Author: Anitha
Cheryl’s Birthday Puzzle
Albert and Bernard just became friends with Cheryl, and they want to know when her birthday is. Cheryl gives them a list of 10 possible dates.
May 15, May 16, May 19
June 17, June 18
July 14, July 16
August 14, August 15 and August 17
Cheryl then tells Albert and Bernard separately the month and the day of her birthday, respectively.
Albert: I don’t know when Cheryl’s birthday is, but I know that Bernard does not know it either.
Bernard: At first I din’t know when Cheryl’s birthday is, but I know now.
Albert: Then I also know when Cheryl’s birthday is.
So when is Cheryl’s birthday?
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Have you found out the answer?
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Do you want to checkout your answer, then you can click on the link to checkout for the solution ….
Feb 30
The month of February has 28 days in common years and 29 days in leap years.
Has February ever had 30 days…..?
We might have got messages of invitation for some events happening on february 30 on our phones or so…and we would have realised it is a prank message…
But has ever Feb 30 existed in our real world?
February 30 was a real date
Sweden’s 30 days of February
Historical records show that February 30 was a real date at least twice in history. Sweden added the date to its 1712 calendar when the Julian calendar was restored and 2 leap days were added that year.
In 1700 Sweden, which included Finland at the time, planned to convert from the Julian calendar to the Gregorian calendar. Therefore 1700, which should have been a leap year in the Julian calendar, was not a leap year in Sweden. However, 1704 and 1708 became leap years by error. This left Sweden out of synchronization with both the Julian and the Gregorian calendars, so the country reverted back to the Julian calendar.
Sweden’s final conversion to the Gregorian calendar occurred in 1753, when a 10-day correction was applied so that February 17 was succeeded by March 1 that year. Some people believed it stole 11 days of their lives.
The Soviet revolutionary calendar
The Soviet Union observed February 30 in 1930 and 1931 in an attempt to cut seven-day weeks into five-day weeks and to introduce 30-day months for every working month.
February 30 existed from 1930-1931 after the Soviet Union introduced a revolutionary calendar in 1929. This calendar featured five-day weeks, 30-day months for every working month, and the remaining five or six days were “monthless” holidays. The abolition of the seven-day week in favor of a five-day week was intended to improve industrial efficiency by avoiding the regular interruption of a non-working day.
However, the Gregorian calendar continued to be used in the Soviet Union during this period. This is confirmed by successive dates found in daily issues of Pravda, the official newspaper of the Communist Party, in which February had 28 days in 1930 and 1931, in accordance with the Gregorian calendar. The Soviet revolutionary calendar was discarded as it was difficult to eliminate the Sunday rest tradition. The original seven-day week was restored in 1940.
Fact or fiction: The Julian calendar
The 13th century scholar Johannes de Sacrobosco claimed that February had 30 days in leap years between 45 BC and 8 BC in the Julian calendar, when February was shortened by Augustus to give the month of August the same length as the month of July named after his adoptive father Julius Caesar. However, historical evidence relating to the Julian calendar refutes Sacrobosco, who was critical of that particular calendar.
Math & Engineering Facts about Eiffel Tower
The Eiffel Tower was built to provide the arched entrance way to the 1889 World’s Fair in Paris. It’s located in Paris on the Champ de Mars. It is an iron lattice tower named after Gustave Eiffel, the engineer whose company designed and built the Eiffel Tower.
It is a cultural and global icon of France. It is also one of the most recognizable man-made buildings in the world. Around 250 million people have visited the tower since it opened since its opening in 1889.
Like all towers, it allows us to see and to be seen, with a spectacular ascent, a unique panoramic view of Paris, and a glittering beacon in the skies of the Capital. If you want to have a panoramic view, please click on the link…Virtual Tour to Eiffel Tower
The Tower also represents the magic of light. Its lighting, its sparkling lights, and its beacon shine and inspire dreams every evening.
As France’s symbol in the world, and the showcase of Paris, today it welcomes almost 7 million visitors a year (around 75% of whom are foreigners), making it the most visited monument that you have to pay for in the world.
Since the 1980s, the monument has regularly been renovated, restored and adapted for an ever-growing public.
Let’s do some data analysis about the Visitors of Eiffel Tower……
The tower is 1063 feet tall. This is approximately equal to the height of an 81 storey building.
There are three visitor’s levels. The first and second levels can be reached by a lift (elevator) or by stairs. The third level is only available by the use of a lift.
The Tower still has two of its original elevators. Each year, the Tower’s elevators travel a combined distance equal to 2.5 trips around the world, or more than 103,000 kilometers (64,000 miles).
There are more than 300 steps to reach the first level by foot. It’s about the same to the second level. Although there are stairs to the third level this option is usually closed to visitors. It has 1710 steps. There are restaurants on the first and second levels for visitors to enjoy. It was assembled from 18,000 parts, held together by 2.5 million rivets.
