Days, Dates, Years - Nostalgia and Number Crunching!

Has it ever happened to you that on a particular date you had wondered that the date had fallen on the same day few years back? Your memory ran down the lane and you were gripped by nostalgia! Well I love numbers and such calculations and obviously I love nostalgia so a way to find out the repeat of a date & day combination led me to a research long time back with the findings diligently noted down with pen on paper. This time it suddenly struck me that a particular date that I remember and which is relevant for me in a sense will be falling on the same day as when it gathered relevance. That was a cue for me to repeat the research work, this time on a digital device (I mean my lappy) and find some more interesting things.

So, just for the sake of fun and nostalgia regarding the birthdays, anniversaries, first meeting, last meeting and all that, here we go:

Following is the list of calendars for the month of February from the year 2015 to the year 1982. The February calendar of each year has been mapped to the next set of calendar horizontally in such a manner that the calendars are same (except the part where 29^th February appears). The leap years have been highlighted with yellow colour.

Now, following is the list of calendars for the month of March from the year 2015 to the year 1982. The arrangement of calendars is same as the previous one.

As we can observe, the series repeats itself after every 28 years. Look closely at the first and last column of both the images to confirm the observation.

Having 365 days per year in a non-leap year with 7 days a week leaves us with 52 weeks and 1 day. This spare 1 day normally shifts the day of a particular date of a particular year to the next day in the next year. For leap years the shift is 2 days as the total number of days increases to 366 however the impact is felt differently by different set of months.

Let me elaborate.

Now the reason that in the above images, we have the same range of years but different months is because of the fact that due to the presence of 29^th February in leap years, the shift in day of a date from the previous year increases by 1 after the 29^th February. For example, 1^st of March, 2012 falls 2 days later i.e. on Thursday than the previous year (1^st of March, 2011 was on Tuesday) instead of 1 day. As evident, the shift for January to February occurs in the year next to the leap year. For example, 1^st of February, 2013 falls 2 days later i.e. on Friday than the previous year (1^st of February, 2012 was on Wednesday) however 1^st of March, 2013 falls 1 day later i.e. Friday.

So, the behavior of dates falling on the same day of some year from 1^st January to 28^th February (not counting 29^th for the leap year gap) can be different from the behavior of dates falling on the same day of some year from 1^st March to 31^st December.

Let’s summarize the behavior for 1^st January to 28^th February from the above images taking the years 2016 to 2000 as the median and 28 years in the future and 28 years in the past. The table in extreme right indicates the difference of any year of any column from its previous column. Hence for the first 5 columns we have 4 differences falling into a pattern which repeats itself in the next 5 columns. Moreover, there exists a pattern vertically as well which repeats itself after every 4 rows.

Now if we summarize the behavior for 1^st March to 31^st December in the same manner, we will get the following:

As we can observe, the pattern for the same set of years is different for the different set of months. However, on a closer look we will be able to find that there are certain sets of year that match for both the set of months. This means the calendar of a particular year would completely match with that of another year and there won’t be any separate calculation for the set of 1^st January – 28^th February and 1^st March – 31^st December.

Let’s see which ones are those with respect to the previous images.

Again there is a pattern to it. The table in extreme right indicates just that. The most interesting observation that can be made from the above image is that leap years seem to repeat the complete calendar after every 28 years which also means that 29^th of February would fall on the same day of a leap year after 28 years. Fascinating!

Now that we have the patterns with us, we can take any of the sets as reference point and start matching the days, dates and years with ease. Let us take few examples before we end this game of numbers!

1^st of January, 2040 would fall on the same day as 1^st of January, 2012 which was Sunday. However, we need not wait that long because 1^st of January, 2017 will also be falling on Sunday.

After 25^th of December, 2015 we will have Christmas falling on a Friday in the year 2020!

If one survives for the next 3 decades then one will be able to see the calendar of 2016 again in the year 2044.

Happy number crunching.

Enjoy. J

P.S. If you can deduce something more from these, do let us know.