## Saturday, 5 January 2013

### Patterns observed from Leap year concept

Consider years from 2001 to 2017. Out of these 17 years, the following 4 are leap years: 2004, 2008, 2012, 2016

Let us consider a property “Equidistant” here. If any day of a year is equidistant from the corresponding day of the previous and next years, then that year is called to exhibit the property ‘Equidistant’. But here, before considering a date, it should be taken care of two scenarios. One scenario is for the dates from Jan’1st to Feb’28th and the other scenario is for the dates from Mar’1st to Dec’31st. These two scenarios result in two different patterns. In brief, scenario-I set of dates is (Jan’1st to Feb’28th) and scenario-II set of dates is (Mar’1st to Dec’31st).
For example, let us consider a year 2007. How many days are there from Jan’1st of 2006 to Jan’1st of 2007? It is 365. Reason is simple.  Being a non-leap year, '2006' has 365 days. Similarly, there are 365 days from Jan’1st of 2007 to Jan’1st of 2008 (here again, being a non-leap year '2007' consists of 365 days). So here, any day (taken from scenario-I set of dates) of 2007 is equidistant from the corresponding days of 2006 and 2008. And hence, we can say that year 2007 possesses the property “Equidistant” if scenario-I set of dates are considered.

Pattern-I
Here we consider a date from Jan’1st to Feb’28th and the pattern looks like this:
To have some fun, fill out the tables on your own...
From here onwards, 'E' stands for the property 'Equidistant' and 'N' for 'Non-Equidistant'.
 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 N E E N N E E N N E 2011 2012 2013 2014 2015 2016 2017 E N N E E N N
*Years in bold are leap years

Pattern-II
Here we consider a date from scenario-II set of dates ie., from Mar’1st to Dec’31st and the pattern looks like this:
 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 E E N N E E N N E E 2011 2012 2013 2014 2015 2016 2017 N N E E N N E
*Years in bold are leap years
The point in common for the two patterns is, for every four consecutive years, there exist two E’s and two N’s.

Exceptions:

If it is to be defined mathematically, Leap year is a number which
-must be a multiple of 4 and
-must not be a multiple of 100 but
-can be a multiple of 400
Sequence of Leap years is not in a perfect pattern. For example, If I start writing down the list of leap years for a period starting from 1895, it goes like this:
1896,1904,1908,1912,1916.....
There is an exception for it to be a perfect 'Arithmetic Progression with a common difference of 4'. The break is between 1896 and 1904. Had 1900 been existing between these two, it would have been so.
As there exist exceptions in the leap year sequence (I mean, it’s not a perfect Arithmetic Progression. Every year which is a multiple of 4 is not a leap year. The exception is due to a year which is being a multiple of 100 (but not a multiple of 400) (and obviously it’s a multiple of 4) is not a leap year), there exist exceptions too in our two patterns. There exist breaches in the pattern at any year which is a multiple of 100 (but not a multiple of 400) and also at one of it’s two adjacent years. To explain this I take an example set of the years from 1895 to 1911:

Pattern-I for this set is:
 1895 1896 1897 1898 1899 1900 1901 1902 1903 1904 E N N E E E E E E N 1905 1906 1907 1908 1909 1910 1911 N E E N N E E
*Years in bold are leap years

Pattern-II for this set is:
 1895 1896 1897 1898 1899 1900 1901 1902 1903 1904 N N E E E E E E N N 1905 1906 1907 1908 1909 1910 1911 E E N N E E N
*Red marked ones are exceptions
*Years in bold are leap years
Had 1900 been a leap year, then 1900 and 1901 of Pattern-I would have been N and N respectively and it would have been in a perfect pattern like two N's followed by two E's. Similarly, for Pattern-II, 1899 and 1900 would have been N and N respectively and also it would have been in a perfect pattern.