The design for the structure was decided by a contest. Contestants had to submit their designs for consideration. Eiffel’s design won. Gustave Eiffel was also responsible for creating the Statue of Liberty’s internal framework. It cost 7,799,401 French gold francs to build the Eiffel Tower in 1889.
It weighs in around 10,000 tons. In the summer the poles expand, causing more weight to be added. The heat causes the metal tower to expand and cold causes it to shrink, the height of the Tower can vary with the outside temperature by 15 centimeters (5.9 inches). Winds can cause the top of the Tower to sway, side-to-side, by up to 7 centimeters (2.8 inches).
It takes approximately 50 – 60 metric tonnes of paint to paint the tower. This is done once every seven years to protect the iron from rust. It takes about 18 months for 25 painters using 1,500 brushes to repaint the entire Tower.
There is a replica of the Eiffel Tower at Paris Las Vegas Hotel in Las Vegas, Nevada (half size), and Tokyo Tower in Tokyo, Japan (full size).
It is also sometimes referred to as La Dame de Fer, meaning the iron lady.
There are 5 billion lights on the tower. It would be hard to miss at night.
The Eiffel Tower is often used in films, especially to establish Paris as the location of the story.
The Tower took 2 years, 2 months and 5 days to build. While being built it became the tallest man-made building in the world as it was taller than the Washington Monument. For 41 years after it opened in 1889, the Eiffel Tower remained the tallest building in the world until 1930, when the Chrysler Building was built in New York City. In 1957, an antenna was added to the Eiffel Tower, once again making it taller than the Chrysler Building.
Eiffel’s building remained the tallest in France until 1973. If you don’t include the antennas in total height, the Eiffel Tower is second tallest in France. The Millau Viaduct is the tallest.
When Gustave Eiffel was making the Eiffel Tower, he had to think about wind resistance. He put a curve on the outer edges so the tower wouldn’t fall. At the base of the Eiffel Tower, four curved pillars tilt inward at an angle of 54 degrees. As the pillars rise, and eventually join, the angle of each gradually decreases. At the top of the Tower, the merged pillars are almost vertical (zero degrees). French engineer Gustave Eiffel calculated that 54° angle as one that would minimize wind resistance. In interviews at the time, Eiffel said that his Tower’s shape was “molded by the forces of the wind,” notes Patrick Weidman. He’s an engineer now retired from the University of Colorado Boulder.
Weidman and a colleague analyzed the shape of the Tower. They also examined Eiffel’s original notes and blueprints. The two experts determined that a single elegant mathematical expression known as an exponential best describes the Tower’s curves. The researchers described their conclusions in the July 2004 issue of the French journal Comtes Rendus Mecanique.
The day after the Tower was inaugurated, Gustave Eiffel installed a meteorology lab on the third floor. He aslo wanted to add an elevator beacuse of the height of the tower. He had a small wind tunnel built at the foot of the tower, to put the elevator in. From August to December 1909 he carried out five thousand tests. In addition, Gustave Eiffel wanted others to perform experiments on the Tower. In fact from 1889 onward, the Eiffel Tower was used as a laboratory for scientific measurements and experiments. Only intended to last 20 years, it was saved by the scientific experiments that Eiffel encouraged, and in particular by the first radio transmissions, followed by telecommunications.
The tower moves in the wind. On days with high, gusting winds, the wind can reach speeds in excess of 100 mph at the top of the tower.
Visitors can feel the tower swaying gently at the top level. Under such wind conditions, the tower is usually closed to the public, although there is always an engineer present at the summit to monitor telecommunications equipment. The magnitude of the sway in the tower, under worse case condition, is about six inches. There is no danger of the tower being damaged by wind-induced movement since it is designed to withstand movements easily five times beyond those produced by the highest winds ever recorded. Today, the movements are monitored by a laser alignment system.
The Eiffel tower was very nearly demolished in 1909. It was saved by its use as a telecommunication tower.
The 1889 World’s Fair to be held in Paris was meant to celebrate the French Revolution’s centennial.
Gustave Eiffel was known for his revolutionary bridge-building techniques, which formed the basis for the Eiffel Tower. One lightweight bridge built by Eiffel over a waterway in Europe supported a 4-ton, single-axle oxcart, deflecting, or bowing, less than 1 inch under the strain. The project included 50 engineers and designers (who produced 5,300 blueprints), 100 ironworkers (who produced 18,038 individual pieces for assembly) and 121 construction workers (who used 2.5 million rivets). In addition to contractor Gustave Eiffel, the effort included engineers Maurice Koechlin and Emile Nouguier and architect Steven Sauvestre.
The doodle, shows the tower being painted by cheery workmen in berets and overalls, swinging from the tower marking 126th Anniversary of the public opening of the Eiffel Tower.
