Why the years 100, 200 and 300 are not leap years?
400th year is a leap year 		How to find the day of any date without using pen and paper?
Example question:
After 1*365 + 1/4 days earth completes 1 round
After 2*365 + 2/4 days earth completes next round
After 3*365 + 3/4 days earth completes next round
After 4*365 + 4/4 which is 1 day earth completes next round
Hence 4th year is a leap year
Hence 8th year is a leap year
..
Hence 96th year is a leap year
Hence 97th year is not a leap year
Hence 98th year is not a leap year
Hence 99th year is not a leap year
But why 100th year is not a leap year?
We can view time taken by earth rotation related comment using wikipedia:
https://en.wikipedia.org/wiki/Gregorian_calendar#:~:text=The%20Gregorian%20calendar%2C%20like%20the,as%20for%20the%20Julian%20calendar.
Our earth taking 1 year to complete one year around our sun.
This is taking APPROXIMATELY 365 days, 05 hours, 49 minutes and 12 seconds.
Once completing 01 year we cannot add one day for extra 05 hours... rotation of our earth to complete one year.
96 years having 96/4 leap years = (100/4)-1 = 24 leap years.
Hence from 1st year to 96th year 0.03*24 days added extra at our calendar = 0.72 days added extra at our calendar.
100th, 200th, 300th year not taken as leap year and the earth rotation during these years(0.2425*4 day each time needs to be deducted from our calendar exception, since 97,98,99,100 not taken as leap year.)
Like the same 101th, 102th, 103th ... 396th years each year earth taking related 0.2425 day calculations.
During first 01-100 years 0.72 days added extra at our calendar and 0.2425*4 day not added
During second 101-200 years 0.72 days added extra at our calendar and 0.2425*4 day not added
During third 201-300 years 0.72 days added extra at our calendar and 0.2425*4 day not added
Assume that we are taking 400th year as not a leap year.
During fourth 301-400 years 0.72 days added extra at our calendar and 0.2425*4 days not added
During 400th year we have added 0.72 days four times and not added 0.2425*4 days 4 times.
=we have added 0.72*4 days and not added (0.2425*4)*4 days.
=we have added 0.72*4 days and not added (0.97)*4 days.
=we have added 2.88 days and not added 3.88 day.
=we have added 2.88 - 3.88 day.
=we have added -01 day.
Due to this -01 day, we are making 2000 as a leap year.
This is the reason why 400/800/1200/1600/2000/2400/... if divisible by 400 it is a leap year and
100/200/300 if not divisibile by 400 is not a leap year.
EARTH ROTATION AROUND THE SUN EXTRA TIMIE TO COMPLETE ONE YEAR SUME OF HOURS/...SECONDS WE MISSED ADDING AT CALENDAR
365 days 05 hours 49 minutes 12 seconds
= 05 hours 49 minutes 12/60 minute
= 05 hours 49 minutes 0.20 minute
= 05 hours 49.20 minute
= 05 hours 49.20/60 hour
= 05 hours 0.82 hour
= 05.82 hours
= 05.82/24 day
= 0.2425 day 		We missed to add 0.2425 day at our calendar once earth is completing FIRST YEAR
365 days 0.2425 day We missed to add 0.2425*2 day at our calendar once earth is completing SECOND YEAR
365 days 0.2425 day We missed to add 0.2425*3 day at our calendar once earth is completing THIRD YEAR
365 days 0.2425 day Once earth is completing four years, we add one day during Feb.
We need to add 0.2425*4 during fourth year.
0.2425*4 = 0.97 day Hence we have added 0.03 day extra during each leap year.
Let us re calculate without assumption.
From 01 to 100 years
24 leap years are there ( from 01 to 96 years)
For these leap years, 0.03 days were added extra for 24 times.
So 24*0.03 were added extra.
Hence 0.72 days were added extra.
However for years 97,98,99 and 100, need to add 0.97 days.
