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ekoshapu

Notes to Myself

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Mathematics

Puzzle

Use digits in the place of letters to solve the above equation. Each letter represents different digit.

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Puzzle: High School Algebra

4 Puzzles

Puzzle of The Week

Very interesting puzzle! On 3 occasions I thought I found the answer…and then found a better solution. Hope I’ve finally got it right…

Source: A puzzle group on internet

National Pi Day 2018

Today is National Pi Day (March 14th or 3/14). I had written a blog post on National Pi Day in 2009 which you can read here (the blog post was imported from my old blog and there seems to be some loss of content or distortion of metadata). I joined Twitter on 13th March 2009, just a day before the 2009 National Pi Day when I wrote that blog post. So I was hardly conversant with Twitter and hadn’t checked Pi Day messages. In fact Twitter itself was new at that time, so I doubt if there was much content around Pi Day.

Anyways, since then I have been following National Pi Day messages on Twitter every year and also on other social media.

So this year I thought of posting a collection of few interesting messages related to “Pi”.


Prof.  Arthur Benjamin is a well known Maths professor who has combined Mathematics and Magic through his program Mathemagic. All mathematicians find “Pi” intriguing and they like to explore various aspects of Pi. Prof. Arthur has composed a nice song on Pi which I recently saw on Youtube. Here is the song:

 

Matt Parker is another Math enthusiast who has several interesting videos on Mathametics on Youtube channel Numberphile. As we know Pi is an irrational number. It doesn’t end and goes on and on. As of now researchers have found out up to 2.7 trillion digits of Pi. That is 2,000,000,000,000,000th digit of Pi!

Coming back to Mike Parker…he printed first million digits of Pi and laid down along a mile strip on airport! He also discussed interesting points about Pi. Here is the video:

A comment on this video summarized the exercise pretty well. It said “Completely unnecessary, but absolutely awesome!!


Here are some interesting messages on Pi and Pi Day.



Pi looks as “PIE” in mirror


Humorous take on Pi – the “irrational number”


Today’s Google Doodle and message on Twitter handle of Google India. There was speculation on what the text message means. One possibility could be the next Android version (version P) could be named “Pi”


Unfortunately today’s Pi Day began with the sad news of demise of great theoretical physicist Dr. Stephen Hawking. Interestingly enough Pi Day also happens to be birthday of another great theoretical physicist, Dr Albert Einstein (or as a funny tweet called him, [mc^2-instein] ). And many people on twitter talked about this coincidence.


Then as usual, there were several offers celebrating Pi Day and offering things for $3.14

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Here is a funny dialog between the “imaginary” number “i” and “irrational” number “pi”


And lastly, here is a funny cartoon on Pi being the lonely number 🙂


Humor for #PiDay Pi Day.

Question: “What did the palindrome and math lover say when she was offered cake?” Answer: “I prefer pi.”


Indian genius mathematician Srinivas Ramanujan had special affinity for Ramanujan. Lot of his equations involved “Pi” in very elegant and artistic manner. Here are a couple of examples:

Pi_Ramanujan1Pi_Ramanujan


I have few more resources on magnificent “Pi”, but will save it for the next year…till then Happy Pi Day!

 

7 Geometry Puzzles

Pink shaded region in above square is what % of area of the square?

Hint: You do not have to know the side of the square to solve this.

Three Puzzles on “2018”

Here are three puzzles related to “2018”. Interesting ones with varying difficulty. Do try and post answers

Mathematically Happy New Year 2018

This New Year crosses 2 years that are the sum of 2 squares: 2017=9²+44², 2018=13²+43².

There was a triple of such years in 1960-1962 but there won’t be again until 2248-2250.

From Twitter

On Calendar

Calendar

Year end is about holidays and New Year wishes. But it is also about Calendars. I don’t know about others but I like to “read” new calendar, immediately check few dates or events, festivals!

I remember having written a couple of blog posts about Calendar. One, written in 2007, was a light take on Calendar – Calendar विषयी थोडेसे… (It’s in Marathi)

The other was written in 2010 about the mathematics behind the Calendar, or how to calculate day of any date. I was not able to find it in Archive – probably it got lost when I imported my old blog into this domain. So posting it here again. It was written on 13th February 2010:


Few years ago I was learning a programming language and was given an assignment to display calendar and tell day of date .