Is Pi = 4 ?
One might think this implies that the circumference of the circle must be the same as the perimeter of the square. After all, the new figures are becoming more circular after every operation of removing corners.
Things of Interest: Troll Pi notes that “the limit of a sequence isn’t necessarily a member of that sequence”. The succession of approximations is not necessarily the same as the final value.
Note that the limit value does not change as the perimeter changes shape. Normally one expects that successive approximations will make the value more appropriate from one operation to the next operation of removing more corners. Since the perimeter has a constant value of four, there is clearly some discontinuity between any new figure’s perimeter and a circle’s circumference.
You can repeat the operation indefinitely, adding more points, but there will always be more points off the line than on the line. Formulated as a limit, the variable controlling the number of steps is an integer, while the length of the line is given by a real number. Thus, the set of steps is, even in the limit case, longer than the diagonal.
Archimedes was trying to approximate a curve with a line, and the curve, unlike the steps, really does look flatter and flatter as you get closer and closer. The steps will always be steps, but a smaller and smaller approximation to a curve resembles a line. The limit case of a polygon really is a circle, while steps are always just steps.
The exact answer for this question comes from mathematical analysis. This zig-zag path does approach the circle, so on the surface of it you’d expect that it would have the same length. However, the length of a curve is more a function of it’s derivative (slope), and less a function of its position.
For example, if you were to throw a jumbled up 10 foot rope into a 1 foot box, you wouldn’t say that the box is 10 feet across. You’d want to straighten the rope out before you draw any conclusions.
The construction at the top (pi=4) merely shows an upper bound & it is not the exact value.
Similarly if we draw a square inside the circle, the diagonal of the circle will be the diagonal of the square…
Diagonal of the square = 1
Side of the square = 1/√2
Perimeter of the square = 4/√2 = 2√2
Now lets change the square such a way that perimeter of the new shape approximates the perimeter of the circle…
Perimeter of the new shape = 2√2
Perimeter of the circle = 2√2
So this picture gives the lower bound for perimeter of the circle
We already know perimeter of the circle = pi
From the upper and lower bounds for perimeter of the circle, we get to know the fact
2√2 < pi < 4
Hilbert’s Grand Hotel – Paradox
A lesson on the theoretical paradox involving infinity…
A never-ending hotel, always full of guests, helps to explain the nature of infinity.
Hilbert’s paradox of the Grand Hotel is a veridical paradox (a valid argument with a seemingly absurd conclusion, as opposed to a falsidical paradox, which is a seemingly valid demonstration of an actual contradiction) about infinite sets meant to illustrate certain counter intuitive properties of infinite sets. The idea was introduced by David Hilbert in a lecture 1924 and popularized through George Gamow’s 1947 book One Two Three … Infinity.
Hilbert’s Grand Hotel has an infinite number of rooms and infinite number of guests occupying those rooms. Whenever a new guest arrives, the manager shifts the occupant of room 1 to room 2, room 2 to room 3, and so on. That frees up room 1 for the newcomer, and accommodates everyone else as well (though inconveniencing them by the move). By repeating this procedure, it is possible to make room for any finite number of new guests.
Now suppose infinitely many new guests arrive to the Grand Hotel. The manager moves the occupant of room 1 to room 2, room 2 to room 4, room 3 to room 6, and so on. This doubling trick opens up all the odd-numbered rooms — infinitely many of them — for the new guests. It is also possible to accommodate a countably infinite number of new guests.
The Grand hotel live up to its motto, “There’s always room at the Hilbert’s Grand Hotel.”
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Manjul Bhargava – Fields Medal Winner
Manjul Bhargava is an Indian-American mathematician. He is known primarily for his contributions to number theory.
Description of the picture: South Korean President Park Geun-Hye, at an awards ceremony for the Fields Medals with Manjul Bhargava, of Princeton University during the International Congress of Mathematicians 2014 in Seoul.
Bhargava was born in Hamilton, Ontario, Canada of parents who had emigrated from India, and he grew up primarily in Long Island, New York. His mother, a mathematician at Hofstra University, was his first mathematics teacher. He completed all of his high school math and computer science courses by age 14. He attended Plainedge High School in North Massapequa, and graduated in 1992 as the class valedictorian. He obtained his B.A. from Harvard University in 1996. For his research as an undergraduate, he was awarded the 1996 Morgan Prize. Bhargava went on to receive his doctorate from Princeton in 2001. He was a visiting scholar at the Institute for Advanced Study in 2001-02, and at Harvard University in 2002-03. Princeton appointed him as a tenured Full Professor in 2003. He was appointed to the Stieltjes Chair in Leiden University in 2010.
Bhargava is also an accomplished tabla player, having studied under gurus such as Zakir Hussain. He also studied Sanskrit from his grandfather Purushottam Lal Bhargava, a well-known scholar of Sanskrit and ancient Indian history. He is an admirer of Sanskrit poetry.
