Category: Mathematics

Winning and Losing

By Anupum Pant

Think about it. A stranger comes up to you while you are queued up for the morning coffee and says this to you – “I’ll toss a coin, you call. If you win, I pay you 10 bucks. If you lose, you pay me 10.”

Would you take the bet? Most wouldn’t. There’s a lot going on here. Firstly, the strange man might have a trick up his sleeve that would tilt the odds in his favour, you’d think. But here’s the deal, even if you are 100% sure that the man is being honest with you and offering you a completely fair coin toss, most still wouldn’t take the bet. Why not? After all it seems like a totally fair deal.

In a social experiment, it has been seen that even if the man offers you a 20 to your 10, most still would not take the bet. Yes, they do have a chance to lose 10, but they might even take home double hat amount. There’s an equal chance, but the gain is clearly in your favour. Mathematically you are getting a great deal. Why wouldn’t people still take it?

That is probably because humans see losing differently as winning. That means, losing 10 would affect you more that gaining ten would. Losing 10 would make you more sad than the amount of happiness you’d experience when you’d ten would make 10. So much that even losing 10 moves you more (downwards) than gaining ten would move you (upwards). So, most people won’t take this bet because throughout the years they’ve been learning (subconsciously) how losing is more painful.

In my view, it has something to do with your attachment to what you own too – materialistic attachment as they’d say. This of course was about money, but as Derek Muller states in the following video, this simple way of how we look at winning and losing affect us is much deeper manner. Or simply, money is just a metaphor here. This comes into play when you make other life decisions also. You’d avoid risks where the difference between the magnitude of gain or loss from the result isn’t much.

10 such bets being offered consecutively is a much favourable choice mathematically, and people mostly would take it, if they had that sort of money in their pockets. When asked to explain why, they base it on intuition. Our minds sure work in a very complex manner.

The Taxicab Number

By Anupum Pant

Srinivasa Ramanujan was a natural genius and had his own way with numbers. With no formal training whatsoever, and immense hard work, he  got his work noticed by the mathematical community. What Ramanujan could do with numbers, indeed was extraordinary. G.H. Hardy a famous English mathematician paid attention to Ramanujan’s letter, invited him to work with him and began a partnership with him. However, Ramanujan was a strict vegetarian and couldn’t sustain a healthy stretch there in England. He often fell sick and passed away at a very young age. In his short time during which he was here, he contributed a lot to the mathematical community.

Ramanujan is not a very well known personality. People from the mathematics community, Indians and others know about him, yet most people haven’t ever heard of him. Some people however still do remember him and have their own way of paying personal tributes to this born genius. One such man is a famous producer and writer, Dr. Ken Keeler – Also an Applied mathematics Ph.D from Harvard. In the popular animated show Futurama, he often brings pseudo-hidden references to certain numbers which are his clear ways to pay tribute to Ramanujan.

One number that often appears in the show is the number 1729. Here’s a story that illustrates Ramanujan’s genius, the importance of this number and the hidden tributes from the writer of Futurama –

G.H. Hardy once rode a taxi to visit Ramanujan at a hospital. Ramanujan was sick and upon the arrival of his advisor, he asked him the number of this cab on which he travelled. Hardy told him it was a rather uninteresting number, the number 1729. Ramaujan didn’t find it uninteresting and said:

No! It is a very interesting number; it is the smallest number expressible as the sum of two [positive] cubes in two different ways.

That is to say, it can be written as (10)^3 + (9)^3 and also as (12)^3 + (1)^3 and also is the smallest number which can be written like that – as sum of 2 cubes in 2 different ways. It was incredible for Ramanujan to have suddenly conjured up such a mathematical visualization out of a seemingly uninteresting number. Since then the number 1729 has been known as the taxicab number and is denoted as  denoted Ta(n) or Taxicab(n) in mathematics.