So (0.97 - 0.72) day needs to be added during the year 100.
which is (0.25) day needs to be added during the year 100.
Like the same, 0.25 day needs to be added during the year 200
for leap years from 101 to 200.
Since old addition 0.25 was not done during 100, adding that for the year 200.
So 0.50 day needs to be added during the year 200.
Like the same, 0.25 day needs to be added during the year 300
for leap years from 201 to 300.
Since old addition 0.50 was not done during 200, adding that for the year 300.
So 0.75 day needs to be added during the year 300.
For years from 301 to 396, 24 leap years are there.
So, 24*0.03 days were added extra.
which is 0.72 days were added extra.
Hence deducting 0.72 from 0.75
deducting from 0.75 since 0.75 was not added for the year 300.
That is equal to 0.75 - 0.72
which is equal to 0.03
This 0.03 day needs to be added during the year 400.
For following years need to add 0.2425 days
97 0.2425
98 0.2425
99 0.2425
400 0.2425 and 0.03
Here adding 0.03 due to exception for the years from 01 to 396
Based on that, for the year 400, need to add the following: 0.2425 + 0.2425 + 0.2425 + 0.2425 + 0.03
This is equal to 1 day.
Hence making 400 as a leap year.
That's why we celebrated year 2000 as a leap year.
Hence next round off of leap year exception will be following year:
2400
All Gregorian calendar description here are based on the assumption => 1 year is equal to 365 days + 97/400       days
1 year is equal to 365 days + 0.2425       days
1 year is equal to 365 days + 0.2425*24 hours
1 year is equal to 365 days + 5.82           hours
1 year is equal to 365 days + 5                hours and 0.82       hours
1 year is equal to 365 days + 5                hours and 0.82*60 minutes
1 year is equal to 365 days + 5                hours and 49.2       minutes
1 year is equal to 365 days + 5                hours and 49          minutes and 0.2       minutes
1 year is equal to 365 days + 5                hours and 49          minutes and 0.2*60 seconds
1 year is equal to 365 days + 5                hours and 49          minutes and 12        seconds
1 year is equal to 365 days + 5 hours 49 minutes 12 seconds.
Also nano seconds taken by earth(if any) to complete one rotation around the sun are not taken into consideration.
 Hence I feel this inclusion might be required during the year 2400.
Used the word might since written taken by earth(if any).
All existing operating systems can be advanced to include the exceptions for handling time exceptions:
1. from birth and even after death,
2. shared market date/timings
3. fixed deposit
4. mutual fund...
5. national savings certificate
6. all types of insurance
7. all types of loans
Selected Programming Languages which is referring programming languages of all operating systems(AIX/HP-UX/Linux(all Linux or UNIX oriented operating system)/SunOS/Windows)
Operating system names displayed here are sorted names.