Bhargava has won several awards for his research, the most prestigious being the Fields Medal, the highest award in the field of mathematics, which he won in 2014.
He won the Morgan Prize in 1996, a Clay 5-year Research Fellowship, the Merten M. Hasse Prize from the MAA in 2003, the Clay Research Award in 2005, and the Leonard M. and Eleanor B. Blumenthal Award for the Advancement of Research in Pure Mathematics in 2005.
He was named one of Popular Science Magazine’s “Brilliant 10” in November 2002. He won the $10,000 SASTRA Ramanujan Prize, shared with Kannan Soundararajan, awarded by SASTRA in 2005 at Tanjavur, India, for his outstanding contributions to number theory.
In 2008, Bhargava was awarded the American Mathematical Society’s Cole Prize.
In 2011, Bhargava was awarded the Fermat Prize for “various generalizations of the Davenport-Heilbronn estimates and for his startling recent results (with Arul Shankar) on the average rank of elliptic curves”.
Bhargava is also a sought-after speaker, having given numerous public lectures around the world. In 2011, he delivered the prestigious Hedrick lectures of the MAA in Lexington, Kentucky. He was also the 2011 Simons Lecturer at MIT.
In 2012, Bhargava was named an inaugural recipient of the Simons Investigator Award, and became a fellow of the American Mathematical Society in its inaugural class of fellows.
Bhargava was also awarded the 2012 Infosys Prize in mathematics for his “extraordinarily original work in algebraic number theory, which has revolutionized the way in which number fields and elliptic curves are counted”.
In 2013, Bhargava was elected to the National Academy of Sciences.
In 2014, Bhargava was awarded the Fields Medal at the International Congress of Mathematicians in Seoul for “developing powerful new methods in the geometry of numbers, which he applied to count rings of small rank and to bound the average rank of elliptic curves”.
In 2015, Bhargava was awarded the Padma Bhushan, the third highest civilian award of India.
Here are a few excerpts from the Interview of his Ideas of Mathematical Teaching
India’s Math Genius Manjul Bhargava talks to NDTV about his love for mathematics, and suggests that the teaching of math in India can be made fun and ‘less robotic’, using even magic tricks as teaching aids. He also touches on the mathematical nature of the tabla and other forms of music.
In his childhood age, he would ask his mom, “is this the answer for the puzzle? Can you tell me how to do it” and her answer always was “you should figure it out itself and then come tell me.” And so, I used to just enjoy just playing around with mathematics and if I ever got stuck, my mother was there, as a resource, and that was something very useful.
The way in which she taught him, the way she inspired him in maths that created his interest in it.
He says, “I used to find my own ways to make mathematics fun. I used to play around with mathematics, do puzzles, ask my Mom for nice suggestions that would be fun for me. My grandfather, who was a Sanskrit scholar, had lots of mathematics textbooks on his shelf from Ancient India, so I used to read those. So, I had a very non-standard mathematics education and I really enjoyed it. I think that contributed a lot.”
So enjoyment is the key to learn math…
When you’re given a problem, be creative; come up with the steps on your own and everyone will come up with a different way to do it. Mathematics is great because there is always one answer, but there are many ways to come to that answer. There’s not one path and everybody has their personal path that they can discover and that’s what makes it fun. That’s the adventurous part of mathematics, the creative part of mathematics and we should follow that in the way mathematics is learnt.
And in mathematics, one of the goals should be to develop intuition about mathematics. But because of memorization, the intuition is not developed within the learners. But if you discover the way to solve problems on your own, you develop the intuition. So when the next problem comes and is not exactly the same kind as first, you will have an intuition to say, oh these are the changes I need to make in order to solve it. That development of intuition is very important, and is very important in our mathematical research. If you will develop the intuition where you can make slight changes, just like when you are painting, you develop techniques, develop an intuition, how to express various emotions by this combination of colours, in the same way you develop an intuition for the kind of techniques and how to combine them to solve different problems when they come up.
If you want to checkout the full Interview of his Ideas of Mathematical Teaching, you can check it out in the link provided below:
How does math guide our ships at sea?
How does math guide our ships at sea? – George Christoph (TED Video)
Explore the beginnings of logarithms through the history of navigation, adventure and new worlds. Check out the above video link from which you can explore about the great mathematical thinkers and their revolutionary discoveries which have made an incredible story in our lives.
This TED Education video shows how mathematicians, inventors and explorers came together to produce a reliable method for navigating open water.
Nature & Numbers
This amazing math video is an inspirational video that shows just how beautiful mathematics can be….
Nature & Numbers
Check out the above video link to know more about the relationship between numbers and nature, featuring Fibonacci numbers, golden spirals, interesting shapes, and mathematical formulas that appear in natural objects such as flowers and shells.
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