There are also other taxicab numbers (smallest numbers of that sort) of higher orders which can be written as the cubes of 2 numbers in 3 ways, or even 4 ways…or more. Here are some of them. 1729 is a taxicab number of order 2. 87539319 is of order three. 6963472309248 is of the order four and so on…

taxicab numbers

I first found out about this from Author Simon Singh. He talks about 1729, Ramanujan and other taxicab numbers in the numberphile video below.

Prime Numbers and The Prime Spiral

By Anupum Pant

There isn’t any defined pattern yet that describes the distribution and spacing of prime numbers among integers. But, the distribution of prime numbers might not be as random as it seems.

PrimeSpiralGrid_800

During one boring lecture, a polish mathematician was doodling in his notebook. He was writing all the numbers in a spiral, shown below. Then, he started circling the prime numbers and noticed there was something different about how prime numbers arranged themselves in this spiral.

primesp

He found that there was a great tendency for prime numbers to come in one of the diagonals. At first, it might look like the human brain is trying to make a pattern out of randomness, it actually isn’t so. Numberphile compares it with a random pattern of dots to clear that up. The video also shows some other startling prime number patterns which have been noted.

[Video] Rienmann Hypothesis Explained

By Anupum Pant

Earning a million dollars is not an easy task in any way. But the most difficult of all the ways to earn a million dollars is probably to solve one of the most difficult problems in mathematics ever – the Millennium problems. Rienmann Hypothesis is one of the seven millennium problems, out of which only one has been solved yet. Numberphile explains what this problem is about.

The Number of the Beast

By Anupum Pant

666 is probably one of the most infamous numbers and is known by many as the number of the beast. That is because the Bible, as translated in English, mentions 666 as the number of the beast. Revelation 13:18 says this…

Let the one who has understanding calculate the number of the beast, for it is the number of a man, and his number is 666.

However, it is very interesting to note that, since the Bible wasn’t written originally in English, the number 666 wasn’t actually there in the original form. In the original Greek manuscript, a language (like Hebrew) which uses letters for numbers, the number is written as 3 letters. I did not know this before. That means, all of the Greek and Hebrew text can also be read as numbers!

This is called isosephy – Meaning a practice of writing text where a text can be a number too. Normally the number values associated with each letter of a word are added to form a number.

So, the letters of my name (A, N, U, P, U, M), in order, would be the numbers: 1, 50, 300, 70, 300, 40. Total: 761 – Which is not very close to 666, I’m not a beast. What about yours? You can look at the table below and calculate.

Nero Caesar written in Hebrew can be converted to numbers and the total is 666. A way of saying, by the author, that Nero Caesar is the root of all evil. At the same time, the author doesn’t end up in trouble for writing this.

ris5en

 

Also, all the 36 numbers on a Monte Carlo roulette wheel add up to 666. Watch more in the video below.

The Birthday Paradox

By Anupum Pant

Imagine you meet a random person in the street and ask him/her when is their birthday, there’s a huge chance that the person’s birthday will not be the same as your birthday. In fact, the probability of both your birthdays being on the same day is around 0.27%. Fat chance. At the back of our heads, this is something that is very clear to all of us.

Again, if you repeat this by asking about 22 people the same question, the chance of you finding someone having the same birthday as yours is still around 5%. Too less. This is too is a very intuitive piece of information.

But consider this. If I put all of the 22 guys and you in a room, there’s a big chance that 2 people in that room will have the same birthday – a 50% chance. Moreover, if there are 70 people in the room, this chance increases to about 99.99%. This is called the birthday problem or the birthday paradox.

So, what changed when 20 people went into the room? It was just the fact that in the room, we are picking 2 people from a group of 23 people. That is equivalent to this – everyone is asking everyone their birth dates. Everyone doing it simultaneously makes the probability much higher. The probability of two people sharing a birth date among a group of 23 people is far higher than you alone going around and asking all the 22 people, and finding someone having the same birthday as your’s.

Suppose there are 200 people in the room. The probability of 2 people sharing their birthday is massive (and yet not definite). There is in fact a 99.9999999999999999999999999998% chance!

