XII CLASS PRE-BOARD - 1 2015 MATHEMATICS PAPER SOLUTION KEY

Dear class XII Students,

Hope this time all of you must have performed well in the exam.

I am providing you with the question paper and the solution key.

Question Paper

Solution Key  

Here is the Sample Paper released by CBSE for 2016 exam.

Click here for the sample papers of the other subjects.
You may also check my Previous Posts where reference material, assignments, book by DOE(govt. Schools) etc.

All the best for your board examination.

Amit Sir

HAPPY BIRTHDAY TO GREAT INDIAN MATHEMATICIAN - RAMANUJAN

Dear Reader,

Srinivasa Ramanujan (22 December 1887 – 26 April 1920) was an Indian mathematician, who made extraordinary contributions to mathematical analysis, number theory, infinite series, and continued fractions. Ramanujan independently compiled nearly 3900 results (mostly identities and equations). His work continues to inspire mathematicians even today! 

Srinivasa Ramanujan, born into a poor Brahmin family at Erode on Dec. 22, 1887, attended school in nearby Kumbakonam. By the time he was 13, he could solve unaided every problem in Loney's Trigonometry, and at 14 he obtained the theorems for the sine and the cosine that had been anticipated by L. Euler.

Ramanujan became so absorbed in mathematics that when he entered the local government college in 1904 with a merit scholarship, he neglected his other subjects and lost the scholarship. Ramanujan married in 1909, and while working as a clerk he continued his mathematical investigations.

In January 1913 Ramanujan sent some of his work to G. H. Hardy, Cayley lecturer in mathematics at Cambridge. Hardy noticed that Ramanujan had rediscovered, and gone far beyond, some of the latest conclusions of Western mathematicians.

In 1914 Ramanujan went to Cambridge. The university experience gave him considerable sophistication, but intuition still played a more important role than argument. In Hardy's opinion, if Ramanujan's gift had been recognized early, he could have become one of the greatest mathematicians of all time. His patience, memory, power of calculation, and intuition made him the greatest formalist of his day.

In 1918 Ramanujan was elected a fellow of the Royal Society and a Fellow of Trinity College, Cambridge.

However, the story goes that, Ramanujan’s health deteriorated greatly while he was in England, and he eventually had to travel back to India in 1919. He died a year later, when he was only 33 years of age, although his work will be remembered for a long time. He dealt with Riemann series, the elliptic integrals, hyper geometric series, and functional equations of the zeta function. Hardy liked to rank mathematicians on a scale of 1 to 100, and he gave himself 25, Littlewood 30, David Hilbert 80, and Ramanujan 100, which shows just how great Ramanujan was.

PRIZE WINNERS - SIXTH INTRA SCHOOL MATHEMATICS OLYMPIAD

Dear Reader,

CRPF Public School has fulfilled its motto of striving towards excellence by organizing and successfully completing its Sixth
Intra School Mathematics Olympiad on 6 November 2015. Around 250 students enthusiastically participated in the Olympiad held in school premises.

Hopefully, it was a great learning experience for all. It surely provided an extended platform for all the participating students and it is wished that it is utilized to its fullest.

Check out the result here:

VI-VIII RANK HOLDERS

IX - XII RANK HOLDERS

VI-XII MERIT HOLDERS

Congratulations to all the rank and merit holders for their brilliant performance in the Olympiad.

You may again like to see the question papers and their solution

5 Reasons Why Math Matters

Dear Reader,

Math is important. It really is.

Despite how you felt about algebra class or geometry class or even basic addition and subtraction, math is a life skill that applies to everyone, not just accountants and engineers. It is the language of many fields, some of which may surprise you. But math matters, in more ways than one. And while numbers and equations are a necessary evil to some and pure enjoyment to others, we need math to live.

Below are five reasons why math is worth knowing. Starting with…

Building Stuff: Balancing a cheque-book? Dividing rent among four roommates? Determining your share of the electric bill? How about taxes and student loans? Paying bills is the worst, no doubt about it. But you need to know some math in order to stay on top of your finances, balance your budget, and avoid all those overdrafts.

Cooking: Surprised? Baking cupcakes and roasting a chicken requires math skills. You need to measure the ingredients in terms of teaspoon, tablespoon, ounces, grams, kilograms, etc. And what if you what to cut a recipe in half? This is serious business. Unless you’re a pro at eyeballing the “this much salt” and “that much flour,” you need math in the kitchen.

Travel: Planning to jump the pond for a backpacking adventure? If so, better prepare yourself for currency confusion, especially if your currency converter is not a physical device. Knowledge of currency conversion is essential when traveling to a foreign country. Your iPhone may not be able to save you, but your math skills sure can.

Building Stuff: There will come a time when you need to assemble a desk, a shelving unit, or perhaps a crib. And you’ll thank your math teacher for helping you become the hero that you are once that desk sits proudly constructed by the window. Putting stuff together requires math aptitude in order to avoid a safety hazard.

Career Readiness: Are you thinking of becoming a nurse? How about a marketing manager or a lawyer? Nurses use math to calculate medication dosages, marketing managers use math to determine profit margin and pricing strategies, and lawyers use math to determine settlements and statistical evidence. Even sports agents and high school English teachers use math on a regular basis.

