Sunday, 29 September 2013

A Day for every Date- beat the calendar!

            One of the most impressive feats of mental calculation: calculate the day of any random date in your head. While it may sound extremely tough, if you are friendly enough with numbers, we reckon you can calculate days within 10 seconds! YES! All you need to do is remember some numbers and tricks and you will be ready.

Each day of the week is assigned a number-
Monday-1
Tuesday-2
Wednesday-3
Thursday-4
Friday-5
Saturday-6
Sunday-0/7

This list is pretty easy to remember and follows a standard chronological order. NOTE that the numbers are 'wrapped' around the number 7; this modulo behavior is extremely important for this trick to work.

Next, we give each month a number. Now, this list doesnt follow a logical order and we recommend that you develop your own method to remember this list-

January-6 (5 for leap years)
February-2 (1 for leap years)
March-2
April-5
May-0
June-3
July-5
August-1
September-4
October-6
November-2
December-4

            Once you have learnt these month codes, you can progress to the all important year codes. Just like months, every year has its own code but remembering year codes is an ordeal in itself and nearly impossible (unless you have an eidetic memory!). Fortunately, there is an efficient way to calculate year codes mentally. We shall demonstrate the method below-
We want to figure out the year code for 2005. Now, 2005 lies in the 21st century and ends with the number 5. We divide 5 with 4 and calculate the quotient (ignore the remainder), which in this case is 1. Then we add the obtained quotient to the last digit. 5+1 = 6. 6 is the year code for 2005. 
Lets find the year code for 2014. 2014 ends with 14. When we divide 14 with 4, the quotient is 3. Adding 3 and 14 we get 17. Next, we find which multiple of 7 that is smaller than 17 is closest to 17. In this case the multiple is 14. Subtract 14 from 17 to get 3; the year code for 2014

            The above method works for only 21st century years, for 20th century years, we have to add 1 to the final code.
1996- 1996 lies in the 20th century and ends with 96. The quotient when we divide 96 and 4 is 24. Add 24 and 96 to obtain 120. The closest multiple of 7 smaller than 120 is 119. 120-199=1. 1 is not the year code. We need to add 1 because 1996 lies in the 20th century. 1+1+2. So, the year code for 1996 is 2, and not 1.

Just like for 20th century years, we add 1; for 19th century years we add 3 and for 18th century years we add 5. The good thing about our calendars is that years are repeated after 400 years, so 18th century and 22nd century calendars are the same.

NOTE- Year codes will always lie between 0 and 6.

            Once you have mastered the trick for calculating year codes, you can proceed to the final step, calculating the day! We shall work out two examples-

1) 2nd March 1997:
   The month code is 2
   Calculate the year code- you will obtain 3
   The date- 2
Now add the three numbers, you will get 7. Now, 7 is our day code which represents Sunday. So, 2nd March 1997 was a Sunday

2) 22nd July 2222:
   The month code is 5
   Calculate the year code- you will obtain 2 (simply obtain the year code for 1822)
   The date-22
Now add the three numbers, you will get 29. Now, 29 is greater than 7 and needs to be reduced. We will now subtract the multiple of 7 closest to 29 but smaller than it from 29, which in this case is 28. 29-28=1. 1 is the day code for Monday. So, 22nd July 2222 will be a Monday.


We hope that you have now gained a clear idea of how to calculate days. If you need any assistance feel free to contact us via our blog.

Monday, 2 September 2013

The 3 Kid Problem

The main problem discussed in The Mathematics Symposium-2 was 'The 3 kid problem', the video for which can be found in the link below:

The 3 Kid Problem

Before you guys start scratching your heads and delve into this problem, there are some things that we want you to take care of, things that will come in your mind while watching the video:

1) The number mentioned on the 'board' is not shown to you for a reason, the main trick of this problem lies in decoding the 'mysterious' number
2) We mention in the end that the eldest kid plays soccer, please don't make any relations between the sport and the child's age. For all we care, he might have been a badminton player; the sport has NO relation with the child's age.

We will soon be posting the solution video!!