1024px-Birthday_Paradox.svg

Finally, if you had 367 people in a room, at least a pair among these 367 people in the room would definitely have the same birth date. The 99.99% chance shoots up to a definite (100%) probability if there are 367 people in the same room. Think about it for a minute.

Wilson Primes

By Anupum Pant

Thanks to the guys at Numberphile for introducing me to Wilson primes. Although the piece of information that describes Wilson primes itself has more or less no practical use, I still think it’s a good thing to know.

The first thing you need to know is that all prime numbers follow this rule – If you take a prime number P and put it in the following equation you get a number that is perfectly divisible by the prime number P.

The equation: (P − 1)! + 1 = Q

Note: ! is a sign used for factorial. That means P! is equal to the product of all natural numbers smaller or equal to P. So, for example, 3! = 3 X 2 X 1

This rule is valid for all prime numbers and no composite numbers follow it. So, for instance, if you take a composite number for P, the number you get after you put it in the above equation is never divisible by the number itself. This is called the Wilson’s theorem.

Wilson primes (P) are a few special numbers which can divide Q in the equation above two times. So, for example, since 5 is a Wilson prime, you get 25 if you put it in the equation above. And 25 can be divided perfectly by 5 once, and the result (quotient 5) can be divided again by 5 to get a whole number.

Now, for Wilson primes here’s the deal – 5, 13 and 563 are Wilson Primes. And a very interesting thing to note here is that, in spite of all the computing technology we have in the world, these are the only three Wilson primes we know yet.

Mathematicians are pretty certain that there are several other Wilson primes waiting to get discovered, probably infinitely many. But one thing is for sure, below the number 20,000,000,000,000 5. 13 and 563 are the only three which exist.

A Piece of Paper as Thick as the Universe

By Anupum Pant

Linear growth is only what we can visualize well. Estimating things that grow exponentially, is something not many of us can do properly.

Here’s what happens when you fold a piece of paper. A paper of thickness 1/10 of a millimetre doubles its thickness. On the second fold it is 4 times the initial thickness and so on. It doesn’t really seem like it would grow a lot after, say, 10 folds, right?

After 10 folds, the paper which was about the thickness of your hair, turns into something that is as thick as your hand.

Without any calculation, how thick do you think would it become if you could fold it 103 times?  (I know, no one has ever folded a paper more than 12 times)

Think about this for a second: How many times do you think would you have to fold a paper to make it 1 kilometre thick? The answer is 23. Yes, it takes just 13 more folds to go from the thickness of a hand to a whole kilometre.

Turns out, if you manage to somehow fold a paper 30 times, it would become 100 km tall. The paper would now reach the space.

For the sake of imagining how exponential growth works, a paper folded 103 times would be about 93 Billion light years thick – which is also the estimated size of the observable universe.

Watch the video below to see one other great example of how exponential growth can mess with you.

An Elegant Proof of the Pythagorean Theorem by a Former US President

By Anupum Pant

Abraham Lincoln, the 16th president of the United States, was probably a math whiz. Doesn’t it sound like an extremely rare combination of things a person could possibly be? – a president and a math whiz. And yet, he was not the only one. James Garfield, the 20th president, was very much into mathematics too.

Garfield wasn’t a professional mathematician. He was a president. But, much like Abraham Lincoln, he was very much into geometry! Before he went into politics, he wanted to become a mathematics professor.

While he was a member of the US house of representatives, five years before he was elected president of the US, he came out with a very elegant and unique proof of the Pythagorean theorem (yes, another one of those Pythagorean theorem proofs). Here’s how he did it with the help of a congruent flipped triangle…

Doing it with a piece of paper is really easy…

Pythagorean theorem proofFold a paper and cut 2 exactly same right-angled triangles out of it. Now, put them together as shown in the image here (click the image). Next, write down the area of the trapezium – (a + b) . ½(a + b) – 1

Now write the area of all the 3 triangles and add them. This is what you’d get – 2 x ½ ab + ½ c2  – 2

Since both these areas are same, just written in a different way, equate them and solve. You’ll end up with the Pythagorean theorem!

a2 + b2 = c2 

Or, simply watch the video to understand better…

Unsolvable Problems – A Math Story With a Moral

By Anupum Pant

True Story

Back in 1939, a first year doctoral student at Berkeley, George Dantzig arrived late for a statistics class one day. On the board, professor Jerzy Neyman, a renowned mathematician, had written two problems, and it wasn’t very clear to George what he had written them were for. As any other student would assume, George assumed them to be homework problems and noted them down.