Think about it. Math really is everywhere. From the kitchen to the courtroom, numbers, graphs, and measurements are all around us. Knowing how to calculate a tip at a restaurant or your fantasy football score is important business! Mortgage payments, caloric intake, and retail discounts are a big part of our everyday lives. And we can’t always rely on our phones to do the work for us when it comes to math. Your brain is a much better, and more reliable tool.

Source: http://onlinemathdegrees.org/

Believe it or not in Mathematics

1. Russian King Peter the Great announced that no member of the Royal Family should marry until they pass in arithmetic, geometry and navigation.

2. There is no equilateral triangle, the coordinates of whose vertices are all integers.

3. There are only five Heronian triangles (triangles having integer sides) such that the perimeter is equal to the area. They are: (5, 12, 13) , (6, 8, 10) , (6, 25, 29) , (7, 15, 20) and (9, 10, 17). The first two triangles are right triangles also.

4. A plane can be filled with squares, equilateral triangles and regular hexagons. No other regular polygons can fill the plane without gaps.

5. Three legged stool is more stable than any other stool. The principle underlying is this fact is we can draw only one circle through three given points which are not collinear. That is why old people walk with the assistance of a stick (third leg) , or even think of tripod.

6. An angle of 3 degrees is the only constructible angle of prime measure.

7. The average human body contains about 10^27 atoms whereas the number of atoms in the whole universe is 10^73.

8. There are certainly at least two peoples in the world with exactly the same number of hairs on their heads! (There are more people than the hairs on any one head).

Sixth Intra School Mathematics Olympiad - Solution Key

Dear All,

Mathematics, often called as language of nature has become an integral part of everyone’s life. In today's highly competitive world, one has to bear a lot of mental stress and also have to get involved in so many things in order to acquire knowledge. This is where co-scholastic activities play a very significant role. They provide the much needed exposure for a student.

CRPF Public School has fulfilled its motto of striving towards excellence by organizing and successfully completing its Fifth Intra School Mathematics Olympiad on 6 November 2015. Around 250 students enthusiastically participated in the Olympiad held in school premises.

Such Olympiads and events provide a great and extended platform for the students to nurture and showcase their ability and talent. It was a towering learning experience for the students of the school from class VI to XII. To encourage students, merit certificates will be given to those scoring at least 60% along with the normal prizes for the top scorers.

We hope that all the participating students must have done well in the exam. To access your performance, check out the solution keys of the question papers below:-

Question papers:-

Class VI      Class VII     Class VIII

Class IX      Class X        Class XI     Class XII

Solutions keys:-

Class VI      Class VII     Class VIII

Class IX      Class X        Class XI      Class XII

The result would be declared on 20 November 2015 and would be available here on the blog.

Wishing you all the best!!!

Ramanujan surprises again

A box of manuscripts and three notebooks. That's all that's left of the work of Srinivasa Ramanujan, an Indian mathematician who lived his remarkable but short life around the beginning of the twentieth century. Yet, that small stash of mathematical legacy still yields surprises. Two mathematicians of Emory University, Ken Ono and Sarah Trebat-Leder, have recently a made a fascinating discovery within its yellowed pages. It shows that Ramanujan was further ahead of his time than anyone had expected, and provides a beautiful link between several milestones in the history of mathematics. And it all goes back to the innocuous-looking number 1729.

Ramanujan's story is as inspiring as it is tragic. Born in 1887 in a small village around 400 km from Madras (now Chennai), Ramanujan developed a passion for mathematics at a young age, but had to pursue it mostly alone and in poverty. Until, in 1913, he decided to write a letter to the famous Cambridge number theorist G.H. Hardy. Accustomed to this early form of spam, Hardy might have been forgiven for dispatching the highly unorthodox letter straight to the bin. But he didn't. Recognising the author's genius, Hardy invited Ramanujan to Cambridge, where he arrived in 1914. Over the following years, Ramanujan more than repaid Hardy's faith in his talent, but suffered ill health due, in part, to the grizzly English climate and food. Ramanujan returned to India in 1919, still feeble, and died the following year, aged only 32. Hardy later described his collaboration with Ramanujan as "the one romantic incident in my life".

To read complete article, click here.

Source: https://plus.maths.org/content/ramanujan

Who is George Boole and why has Google made a doodle on him?

George Boole (2 November 1815 – 8 December 1864) , the British mathematician whose work on logic laid many of the foundations for the digital revolution, has been honoured on the 200th anniversary of his birth with a special Google Doodle.

He was an English mathematician, educator, philosopher and logician. He worked in the fields of differential equations and algebraic logic, and is best known as the author of The Laws of Thought which contains Boolean algebra. Boolean logic is credited with laying the foundations for the information age.

George Boole was the son of a shopkeeper. This working class of people was not given a high level of education. He had common schooling and a commercial course. His father, who had studied some mathematics privately, tutored George in the subject.

George wanted to learn Latin and Greek so that he could advance in society. He was given some basic tutoring from the local bookseller, a friend of his father. George managed to learn Latin by himself. At the age of 12, he translated an ode of Horace into English. His father was proud and had his work printed in the local paper. Several critics denied a boy of his age could do such work, but they also pointed out his errors. George was humiliated.