Friday, 26 July 2013

THE MATHEMATICS SYMPOSIUM - II 2013

Poster for the Symposium
After organising a successful Mathematics Symposium I 2012, Varun and Yuvraj, two Students of Class 12, this time along with five of their friends organised The Mathematics Symposium II 2013. The event was Bigger and Better, as it was evident by the attendance. While the first event was attended by about 35 students from classes 9th to 10th, the second one witnessed an attendance of an overwhelming 100+ students, with active participation by all the students.
The organisers (all in white, from left to right): Harkirat, Akhilesh, Sidhant, Kartike, Varun, Shreyjeet and Yuvraj


Students discussing in groups
Organisers looking at students discussing
The students were first shown an introductory video and then given a 5-Question quiz which consisted of interesting Mathematics problems. The students were segregated into groups in which they had discussions about the problems. After the discussion within the groups, Sidhant, Varun and Yuvraj explained the problems to the students. Harkirat Singh Randhawa and Akhilesh Sharda were a part of the Technical team and helped in organising the event. Kartike Sachdeva was the head of the Technical team. Shreyjeet Singh Ahuja solely looked after the Creative Department.
 



 Mrs. Payal Makhija, teacher of class 9 and 10, was very helpful in making this event the success it was and she was the only Maths Teacher from the school to attend the event (unlike the first one, which was attended by 4 Maths teachers) as the organisers were able enough to manage the students and did not need any help from the teachers. 


Mrs. Payal Makhija (extreme left)




 After the quiz was done with. There was a discussion about the Nested Radicals and a Mathematics Problem named “The 3-Kids problem”, which will be put up on this blog very soon.
A student explaining a problem

Students keenly taking part in discussion

Following was the quiz prepared by the organisers:

Q1. After losing her basket of eggs, a mathematician files a complaint with the police. The policeman asks her for the number of eggs she had. She decides to give him a riddle and says that if she took three at a time out of it, she was left with 2, if she took five at a time, she was left with 3, and if she took seven at a time, she was left with 2. She then asks the policeman what is the minimum number of eggs she might have had?

Q2. In a 43*43 grid. what is the total number of squares in it?

Q3. Derive a relation between terms of Fibonacci and Bumped Fibonacci series. And hence give a formula to calculate the 100th term of the Bumped Fibonacci series.  Bumped Fibonacci problem

Q4. A box contains 100 balls of different colours: 28 red, 17 blue, 21 green, 10 black, 12 white and 12 yellow
balls. What is the smallest number n such that any n balls drawn from the box will contain at least 15 balls of the same colour.

Q5. A person born in 19th century was x years old in year x^2. In which year was he born?


Tuesday, 16 April 2013

THE MATHEMATICS SYMPOSIUM - I 2012

The year 2012 was decleared National Mathematical Year in honor of Mathematician Srinivasa Ramanujan's 125th birth anniversary. To celebrate the event, Varun Goel and Yuvraj, students of Grade 11 of Spring Dale Senior School, Amritsar, organised a Mathematical Symposium to gather children passionate about Mathematics. 
Varun explaining a problem


Yuvraj explaining a problem 
The event was held on 21st December 2012 and was a success, with students debating and discussing interesting Mathematical problems. There were around 35 participants from class 9th to 11th. The event was attended by 4 Maths teachers from the school.
Some students having fun

The teachers 


Following was a quiz prepared by Varun and Yuvraj, and the students solved it and had a discussion:


Q1.  What is the total number of possible starting moves in a game of chess for both the players?


Q2.  1, 4, 9, 7, _ ? Find the missing number


Q3.  A painter has been assigned a job of painting the square tiles of the external boundary of a 16 X 16 floor. If side of each tile is 4 inches, then how much paint does he need if one tile needs 200 mL paint. (Give your answer in liters.)





Q4.  8+13 = 89+?   [HINT:  6, 9,2,15,14,1,3,3,9 (DECODE TO   GET THE MATHEMATICIANS NAME)]


Q5. A polygon has 54 diagonals. How many sides does this polygon have?  