He went back and started working really hard on those problems. They seemed a little harder than usual to him. Nevertheless, George was determined enough. After a couple of days, when George was satisfied with his solution, he went to his professor and apologized to him for taking so long to finish the homework. Without looking at what he had done, the professor told him to put the work on his table, and he’d see it later. George did exactly that.

Six weeks later, on an unsuspecting Sunday morning, at 8:00 in the morning, George was awakened by a frantic knock on the door. It was professor Neyman. With a pile of papers in his hands, he seemed very excited. It was only then, through professor Neyman, that George came to know what he had done on those papers six weeks back.

Six weeks back, those two problems which George mistook for homework turned out to be two examples of unsolved statistics problems Neyman had written on the board. George had unknowingly noted them as homework, and ended up solving the 2 unsolved statistics problems.

Later the papers on these problems were published. However the second one was published much later, in the year 1950.

Moral: When people are not tied down by prejudice, by putting in good work, they often manage to achieve extraordinary things.

Via [Snopes]

Cutting a Round Cake on Scientific Principles

By Anupum Pant

Background

For years the phrase “cake cutting” has conjured up just one image in my brain – A triangular section of the cake. This way of cutting a cake is so normal that even the tools (especially the spatula) that are made for cake cutting are made in a way that’d work with best when you are making that traditional triangular cut. Turns out, this method of cutting a cake which we’ve all know for years is totally wrong.

Why is it wrong?

It’s wrong mostly for mathematical loners. People who, on their birthday, have no one around to share the cake with, and cannot finish off the whole cake. For them and the ones who have to store the cake after cutting it, are extremely careful about how moist the edges remain when they next eat it, this right way to cut a cake might be of great importance.

The way we’ve always know is “wrong” because when you cut off, say a single section of the cake and decide to store the larger piece in the fridge, some internal part of the cake remains exposed and it dries off. So, the next time you cut off a piece near the area where you started, you’d get a freshly cut moist wall of cake on one side, and a repulsively hard dried up wall on the other. That, some think, is extremely unpleasant.

What’s the Right way?

About 100 years back, a brilliant Polymmath (and a mathematician), Sir Francis Galton, faced a similar annoyance. So, instead of cursing others for having invented an absurdly inefficient way to cut a cake, he decided to develop his own. He ended up developing a very simple and efficient cut which helped him keep the cake wall relatively moist. Here’s how the cut works. (Cut along the dotted line)

the right way to cut a cake

Describing his new way of cutting cakes, he got an article published in the Nature magazine (dated December 20th, 1906). “Cutting a Round Cake on Scientific Principles”

Alex Bellos from the Numberphiles describes it in a video below:

Six Weeks and Ten Factorial – Bizarre Math Coincidence

By Anupum Pant

The number of seconds in 6 weeks might be of little importance to anyone. However there is an interesting bit of trivia related to it, or call it a bizarre mathematical coincidence. Here it is…

The number of seconds in 6 weeks, or 42 days (The answer to life universe and everything) equates to:
6 X 7 (days) X 24 (hours) X 60 (minutes) X 60 (seconds) = 3,628,800 sec
The number 362,880, on the first glance, looks like very random number. Now here is what this number is equal to…

10 factorial (denoted by 10!).
Or simply, 10 X 9 X 8 X 7 X 6 X 5 X 4 X 3 X 2 X 1 = 3,628,800

Down to a single second, the number of seconds in 6 weeks is exactly equal to the numerical 10! Very strange!