George spent the next two years studying Latin and Greek. At the age of 16 he was ready to find a profession that would allow him to support his aging parents. Boole worked as an assistant teacher at two schools over the next four years. He was not satisfied with the low wages and looked for another profession. He could not afford the Army or the Law, and he didn't like the teacher's wages, so he focused on the Church.

After four years of preparation to be a clergyman, his parents persuaded him back to teaching. He did learn French, German and Italian while studying to become a clergyman, languages that would help him later in mathematics.

At age 20, George Boole opened his own school. He had to begin teaching mathematics to his pupils, which sparked his own interest in math. Dissatisfied with the textbooks, he began reading Laplace and Lagrange for ideas. Inspired by ideas in their work, he wrote his first mathematical paper on the calculus of variations. During this time, Boole also discovered invariants.

Boole began submitting his work to the Cambridge Mathematical Journal. The editor, Duncan Gregory liked his papers and published them in the journal. Gregory suggested that Boole study at Cambridge, but he could not quit teaching because he supported his parents financially.

Boole began studying algebra as Gregory suggested. His work was soon published and awarded. In August 1849, Boole was appointed as a professor of mathematics at Queens College, Cork. Within two years, he was named Dean of Science.

In 1854, Boole published An Investigation into the Laws of Thought, on Which are founded the Mathematical Theories of Logic and Probabilities. Boole suggested that logic and algebraic symbols were similar. By tying logic and algebra, Boole allowed algebra to be viewed as purely abstract. Today, computer programming is based upon Boolean algebra.

George Boole married Mary Everest (daughter of George Everest, for whom the mountain is named) in 1855. Boole encouraged his wife to study at the college. They had five daughters.

George Boole died on December 8, 1864, after several weeks of fighting a lung infection. George had walked to college in the rain, lectured, and returned home which prompted the sickness.

George Boole's contributions to mathematics have very modern applications: computer programming, electrical engineering, satellite pictures, telephone circuits and even Einstein's theory of relativity.

His legacy was Boolean logic, a theory of mathematics in which all variables are either "true" or "false", or "on" or "off". The theory proceeded the digital age, with American Claude Shannon applying Boolean logic to build the electrical circuits in the 1930s that led to modern computers.

First Term Mathematics Paper and Solution Key 2015-16

Dear class XI Students,
I hope that you must have performed well in the Mathematics exam held today i.e. 26 September 2015.
I am providing you with the question paper and the solution key.
Question Paper
Solution Key
All the best for your board examination.
Amit Sir

First Term Mathematics Paper and Solution Key 2015-16

Dear class XII Students,

I hope that you must have performed well in the Mathematics exam held today i.e. 18 September 2015.

I am providing you with the question paper and the solution key.

Question Paper

Solution Key

All the best for your board examination.

Amit Sir

Ramanujan (2014) - A movie

Dear Reader,
Today I watched the movie Ramanujan (2014) , the only movie ever made about life of genius Indian mathematician Srinivasa Ramanujan, . Though it was in tamil but to one who worship him like a God , language was not a barrier. Well, thanks to english subtitles. It is really a very nice movie. Every Mathematics lover should watch this movie.
Also those who have not read his biography 'A man who knew infinity' by Robert Kanigel, should spare some time and must read that book.
Amit Bajaj

Information regarding various Mathematics Olympiads

Class XII - Weekly Test (Paper with it's Solution Key)

Dear Students,

Hope you have done well in Math’s XII class weekly test held today. I am providing Question paper and its solution key for your ready reference.

Question Paper            Solution Key

All the best!

Amit Sir

Happy pi approximation day

Dear Reader,

Well, today is Pi Approximation Day.

Pi Approximation Day celebrates the mathematical constant π (pi). It is observed on July 22, due to π being roughly equal to 22/7.

The fractional approximation of π ,22⁄7, resembles the date July 22 in the day/month format, where it is written 22/7. Pi Approximation Day is therefore celebrated on July 22.

In geometry, pi is defined as the ratio of a circle's circumference to its diameter. July 22, often written as 22/7, is an appropriate day to explore pi, since 22 divided by 7 is an approximation of pi.

Pi Approximation Day is celebrated in a number of ways, including contests to memorize pi to the most decimal places, solving math problems involving pi, discovering the history of pi, ruminating on how life would be different without pi and—of course—eating pie.

So, enjoy this mathematical day ...

Amit Sir

Some Great Mathematicians

Movies Related to Mathematics

1. A Beautiful Mind (2001)
2. A Brilliant Young Mind (2014)
3. Good Will Hunting (1997)
4. IQ(1994)
5. The Number 23 (2007)
6. Proof (2005)
7. Contact (1997)
8. Sphere (1998)
9. Stand and Deliver (1988)
10. cube(1997)
11. 21 (2008)
12. The 4th Dimension (2006)
13. Bianca (1984)
14. Fermat's Last Tango (2001 Video)
15. Fermat's Room (2007)
16. Hotel Hilbert (1996 TV Movie)
17. Numb3rs (2005)
18. The Man Without a Face (1993)
19. A Brief History of Time (1992)
20. The Theory of Everything (2014 film)
21. Primer (2004)
22. Enigma (2001)
23. Sneakers(1992)
24. 21 grams (2003)
25. Pi(1998)
26. Straw dogs (1971)
27. A summer's tale (1996)

What's the next number?