murugesanoepnssl@CYGWIN_NT-6.1-WOW OS CYGWIN_NT REL 2.8.0 MODEL i686
murugesanoepnssl@TechnicalEnterpriseSolutions /home/murugesanoepnssl [ 0 ]
$ sample_bday.sh
Assume that I born on: Mon 01-Mar-1999 10:00:00 AM
My second birthday = 01-Mar-1999 10:00 AM + 365 days + 5 hours 49 minutes 12 seconds minus 1 day(29-Feb-2000)
Hence my second birthday on Tue 29-Feb-2000 03:49:12 PM
My third birthday = 01-Mar-1999 10:00 AM + 365+365 days + 10 hours 98 minutes 24 seconds minus 1 day(29-Feb-2000)
Hence My third birthday on Wed 28-Feb-2001 09:38:24 PM
$ echo "AIX/HP-UX/Linux(all Linux or UNIX oriented operating system)/SunOS/Windows" | /usr/bin/tr "/" "\n" | /bin/sort -u
AIX
HP-UX
Linux(all Linux or UNIX oriented operating system)
SunOS
Windows
murugesanoepnssl@TechnicalEnterpriseSolutions /home/murugesanoepnssl [ 0 ]
$
Sample birthday:
AGEBIRTHDAY TIME DURING EACH YEAR
00Mon 01-Mar-1999 10:00:00 AM
01Tue 29-Feb-2000 03:49:12 PM
02Wed 28-Feb-2001 09:38:24 PM
03Fri 01-Mar-2002 03:27:36 AM
04Sat 01-Mar-2003 09:16:48 AM
05Sun 29-Feb-2004 03:06:00 PM
06Mon 28-Feb-2005 08:55:12 PM
07Wed 01-Mar-2006 02:44:24 AM
08Thu 01-Mar-2007 08:33:36 AM
09Fri 29-Feb-2008 02:22:48 PM
10Sat 28-Feb-2009 08:12:00 PM
11Mon 01-Mar-2010 02:01:12 AM
12Tue 01-Mar-2011 07:50:24 AM
13Wed 29-Feb-2012 01:39:36 PM
14Thu 28-Feb-2013 07:28:48 PM
15Sat 01-Mar-2014 01:18:00 AM
16Sun 01-Mar-2015 07:07:12 AM
17Mon 29-Feb-2016 12:56:24 PM
18Tue 28-Feb-2017 06:45:36 PM
19Thu 01-Mar-2018 12:34:48 AM
20Fri 01-Mar-2019 06:24:00 AM
21Sat 29-Feb-2020 12:13:12 PM
22Sun 28-Feb-2021 06:02:24 PM
23Mon 28-Feb-2022 11:51:36 PM
24Wed 01-Mar-2023 05:40:48 AM
25Thu 29-Feb-2024 11:30:00 AM
How to find the day of any date without using pen and paper?
I am taking 15-Aug-1947 as an example to find the day?
year = 1947
Subtract 1600 from 1947 (calendar repeated again 400 years once).
year = 1947 - 1600
year = 347
Deduct 1 since I am taking months for last year.
year = year - 1
year = 347 - 1
year = 346
Remember that each year increases by one day.
For 100 years 24 leap years are there => 100day extra + 24 days extra = 124 days extra => 124 modulus 7 (Since each week having 7 days)
         => 124%7(divide by seven and find the remainder) => Deduct 12 by seven giving 5. Hence taking 54 which when divided by 7 remainder is 5.
Hence for 300 years 5*3 days extra => 15 days extra => 15%7 days extra => 1 day extra.
Here after we can deduct 300 and add 1 if year less than 2000
Hence
year = 346 => 300 + 46
year = 346 => (1 day extra) + 46
year = (1 day extra) + 46 + (all leap years during 46 years)
year = (1 day extra) + 46 + (46/4)
year = 1 + 46 + (46/4)
year = 1 + 46 + 11
year = 58
year = 58%7 (since each week having 7 days)
year = 2 (Keep 2 in mind)
Fold your fingers to have a format of finger bottom like /\_/\_/\_/\
When the month at bottom of the finger is up 3 days extra(31 days % 7)
When the month at bottom of the finger is down 2 days extra(30 days % 7)
During february if the year is leap year, bottom of Feb is 1 day(29%7) else 0(28%7)
Hence I am adding using my fingers first(upward) locations => Jan+Mar+May+Jul => 3 + 3 + 3 + 3 => 12%7 => 5
followed by adding using my fingers first(downwards) locations => Feb+Apr+Jun+Aug => 0 + 2 + 2 + 15 => 19%7 => 5
At last I am adding 2(my old statement was: keep 2 in mind)+5+5 = 12
=> 12 % 7 => 5
Remainder 1 => Mon
Remainder 2 => Tue
Remainder 3 => Wed
Remainder 4 => Thu
Remainder 5 => Fri
Remainder 6 => Sat
Remainder 0 => Sun
Hence 15-Aug-1947 is Fri
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