One thing you could do is split the 6 weeks calculation into factors, and see it for yourself. The result is all numbers from 1 – 10. The most amazing factoring I’ve ever seen.

If you are too lazy to calculate it yourself, go to this WolframAlpha calculation and see it for yourself. It subtracts 10! seconds from 6 weeks (the result is exactly 0). Apples and Oranges, I know, but the 6 weeks refers to seconds in 6 weeks, here.

6-weeks-and-10-factorial

Salutes to the person who discovered this.

First seen on [Reddit]

The New York Pizza and Subway Ride Connection

By Anupum Pant

This is funny and interestingly true at the same time. According to Eric M. Bram of New York, there’s an unspoken rough law that says – there’s a connection between the price trends of a pizza* and a subway ride ticket in the New York city.

Their prices have remained more or less same for about 50 years now, moving up and down together! If one goes up, the other has been seen to move in the same direction too. The trend has been observed to be more or less valid for the last half century. 

Don’t ask why. I say, just go with the flow.

It was first observed in the year 1980 by Eric M. Bram. He observed that there was an uncanny similarity in the price of a slice of a pizza and a subway token in the year 1960, in the NYC – Both 15 cents.

In the 70s pizza started costing about 35 cents. So did the subway token! and so on…

Fairly recently, in the year 2002, when the price of pizza was about $2.00, and the price of a subway token was $1.50, it was predicted that the price of the subway ticket would rise. And as the “law” states, it did!

Again in the year 2005, the price of pizza went up to $2.25 and it was predicted that the subway token price would rise. It did again!

Once again in the year 2007. In 2013 the fare moved to $2.50, again just after the average pizza price had risen to the same price. No one probably knows why.

With 50 years of record, I’m pretty sure that the trend is going to keep up. Some one from the future, living in New York could confirm it in the comments section below.

So, if someone starts mass producing pizzas, floods the pizza market in the NYC, and drops the price, will the Subway ticket drop too? Someone should try that. 

*When we say pizza here, specifically, we are talking about a plain tomato + mozarella + crust pizza only.

A Fun Way to Multiply Numbers

By Anupum Pant

Please note, in the heading I said, a fun way to multiply numbers, not necessarily a quick way. Widely touted as an “amazingly quick Japanese method to multiply”, I think firstly, it really is not a very quick method. Secondly, I couldn’t find any sources confirming that it is a method developed by the Japanese. In fact, I’m not even sure if there’s anything Japanese about it. Nevertheless, the method sure is fun and should work great for people who don’t remember the multiplication tables well.

Another great thing about it is that it is a multiplication problem turned into a visual counting  problem. Since multiplication exercises don’t really make kids happy, they’d definitely love to count intersections instead (multiplication disguised intelligently).

Of course the counting can be used for single digit numbers too, but that won’t be too useful. For slightly more complex problems involving 2 digits like 32 X 42, it could be a life saver. It’s a fairly simple 3-step process. Here’s how you do 32 X 42 with it…

Step 1Step 1: The best way to go about it is by starting from the top left. First, you draw the 3 lines for the 3 of the number 32. And then you make 2 lines for the number 2, as shown.

Step 2

Step 2: Next make 4 lines and 2 lines intersecting the previously made lines as shown. Clearly, 4 lines for the 4 of 42 and 2 lines for the units place of 42.Step 3

 Step 3: Count the number of intersections in the far left (a), centre (b), and the far right (c). (a), (b) and (c) are 12, 14 and 4 respectively, for this problem.

The 1 from 14 gets carried to the number just at the right of it – 12 of (a), and (a) becomes 13. A similar carrying of the ten’s place to the immediate right column happens if there are any 2 digit numbers. So you are left with 13, 4 and 4 now. 1344 is the answer to 32 X 42.

This can be done for 3 digits too and more…
If there’s a zero, you could make a line and not count any intersections with it. As it has been shown in the video below…

Please hit like if you learnt something from this article.