How would you continue the sequence 1,2,3,4,5,6,7,8,9?
The next number might be:
• 10, if this is the sequence of natural numbers;
• 1, if this is the sequence of the digital sums of natural numbers;
• 11, if this the sequence of palindromes;
• 0, if this is the sequence of digital products of natural numbers;
• 13, if this is the sequence of numbers such that 2 to their powers doesn’t contain 0;
• 153, if this is the sequence of numbers that are sums of fixed powers of their digits;
• 22, if this is the sequence of numbers for which the sum of digits equals the product of digits;
or • any number you want.

Special properties of number 30 (Today’s Date)

1. 30 is the sum of the first four squares, which makes it a square pyramidal number.
2. The icosahedron and the dodecahedron are Platonic solids with 30 edges. 
3. 30 is a Harshad number ( an integer that is divisible by the sum of its digits)
4. The atomic number of zinc is 30
5. The minimum age for United States senators.
6. February 30 is usually used as a sarcastic date for referring to something that will never happen or will never be done.

Special properties of number 29 (Today’s Date)

1. None of the first 29 natural numbers have more than two different prime factors. This is the longest such consecutive sequence.
2. 29 is the sum of three consecutive squares i.e. 29 =  2^2 + 3^2 + 4^2
3. The number of days February has in leap years.
4. The atomic number of copper
5. Saturn requires over 29 years to orbit the Sun.
6. Number of states in india

Special properties of number 28 (Today’s Date)

1. 28 is a triangular number since it is the sum of the first seven counting numbers: 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28.
2. 28 is also a perfect number since the sum of its aliquot divisors (factors that are less than the number itself) is also 28. The aliquot factors of 28 are 1, 2, 4, 7, and 14, and 1 + 2 + 4 + 7 + 14 = 28. The next smallest perfect number is 6, and the next highest perfect number is 496.
3. 28 is an example of a happy number. When you square the digits, add them, and repeat this process on the resulting number, you eventually end up with 1:
2^2 + 8^2 = 68
6^2 + 8^2 = 100
1^2 + 0^2 + 0^2 = 1
4. The sum of the first five prime numbers is 2 + 3 + 5 + 7 + 11 = 28.
5. 28 is a hexagonal number, but not a centered hexagonal number. Every hexagonal number is also a triangular number, but not every triangular number is a hexagonal number.
6. A polygon with 28 sides is called an icosikaioctagon.
7. Between the years 2000 and 2099, the calendar repeats every 28 years. It also cycles every 28 years between 2101 and 2199. 

Special properties of number 27 (Today’s Date)

1. Twenty-seven is a perfect cube = 3^3.
2. In base 10, it is the first composite number not evenly divisible by any of its digits.
3. 27 is the only positive integer that is 3 times the sum of its digits.
4. 27 is a Harshad number (a number divisible by sum of its digits).
5. 27 is the smallest 2-digit number in which the sum of digits is equal to the sum of prime factors, e.g.: 27 = 3 x 3 x 3 and 2 + 7 = 3 + 3 + 3 = 9
6. 23 is the largest integer which is the sum of the digits of its cube:
27^3 = 19,683 and 27 = 1 + 9 + 6 + 8 + 3
7. 27 is the number of "cubies" in a Rubik's cube.
8. The numbers of atoms in the body of an average male (about 80 kg) is about 10^27.
9. The Chemical Element Cobalt has an atomic number of 27.
10. The primitive Greek alphabet had 27 letters.
11. There are 27 signs of the zodiac in Indian astrology

Special properties of number 26 (Today’s Date)

1. 26 is the only number sandwiched between a square (5^2) and a cube (3^3).
2. There are 26 letters in the English alphabet.
3. In base ten, 26 is the smallest number that is not a palindrome to have a square (26^2 = 676) that is a palindrome.
4. In a normal deck of cards, there are 26 red cards and 26 black cards.
5. The Chemical Element Iron has an atomic number of 26.
6. 26 is the number of bones in the normal human Foot and Ankle.
7. There are 26 weeks in half a year.

Special properties of number 25 (Today’s Date)

1. It is a square number, being 5^2 = 5 × 5. It is the smallest square that is also a sum of two (non-zero) squares: 2^5 = 3^2 + 4^2.
2. 25 is the sum of the first 5 odd numbers(= 1 + 3 + 5 + 7 + 9).
3. Couples who have been married for 25 years celebrate their Silver Wedding.
4. The Chemical Element Manganese has an atomic number of 25.
5. 25 is the number of days approximately that takes the sun to do a complete rotation on itself.

Special properties of number 24 (Today’s Date)

1. 24 is the factorial of 4 (24 = 4!).
2. 24 is an Abundant Number. In number theory, an abundant number or excessive number is a number for which the sum of its proper divisors is greater than the number itself. The first few abundant numbers are 12, 18, 20, 24, 30, 36, ... There are only 21 abundant numbers less than 100, and they are all even.
3. 24 is the largest number divisible by all numbers less than its square root.
4. It is a highly composite number, having more divisors than any smaller number.
5. 24 is a semiperfect number, since adding up all the proper divisors of 24 except 4 and 8 gives 24.
6. Subtracting 1 from any of its divisors (except 1 and 2, but including itself) yields a prime number; 24 is the largest number with this property.
7. 24 is a Harshad number (number divisible by its sum of digits).
8. The product of any four consecutive numbers is divisible by 24.
9. 24 is the only nontrivial solution to the cannonball problem, that is: 1^2+2^2+3^2+...+24^2 is a perfect square (=70^2). (The trivial case is just 1^2 = 1^2.)
10. 24 is the number of Tirthankaras (in Janism)
11. 24 is the number of spokes in the Ashok Chakra
12. 24 carats represents 100% pure gold.
13. There are 24 hours in a day.
14. 24 is the number of letters in both the modern and classical Greek alphabet.
15. The Chemical Element Chromium has an atomic number of 24.