There was also a discussion about the famous Monty Hall Problem...........Now, the next Mathematics Symposium will be held on 15th May 2013!


Monday, 25 March 2013

Paul Erdős- Mathematician Extraordinaire

'If numbers aren't beautiful, I don't know what is.'

Paul Erdős, one of the greatest mathematicians to have graced the world would have completed a 100 years today had he been alive. In stature and number of mathematical papers published, he is comparable only to the legendary Leonhard Euler. Paul Erdős has written 1525 papers in his lifetime! Unlike other mathematicians, all his papers weren't authored by him; most of them were coauthored, he viewed and practiced Math as a social activity, for the betterment of the society.

It is safe to assume that he was a child prodigy, at the age of four, he could calculate, in his head the number of seconds a person had lived, given their age! Born to high school Mathematics teachers, Paul Erdős was always surrounded by beautiful Mathematical stuff. At 16, his father introduced him to Infinite Series and Set Theory, subjects that later went on to become his favorite. In 1934, when he was just 21 years old he was awarded a doctorate in Mathematics.

Paul Erdős had the eccentric personality that most geniuses of this stature are famous for. Worldly possessions meant nothing to him, and most of his rewards and earnings were donated to the needy. Although he was an atheist, he had a strange belief about 'THE BOOK', a book in which GOD (or Supreme Fascist as Paul Erdős referred to him) kept the best and most beautiful Mathematical proofs to himself! He spent most of his life as a wanderer, roaming around in search of the Truth; the Mathematical Truth. He would often show up at a colleague's house without informing them, and say, "My brain is open", indicating that he was ready to write a Mathematical proof along with the colleague. He would spend a week in their house before shifting to another house. Another one of his famous quote referring to this habit of his- 'Another roof, another proof'

Because he had worked with close to 511 people on Mathematical papers, his friends created the Erdős number as a humorous tribute to him. The Erdős number basically describes the closeness of a person to Paul Erdős. So, Erdős had a number 0, his closest collaborators had a number 1. See the list - LIST of people by Erdos Number


Ultimately it can be said, that Paul Erdős was more than what his achievements tell us. He was a genius, no doubt but it was his passion for Mathematics, the quest for the truth that truly defined him. He wanted the make the world a better place by spreading the beauty of Math, and we at Mathemating with Numbers salute this great personality on his 101st birthday.


Another one of his quotes!




   


Thursday, 14 March 2013

Pi day celebrations!!

  It is irrational, transcendental and highly enigmatic. Yet, we all love it and can't possible imagine our lives without, The oh so adorable Pi! And today all of us Math lovers are celebrating Pi day...

While the origins of Pi have been subject to a lot of speculation, it is thought that Pi was first used actually by the people who built the Great Pyramid at Giza. The reason behind this belief is that these pyramids were built with a perimeter of 1760 cubits and a height of 280 cubits the ratio of which approximates to 6.2857 which is equal to 2 times the value of Pi. However the symbol of Pi which we most commonly use is said to have been first used by mathematician William Jones in 1706. But what is exactly Pi? Pi is the ratio of a circle's circumference and diameter.

Over the years, many people have tried to give their own versions of Pi; using infinite series,continued fractions, polygon approximations and many more unique methods. Check them out:

Besides finding massive use in geometry, calculus and almost every known field to man, Pi has also made appearances in very special Mathematical problems that have actually the changed the way the world thinks about Math- 
Basel problem- This particular problem was solved by Leonhard Euler

The charm of Pi lies in its enigma, its mystery, no one knows where it will end. Because of this annually, people compete to find the most accurate value of Pi and see who can remember most digits of Pi

Here are some other cool Pi facts we think you would like-

1. The first 144 digits of pi add up to 666 (which many scholars say is “the mark of the Beast”). And 144 = (6+6) x (6+6)

2. In the Greek alphabet, π (piwas) is the sixteenth letter. In the English alphabet, p is also the sixteenth letter.

3. Pi Day” is celebrated on March 14 (which was chosen because it resembles 3.14). The official celebration begins at 1:59 p.m., to make an appropriate 3.14159 when combined with the date. Albert Einstein was born on Pi Day (3/14/1879) in Ulm Wurttemberg, Germany.

