit of time for many
purposes, but, no surprise, the year
does not contain a whole number of months.
Here is a way to keep track of the
calendar pattern. Since
365=52*7+1, we have (365 remainder 7)
is 1. This means that for every ordinary
year, the day of the week will advance
by one day from the previous year. However,
after a leap year the day of the week will
advance by two days.
Since leap year happens every 4th year (exception:
see note below), there is a 4*7=28 year cycle
where the calendar repeats the same pattern.
This means the 2017 calendar is the same as it
was 28 years ago in 1989, and it will be the same
again in 2045. If it’s a leap year, you have to
wait 28 years until the calendar will
be exactly the same, but in a non-leap year you
don’t have to wait as long.
If y is the year, here are the steps to determine
the day of the week for January 1 (and you can
determine the rest of the days from that).
Step 1: let k1 = y remainder 28
(if the result is zero, then set k1 equal to 28)
Step 2: let k2 = k1 + (k1-1)/4
(k2 tells you where you are in the leap year cycle;
ignore any remainder in the division)
Step 3: the year starts on day [(k2-1) remainder 7]
where 0= Sunday, 1= Monday, and so on.
For 2017:
k1 = 2017 remainder 28 = 1
k2 = 1 + (1-1)/4 = 1
the year starts on (1-1) remainder 7 = 0 (Sunday)
Here is a table showing how this works for some other years:
Year k1 k2 year start day (Jan 1) 2000 12 14 6 Sat 2001 13 16 1 Mon 2002 14 17 2 Tue 2003 15 18 3 Wed 2004 16 19 4 Thur 2005 17 21 6 Sat 2006 18 22 0 Sun 2007 19 23 1 Mon 2008 20 24 2 Tue 2009 21 26 4 Thur 2010 22 27 5 Fri 2011 23 28 6 Sat 2012 24 29 0 Sun 2013 25 31 2 Tue 2014 26 32 3 Wed 2015 27 33 4 Thur 2016 28 34 5 Fri 2017 1 1 0 Sun 2018 2 2 1 Mon 2019 3 3 2 Tue 2020 4 4 3 Wed 2021 5 6 5 Fri 2022 6 7 6 Sat 2023 7 8 0 Sun 2024 8 9 1 Mon 2025 9 11 3 Wed 2026 10 12 4 Thur 2027 11 13 5 Fri 2028 12 14 6 Sat 2029 13 16 1 Mon 2030 14 17 2 Tue
See also
http://myhome.spu.edu/ddowning/percal.htm
for how this works for other dates.
The other complication is that not every fourth
year is a leap year, according to the Gregorian calendar.
The next time this will be an issue will be the
year 2100, which will not be a leap year.
In that case you would have to make an adjustment,
but the formula given above will work for the years
from 1901 to 2099.
……………..
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