Special properties of number 23 (Today’s Date)

1. According to the birthday paradox, in a group of 23 (or more) randomly chosen people, the probability is more than 50% that some pair of them will have the same birthday. For 60 or more people, the probability is greater than 99 per cent!
2. 23 is the smallest prime number with consecutive digits.
3. 23! is 23 digits long.
4. Twenty-three is the sum of three other, consecutive, prime numbers; 5, 7 and 11. It is the first prime number showing this characteristic.
5. The sum of the first 23 primes is 874, which is divisible by 23, a property shared by few other numbers.
6. Repeat the digit 1, 23 times like this: 11,111,111,111,111,111,111,111 and you obtain a prime number.
7. Every positive whole number can be written as the sum of eight cubes (including 0^3 when necessary) except 23 (and 239). Those two numbers require 9 cubes.
8. Nobel Prize-winning economist John Forbes Nash, the inspiration for the film A Beautiful Mind, was obsessed with the number 23 and it featured prominently in his nervous breakdown. He claimed that Pope John XXIII was in fact himself, the evidence being that 23 was his favorite number. Nash also published only 23 scientific articles.
9. The axis of the planet Earth is 23.5 degrees to the vertical. The tropics of Cancer
and Capricorn are at 23.5˚ north and south respectively.
10. The Chemical Element Vanadium has an atomic number of 23.
11. Normal human sex cells have 23 chromosomes. Each parent contributes 23 chromosomes to the start of human life. The nuclei of cells in human bodies have 46 chromosomes made out of 23 pairs.

Special properties of number 22 (Today’s Date)

1. Twenty-two is a pentagonal number.
2. 22 is a palindromic number in decimal system
3. 22 divided by 7 approximates the irrational number π, the ratio of the circumference of a circle to its diameter.
4. There are, in fact, precisely 22 ways to express 8 as a sum of positive integers. i.e. 8 can be partitioned in 22 ways:
8, 7+1, 6+2, 6+1+1, 5+3, 5+2+1, 5+1+1+1, 4+4, 4+3+1, 4+2+2, 4+2+1+1, 4+1+1+1+1, 3+3+2, 3+3+1+1, 3+2+2+1, 3+2+1+1+1, 3+1+1+1+1+1, 2+2+2+2, 2+2+2+1+1, 2+2+1+1+1+1, 2+1+1+1+1+1+1, and 1+1+1+1+1+1+1+1
5. 22 is the smallest number which can be expressed as the sum of 2 primes in 3 ways: 3 + 19, 5 + 17, and 11 + 11
6. 22 is the number of different ways of linking 5 hexagons together.
7. The Chemical Element Titanium has an atomic number of 22.
8. The Human head is constituted of 22 bones: 8 for the cranium and 14 for the face.
9. By the old English measuring system there are 22 yards in a chain. This is the
length of a cricket pitch.
10. There are 22 players on the field in American Football, soccer and field hockey.

Special properties of number 21 (Today’s Date)

1. Twenty-one is a Fibonacci number, a Harshad number, triangular number and an octagonal number, as well as a composite number, its proper divisors being 1, 3 and 7.
2. 21 is the sum of the first six natural numbers (1+2+3+4+5+6=21), making it a triangular number. Also, the number of spots on a standard cubical (six-sided) die (1+2+3+4+5+6)
3. 21 is the smallest non-trivial example of a Fibonacci number whose digits are Fibonacci numbers and whose digit sum is also a Fibonacci number.
4. 21 is a repdigit in base 4 (1114).
5. 21 is the third star number. Star numbers can be represented by a square with a triangle on each side.
6. 221 - 21 is prime.
7. The current century—spanning the years from 2001 to 2100—is referred to as the 21st century.
8. 21 grams is the weight of the soul, according to research by Duncan MacDougall, generally regarded as meaningless.
9. The Chemical Element Scandium has an atomic number of 21.
10. There are 21 letters in the Italian alphabet.

Special properties of number 20 (Today’s Date)

1. 20 = 2 + 4 + 6 + 8
2. 20 = 1 + 3 + 6 + 10. i.e. 1 + (1 + 2) + (1 + 2 + 3 ) + (1 + 2 + 3 + 4 )
3. 20 is an Abundant Number. In number theory, an abundant number or excessive number is a number for which the sum of its proper divisors is greater than the number itself. The first few abundant numbers are 12, 18, 20, 24, 30, 36, ... There are only 21 abundant numbers less than 100, and they are all even. 
4. 20 is a Tetrahedral Number.
5. One month in the religious Mayan calendar contained twenty days.
6. 20 is the base of the ancient Mayan numeral system.
7. A set of 20 units may also be referred to as a score.
8. The Chemical Element Calcium has an atomic number of 20.