4. A Web site titled “The Pi-Search Page” finds a person’s birthday and other well known numbers in the digits of pi

5. The Pi memory champion is Hiroyoki Gotu, who memorized an amazing 42,000 digits.

6. Taking the first 6,000,000,000 decimal places of Pi, this is the distribution:
0 occurs 599,963,005 times,
1 occurs 600,033,260 times,
2 occurs 599,999,169 times,
3 occurs 600,000,243 times,
4 occurs 599,957,439 times,
5 occurs 600,017,176 times,
6 occurs 600,016,588 times,
7 occurs 600,009,044 times,
8 occurs 599,987,038 times,
9 occurs 600,017,038 times.



7. Computing pi is a stress test for a computer—a kind of “digital cardiogram.n the Star Trek episode “Wolf in the Fold,” Spock foils the evil computer by commanding it to “compute to last digit the value of pi.

8. The earliest known official or large-scale celebration of Pi Day was organized by Larry Shaw in 1988 at the San Francisco Exploratorium, where Shaw worked as a physicist, with staff and public marching around one of its circular spaces, then consuming fruit pies. The Exploratorium continues to hold Pi Day celebrations.


HAPPY Pi DAY fellow MATH LOVERS !



 
 


Sunday, 24 February 2013

MULTINOMIAL EXPANSION TUTORIAL


Hello readers, we have recorded our first tutorial video and you can check it out on YouTube :


We hope you guys like it. Feel free to leave your valuable remarks and suggestion in the comment box below.

PS- To check out more tutorials (non Math), check out our friend's blog : LINK

KEEP ON MATHEMATING!!

Wednesday, 13 February 2013

DIGITAL ROOT- ADD EM' UP!!



The concept of digital root is one of the most neat and easy concept in Mathematics. It helps explain many problems and further exemplifies the beauty of Math.

Digital Root of any number is the smallest single digit number that one obtains upon adding all the numbers present in a number. The digital root of 39, for instance is 3+9=12=1+2=3.

Digital Root showcases the beauty of Mathematics because the numbers of all the major sequences and series, like squares, cubes, Fibonacci series et al show a repetitive pattern in their digital root:                      

SQUARES
SQUARE
DIGITAL ROOT
 1
1
4
4
9
9
16
7
25
7
36
9
49
4
64
1
81
9
100
1
121
4
144
9
169
7
               
If one observes closely, the digital root starts showing a repetition after 9, i.e. a set of 9 consecutive square numbers shows this repetitive pattern. So, the next set showing this pattern will be from 10-18, then 19-28 and so on...

CUBES
CUBE
DIGITAL ROOT
 1
1
8
8
27
9
64
1
125
8
216
9
343
1
512
8
729
9
1000
1
1331
8
1728
9
Cubes follow an even shorter pattern; a set of only 3 consecutive cubes i.e. 1-3, 4-6 and so on...   

INTERESTING FACTS
·     
  •     If you are careful enough, you might have seen that if 9 is added to any number ‘n’, the digital root of the sum will be equal to digit root of ‘n’. In simpler words, the addition of 9 to a number doesn't produce a change in the digital root.

  •     The above fact helps us locate a multiple of 9. So if someone asks you for a multiple of 9 closest to the number 37218, simply obtain the digital root of 37218 (3) and then you have two options, you can either add 6 to get 9, or subtract 3 from the number; but because we are concerned with the closest number, we must subtract 3, which gives us 37215 whose digital root is 9, hence proving that it is a multiple of 9. This particular method is very useful in larger numbers

·       FUN FACT- 
      After lots of adding up and messy calculations, we both found out that 14 or FOURTEEN is the first number whose digital root is equal to the digital root of the numerical values of the alphabets of its spellings!!! They both equal 5!

We encourage our readers to perform their own operations and share any observations with the Mathematics world via our blog!!