Instead of using library function I wrote the logic to come up with the day of that date. I had worked it out long back when I was in school – when I had read about Shakuntala Devi. I was fascinated by how to do mathematical calculation in head so quickly, or tell day of any past or future date. For mathematical calculations I read some books on Vedic Maths and learnt few things. As for date, I worked it out purely on my own. Don’t know if that is how those people do it or is there some other/better way. But this method works fine. And its very easy to do, once you understand how calendar works.
Let me explain how.
First few basics about Calendar:
  1. There are 365 days in a normal year and 366 in a Leap year.
  2. If the year is divisible by 4 (but not by 100) then the year is a Leap year e.g. 1988, 1876, 2024 etc.
  3. If the year is divisible by 100 and also by 400 then the year is a Leap year. e.g. 2000, 1600, 2400.
  4. If the year is divisible by 100 but not by 400, then the year is NOT a Leap year. e.g. 1800, 1900, 1700, 2100 etc.
Now with these basics we can go ahead to discuss the logic:
Let’s understand the concept of ‘odd day’.
The weekdays are Sunday through Saturday. And then the cycle repeats continuously. So if there are exactly 4 weeks then thee is no ‘odd day’ i.e. extra day. In a 30-day month there are 4 complete weeks and 2 ‘odd days’ (i.e. extra days). In a 31-day month, there are In a leap month (February of a Leap year) there is one odd day.
So lets first see how many odd days are there in each month. (All you have to do is take modulus of 7 for each month…)
  • January – 3
  • February – 0
  • February in a Leap year – 1
  • March – 3
  • April – 2
  • May – 3
  • June – 2
  • July – 3
  • August – 3
  • September – 2
  • October – 3
  • November – 2
  • December – 3
In a normal year there are 365 days i.e. 52 weeks and 1 odd day. In case of a Leap year there are 2 odd days.
So now we are good to find day of any date. But before that, to make life easy we need a a reference day (it’s not mandatory but beneficial).
1st January 1900 was Monday
1st January 2000 was Saturday
Note: For any day in 20th and 21st century these 2 reference days respectively are enough)
——————————————————————
Lets work out some examples:
Example 1. 15th Aug 1947 (India’s Independence Day)
Reference Date: 1st January 1900 was Monday i.e. Day ZERO
Step 1: 46 completed years since 1900. i.e. 46 odd days (1 odd day per year, not counting leap years). This is further equivalent to 7 weeks and 4 odd days
Step 2: There were 11 Leap years between 1900 and 1946 (i.e. 1946-1900 MOD 7). So 11 more odd days. i.e 1 week and 4 odd days
Step 3: Odd days for Year 1947 (till July 1947) are 16 (3+0+3+2+3+2+3) i.e. 2 weeks and 2 odd days
Step 4: As on Aug 15, there was 1 odd day
 
Adding odd day for Steps 1 through 4 we get 4+4+2+1 = 11 odd days i.e. further equal to 4 odd days.
 
Since reference day 1st January 1900 was Monday, counting 4 days from that day we get Friday.
So 15th August 1947 was Friday!
————————————————————————————————
Example 2. 11th September 2001 (World Trade Center Attack)
Reference: 1st January 2000 was Saturday
Step 1: 0 completed year since 2000. So 0 odd days
Step 2: 1 Leap year since 2000. So 1 odd days
Step 3: Odd days till August 2001 (3+0+3+2+3+2+3+3) are 19 i.e. 2 weeks and 5 odd days
Step 4: As on September 11, there were 4 odd days
Adding odd day for Steps 1 through 4 we get 0+1+5+4 = 10 i.e. 3 odd days.
Since reference day 1st January 2000 was Saturday, counting 3 days from that day we get Tuesday.
So 11th September 2001 was Tuesday!
————————————————————————————————
Example 3. 26th November 2008 (Mumbai Attack)
Reference: 1st January 2000 was Saturday
Step 1: 7 completed years since 2000. i.e. 7 days. So 0 odd days
Step 2: 2 Leap years since 2000. So 2 odd days
Step 3: Odd days till October 2001 (3+1+3+2+3+2+3+3+2+3) are 25 i.e. 3 weeks and 4 odd days
Step 4: As on November 26, there were 5 odd days
Adding odd day for Steps 1 through 4 we get 0+2+4+5 = 11 i.e. 4 odd days.
Since reference day 1st January 2000 was Saturday, counting 4 days from that day we get Wednesday.
So 11th September 2001 was Wednesday!
————————————————————————————————
Example 4: 25th June 1983 (India’s World Cup Cricket Win)
Reference Date: 1st January 1900 was Monday i.e. Day ZERO
Step 1: 82 completed years since 1900. i.e. 82 odd days –> 5 odd days
Step 2: There were 20 Leap years between 1900 and 1982. So 20 more odd days. i.e 2 week and 6 odd days
Step 3: Odd days for Year 1983 (till May 1983) are 11 (3+0+3+2+3) i.e. 4 odd days
Step 4: As on June 25, there were 4 odd days
 
Adding odd day for Steps 1 through 4 we get 5+6+4+4 = 19 odd days i.e. further equal to 5 odd days.
 
Since reference day 1st January 1900 was Monday, counting 5 days from that day we get Saturday.
So 15th August 1947 was Saturday!
————————————————————————————————
Example 5: Today i.e. 13th February 2010
Reference: 1st January 2000 was Saturday
Step 1: 10 completed years since 2000. i.e. 10 days. So 3 odd days
Step 2: 3 Leap years since 2000. So 3 odd days
Step 3: Odd days till January 2010 are 3 i.e. 3 odd days
Step 4: As on February 13, there were 6 odd days
Adding odd day for Steps 1 through 4 we get 3+3+3+6 = 15 i.e. 1 odd day.
Since reference day 1st January 2000 was Saturday, counting 1 day from that day we get Sunday.
Yes, today is Sunday! 🙂

​​

Let’s update this for today’s date i.e. 27th December 2017.

Reference: 1st January 2000 was Saturday
Step 1: 16 completed years since 2000. i.e. 16 days. So 2 odd days
Step 2: 5 Leap years since 2000. So 5 odd days
Step 3: Odd days till end of November 2017 are 26 i.e. 5 odd days (26 mod 7)
Step 4: As on December 27, there were 6 odd days (27 mod 7)
Adding odd day for Steps 1 through 4 we get 2+5+5+6 = 18 i.e. 4 odd days.
Since reference day 1st January 2000 was Saturday, counting 0th day from that day we get Wednesday.
Yes, today is Wednesday! 🙂

​​

The reason I remembered this blog post was this very interesting video: A Tale of Two Calendars
I will write about this in subsequent blog. Meanwhile, do watch this video. Also do some exercises about finding the day given a date…hope you find it entertaining and useful!

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