Special properties of number 19 (Today’s Date)

1. 19 is a prime number.
2. 19 is the maximum number of 4th powers needed to sum to any number
3. 19 is the largest prime number that is palindromic in Roman numerals (XIX).
4. 19 is an Octahedral Number.
5. The Chemical Element Potassium has an atomic number of 19.
6. "The Sun" is numbered with 19 in Tarot cards.

Special properties of number 18 (Today’s Date)

1. 18 is a Heptagonal Number.
2. 18 is an Abundant Number. In number theory, an abundant number or excessive number is a number for which the sum of its proper divisors is greater than the number itself. The first few abundant numbers are 12, 18, 20, 24, 30, 36, ... There are only 21 abundant numbers less than 100, and they are all even. 
3. 18 is a Lucas Number (No. from the sequence 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, … where first two numbers are fixed as 1 and 3, next number is the sum of its two preceding terms).
4. The 18 Regular Solids: the 5 Platonic Solid + the 13 Archimedean Solids.
5. 18 is the area and also the perimeter of a rectangle with 6 and 3-unit sides.
6. 18 is the only number that is twice the sum of its digits.
7. On Earth, the length of a day 900 million years ago was only about 18 hours.
8. There are 18 chapters in the Hindu epic tale Bhagavad Gita.
9. The Chemical Element Argon has an atomic number of 18.

Special properties of number 17 (Today’s Date)

1. 17 is a prime number.
2. 17 is the sum of the first 4 prime numbers: 17 = 2 + 3 + 5 + 7
3. 17 is a palindromic number in binary system (10001).
4. 17 is a Leyland number, 17 = 3^2 + 2^3(In number theory, a Leyland number is a number of the form x^y + y^x where x and y are integers greater than 1. They are named after the mathematician Paul Leyland. The first few Leyland numbers are
8, 17, 32, 54, 57, 100, 145, 177, 320, 368, 512, 593, 945, 1124…
5. 17 is the 7th prime number and the only prime of the form p^q + q^p, where p and q are prime: 17 = 2^3 + 3^2
6. 17 is the lowest possible number of givens for a Sudoku puzzle with a unique solution (this was proven in 2012 by Gary McGuire, Bastian Tugemann, and Gilles Civario).
7. No odd Fibonacci Number is divisible by 17.
8. abcdefghabcdefgh is divisible by 17 (replace each letter with a digit. E.g. 1234567812345678). The reason is that any number of that form is a multiple of 100,000,001 - which is divisible by 17.
9. A regular 17-gon can be constructed by using rule and compasses only.
10. There are exactly 17 ways to express 17 as the sum of 1 or more primes i.e. 17 is the only integer which is equal to the number of prime partitions of itself.

11. The famous problem of the 17 camels: a sheik has 3 children and owns 17 camels. His will stipulates that the eldest is to receive half his property; the second son is to receive the third of his property; and the third one, the ninth of his property. On his death, how would the sons share out the inheritance? Solution: They borrow a camel, share out, and give back the camel: (17+1)/2 + (17+1)/3 + (17+1)/9 = 17 camels.

12. The Chemical Element Chlorine has an atomic number of 17.
13. There are 17 muscles in the Human tongue.
14. A day on Uranus lasts for 17 hours.

Special properties of number 16 (Today’s Date)

1. 16 is the only number of the form x^y = y^x with x and y being different integers.
2. 16 is a Square Number (4x4).
3. 16 is the maximal number of regions into which 5 lines divide a plane.
4. 16= 1 + 3 + 5 + 7 i.e. sum of first four odd numbers
5. The base 16 notational system for representing real numbers, called hexadecimal, is particularly important and used extensively in computer science, since four bits (each consisting of a 'one' or 'zero') can be succinctly expressed with just a single hexadecimal digit. The 16 distinct digits used to represent numbers in hexadecimal notation are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F.
6. The Chemical Element Sulfur has an atomic number of 16.

Special properties of number 15 (Today’s Date)

1. 15 is a triangular number  as 15 = 1 + 2 + 3 + 4 + 5
2. 15 is a hexagonal number
3. 15 is a palindromic number in binary system (1111)
4. 15 is the magic constant for a 3×3 magic square, the smallest possible nontrivial magic square. An N×N magic square consists of the N2 integers from 1 to N arranged in an N×N square grid, such that the sum of any row, or any column, or of one of the two diagonals, is equal to any other. There is only one solution for a 3×3 square, not counting its rotations and reflections:
  4  9  2
  3  5  7
  8  1  6
This magic square was known to the Chinese at least 3000 years ago and is called the Lo Shu.
5. The Chemical Element Phosphorus has an atomic number of 15.
6. The only 15-letter word that can be spelled without repeating a letter is "uncopyrightable".

Special properties of number 14 (Today’s Date)

1. 14 is a square pyramidal number as 14 = 1 + 4 + 9 = 12 + 22 + 32 
2. In September 1752, Great Britain switched from the Julian Calendar to the Gregorian Calendar. In order to achieve the change, 11 days were 'omitted' from the calendar: the day after 2 September 1752 was 14 September 1752.
3. The Chemical Element Silicon has an atomic number of. 14.

Special properties of number 13 (Today’s Date)

1. 13 is a Fibonacci number
2. 13 is the number of Archimedian solids.
3. 13 is a Prime Number. 13 is a Fibonacci Number.
4. The Chemical Element Aluminum has an atomic number of 13.
5. The card deck includes 13 hearts, 13 spades, 13 squares, 13 clubs.
While the Earth revolves once, the Moon revolves 13 times.
6. There is always at least one Friday 13th in each year.
The probability of being born on a Friday the 13th is about 1/214
7. According to historians, there were 13 people at Christ's Last Supper and Christ was crucified on Friday 13th. So, Friday 13th is considered as unlucky.
8. The fear of the number 13 is called 'Triskaidekaphobia'. 

Special properties of number 12 (Today’s Date)

1. 12 is a pentagonal number
2. 12 is the smallest abundant number.
3. 12 is a pentagonal number.
4. 12 is the smallest number with exactly six divisors.
5. 12 is the largest known even number expressible as the sum of two primes in one way (5 + 7).
6. A dozen is a group or set of 12.
7. The Cube is a Platonic Solid with 12 Edges.
8. The Dodecahedron is a Platonic Solid with 12 Faces.
9. The Icosahedron is a Platonic Solid with 12 Vertices.
10. The Chemical Element Magnesium has an atomic number of 12.
11. The 12 Star signs: Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpio, Sagitarius, Capricorn, Aquarius and Pisces.

Special properties of number 11 (Today’s Date)

1. 11 is a palindromic number in decimal system. It is only palindromic prime with an even number of digits.
2. 11 is a Lucas Number.
3. 6 can be Partitioned in 11 ways.
4. 11 is the third honest number, because 11 = "two plus nine".
5. An average room holds 11 x 100 pounds of air!
6. A million seconds is 11 days and a half.
7. The Chemical Element Sodium has an atomic number of 11.
8. The sun spot cycle repeats about every 11 years.

Special properties of number 10 (Today’s Date)

1. 10 is the base of our number system.
2.10 is the sum of the first 3 prime numbers: 10 = 2 + 3 + 5
3. 10 is both a triangular (1 + 2 + 3 + 4 = 10) and tetrahedral number.
4.10 is asemiprime number. A semiprime is a natural number that is the product of two (not necessarily distinct) prime numbers. The semiprimes less than 100 are 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 38, 39, 46, 49, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, and 95.
The Chemical Element Neon has an atomic number of 10.
5. 10! = 6! x 7! = 3! x 5! x 7! (Unique solution to the factorial equation n! =a! x b! x c! with consecutive prime factors)
6. In most advertisements, including newspapers, the time displayed on a watch is approximately 10:10. The reason for this is to "frame" the logo of the watch maker. The hands at that number cause the eyes to look right at the logo!
7. Marion's theorem states that the area of the central hexagonal region determined by trisection of each side of any triangle ABC and connecting the corresponding points with the opposite vertex is exactly one tenth (1/10) the area of the whole triangle ABC.

Special properties of number 9 (Today’s Date)

1. 9 is the largest single-digit number.
2. 9 is a Square Number (3x3).
3. 9 is a palindromic number in decimal system
4. 9 is a palindromic number in binary system (1001)
5. The Chemical Element Beryllium has an atomic number of 9.
6. Water expands by about 9% as it freezes.
7. A full moon is nine times brighter than a half moon.

Special properties of number 8 (Today’s Date)

1. The smallest "non-trivial" cube: 8=23=2×2×2.
2. 8 is a Fibonacci number. Also it is the largest cube in the Fibonacci sequence.
3. 8 is an Octagonal Number.
4. 8 is the smallest sum of two factorials of distinct primes: 2! + 3!
5. When specifying directions on a map, most people choose from one of these 8 directions: north, northeast, east, southeast, south, southwest, west, and northwest.
6. In three-dimensional space there are 8 "diagonal" ways to move, corresponding to the eight "octants" you get if you divide the three-dimensional space with three mutually-perpendicular planes.
7. The Chemical Element Oxygen has an atomic number of 8.

Special properties of number 7 (Today’s Date)

1. 7 is a Prime Number.
2. 7 is a palindromic number in decimal system. Also 7 is a palindromic number in binary system (111).
3. 7 is a Mersenne Number (number of the form 2^n -1)
4. 7 is an Octahedral Number.
5. 7 is a Lucas Number.
6. Sum of the first 4 Fibonacci numbers= 1 + 1 + 2 + 3 = 7
7. 7 is the smallest number of sides of a regular polygon that is not constructible by straightedge and compass.
8. Topologists have been able to prove that 7 colors may be needed on a donut shaped map to ensure that no adjacent areas are the same.
9. The smallest positive integer whose reciprocal has a pattern of more than one repeating digit: 1/7 = 0.142857142857...
10. The opposite sides of a die cube always add up to 7.
11. A square piece of paper cannot be folded in half more than 7 times... However, Britney Gallivan (of Pomona, California) found a formula that tells us to successfully fold paper 12 times, we would need about 1.2 km of paper... and she proved it!
12. The 7 colors of the rainbow.
13. There are 7 days in creation, 7 days in the week, 7 phases of the moon etc
14. The 7 wonders of the ancient world are:
- Great Pyramid of Giza,
- Hanging Gardens of Babylon,
- Temple of Artemis at Ephesus,
- Statue of Zeus at Olympia,
- Mausoleum of Maussollos at Halicarnassus,
- Colossus of Rhodes,
- Lighthouse of Alexandria.
15. Adolf Hitler was reputed to take his coffee or tea with seven teaspoons of sugar.
16. Nitrogen is a chemical element with atomic number 7.

Special properties of number 6 (Today’s Date)

1. 6 is the smallest perfect number, that is a number whose divisors add up to itself, e.g.: 1 x 2 x 3 = 1 + 2 + 3 = 6
2. 6 (= 1 + 2 + 3) is a triangular number
3. 3! = 6
4. 6 is a Hexagonal Number.
5. 6 is a congruent number because it is the area of a 3, 4, 5 triangle (a congruent number is an integer that is the area of a right triangle with three rational number sides).
6. 6 is the smallest number of colors needed to color the regions on a Möbius strip. A Möbius strip is a continuous closed surface with only one side; formed from a rectangular strip by rotating one end 180 degrees and joining it with the other end.
7. 6 circles of the same size (try this with 6 coins of the same denomination) will always perfectly surround, all touching, without gaps, 1 circle of that same size.
8. The Chemical Element Carbon has an atomic number of 6.

Special properties of number associated with Today’s Date

Dear All,
I have been collecting information from various resources regarding the special properties being associated with today's date.
It's already 5th June today. So, today I will share properties from the dates (numbers) 1 to 5.
In future I would prefer to post day wise.
In case you also wish to share something interesting kindly do it.
Any suggestions are also welcome.
Amit Bajaj

Special properties of number 5 (Today’s Date)

1. 5 is the only prime number that ends in 5.

2. there are 5 Platonic solids

3. A five-sided polygon (pentagon) has 5 diagonals. This is the only shape for which the number of sides and diagonals is the same 

4. 5 is a Fibonacci Number.

5. 5 is a Pentagonal Number.

6. The Chemical Element Boron has an atomic number of 5.

7. The five natural senses of sight, hearing, smell, taste, and touch.

8. The 5 Elements: Fire, air, water and Earth and Spirit.

Special properties of number 4 (Today’s Date)

1.  Four is the smallest composite number, its proper divisors being 1 and 2.

2. 4 is the smallest number of colors sufficient to color all planar maps.

3.  4 is a palindromic number in decimal system.

4. The word 'four' has 4 letters and is the smallest honest number. Honest numbers are numbers n that can be described using exactly n letters in standard mathematical English.

5. A 4 x 4 square has an area equal to its perimeter.

6. The atomic number of beryllium.

7. Using four 4's and any operations, try to write equations that have the integers from 1 to 100 as the answer (see example below):

1 = 44/44

2 = 4/4 + 4/4

3 = (4 + 4 + 4)/4

4 = 4(4 - 4) + 4, etc...

8. In Puruṣārtha, there are four aims of human life: Dharma, Artha, Kāma, Moksha.

9. The four stages of life Brahmacharya (student life), Grihastha (household life), Vanaprastha (retired life) and Sannyasa (renunciation).

10. Four seasons: spring, summer, autumn, winter

11. Four cardinal directions: north, south, east, west.

12. Four suits of playing cards: hearts, diamonds, clubs, spades.

13. There are four basic states of matter: solid, liquid, gas, and plasma.

Special properties of number 3 (Today’s Date)

1. 3 is a Fibonacci number

2. 3 is a triangular number3 is a palindromic number in decimal system

3. 3 is a palindromic number in binary system (11)

4. Three is the number of dimensions that humans can perceive. Humans perceive the universe to have three spatial dimensions

5. There are three geometric constructions you cannot build using just a ruler and compasses: 1. You cannot trisect - divide into three equal parts - a given angle; 2. Double a cube; and 3. Square a circle. (Geometric Problems of Antiquity).

6. Three non-collinear points determine a plane and a circle.

7. 3 is the smallest number of sides that a simple (non-self-intersecting) polygon can have.

8. Gauss proved that every integer is the sum of at most 3 triangular numbers.

9. Three is the atomic number of lithium.

10. The triangle, a polygon with three edges and three vertices, is the most stable physical shape. For this reason it is widely utilized in construction, engineering and design.

 

Special properties of number 2 (Today’s Date)

1. 2 is the only even prime.

2. 2 is a Fibonacci number

3. 2 is a palindromic number in decimal system.

4. For any Polyhedron , V – E + F = 2 where V is the number of vertices, E is the number of edges, and F is the number of faces.

5. The atomic number of helium.

6. A honey bee must tap TWO million flowers to make ONE pound of honey!

Special properties of number 1 (Today’s Date)

1. 0! = 1

2. 1 is the atomic number of hydrogen

3.   Philo of Alexandria (20 BC – AD 50) regarded the number one as God's number, and the basis for all numbers

4. 1 is a Fibonacci number

5. 1 is a triangular number

6. 1 is a pentagonal number

7. 1 is a hexagonal number

8. 1 is a heptagonal number

9. 1 is a octogonal number

10. 1 is a palindromic number in decimal system

 11. A list of primes to 10,006,721 published in 1914 by Derrick Lemher includes 1 as prime.

12. A, B , C , D , E, M, T, U, V , Y has one line of symmetry

13. Till 15th century 1 was not included in set of natural number. Simon Stevin included it in Natural number.