Some call it the Potato paradox, but I prefer calling it a puzzle. It isn’t really a paradox. It’s just that we tend to get confused easily when working with such problems and most times end up with the wrong answer. Here’s the question:
You have 100 kg of potatoes. Assume that they are made up of 99% water. Now, you keep them outside to dry for a while. When measured for water content now, they contain 98% water. What do you think is the weight of these dried potatoes now?
Answer quickly first. Then, try and calculate. It isn’t really tough.
Take 15 minutes, or more if you have to. Use papers, pens and calculators if you have to. But whatever you do, stay honest. Don’t search for the solution on the internet, or don’t read further if you haven’t done it yet. Trust me, your brain won’t like the answer.
Couple of days back, I read about a puzzling geometrical conundrum, probably on Quora. It might not sound amusing to you math geeks out there, but to me, it sounded like an impossible thing at first. The sad part is, I did not save the source link in my notes. Thankfully, I did care to note down the idea. Let me call it the “Thread wound Around the Earth” puzzle.
Here is a simple question first. Try answering it without any calculation. Just guess. Be honest to yourself, don’t see the answer just below it. Scroll slow.
Imagine a piece of string wrapped around the Earth’s equator – that’s about 40,000km. How much MORE string would you need for it to sit 15cm above the ground, all the way around?
A) 1 metre, B) 1 kilometre or C) 1,000 kilometres
Thread wound around the Earth
The answer is A) 1 meter. Yes, just 1 meter of extra rope.
Suppose, you have an outrageously long thread with you. You tie it around the base of a tree, somewhere at the equator. Now, you go around the earth, along the equator, carrying the thread with you, till you come back to that tree where you started. At this point, you’ll have a thread that goes around the earth in a circle. At every point, let us imagine that the rope is taut and touching the ground (there are no mountains or valleys in between). It’s a perfect circle (assume).
Suppose, you still have an extra meter of the rope left now. So, you break the wound rope at one point and add the extra meter to it. That of course slackens the wound rope. For this rope to be taut again, it has to be lifted up by some amount. What do you think that distance would be from the ground? Assume that the rope still makes a huge circle just above the ground and lifts by equal amount at every point along the equator.
Just the extra meter of rope, causes the rope to rise by ~15 cm all around the earth (actually 15.9 cm). For a single meter of rope added to a 40,000 km of rope, that sure seems like a huge lift! But that isn’t all…
The most amazing part is that, no matter what the size of the circle, a meter of increased circumference will increase the radius by ~15 cm. Always!
Try tying a rope around a golf ball, or even try doing that around the sun. It’s always that – 1 meter increase in circumference, always increases the radius by ~15 cm.
The Math is so straightforward.
If you think about it mathematically, it is completely straightforward.
Radius X 2 X Pi = Circumference
That means, the Radius is directly proportional to the circumference of a circle. Everyone knows that. So, the amount of change in the radius is reflected proportionally in the circumference, the magnitude of radius can be anything, really. So it’s pretty natural that just a single meter of rope is required to lift the rope by 15.9 cm around any circle. The size doesn’t matter. But practically thinking, the above question makes it seem impossible.
Please hit like if you learnt something from the article.
Today is pi day. Pi day is celebrated on March 14 at the Exploratorium in San Francisco (March 14 is 3/14) at 1:59 PST which is 3.14159.
Since pi day is today’s date written in the mm.dd (03.14) format, it could not be a pi day for most of you because dd.mm is the format used for writing dates in most countries around the world. In fact, those countries where more than half of the world’s population resides, will never have a pi day because you know, we can’t have a 14th month! Pi day is a valid celebration for people living only in the United States (including the 49th and northernmost state, Alaska and Hawaii of course) and Belize. Everywhere else people get zilch today?
Only the purple parts in the map use the mm.dd.yy format to write dates
Firstly, there is always the pi approximation day, which is celebrated on 22nd July (22/7) and uses the dd.mm format. Talking about March 14th, there is much more to pi day than just the date format itself. Let’s see…
Birthdays
I know, Eugene Cernan – The NASA astronaut who was the last man on the moon, and the one you can hear speaking in a popular Daft Punk Track – is one famous man who was born on pi day, 79 years from now, is an American too.
But guess what? Albert Einstein, one of the most genius men of recent times, was born on pi day too. He was a German born physicist (He did live in the US for more than 15 years and in fact, even took his last breath in New Jersey)
Left to Right: Albert Einstein, Gene Cernan and the commander of Apollo 8, Frank Borman, have their birthdays on March 14, Pi Day.
So, you see there is a little bit of pi day for every one around the world today. It is not just an American thing. Now moving on the most amazing things about pi.
Irrational pi
Firstly, pi, unlike what we all are taught in school, isn’t 22/7. 22 divided by 7 is just an approximation of pi – it is only 99.95975% accurate. As we all know, pi is actually the ratio of a circle’s circumference to its diameter. A slightly better approximation of pi would be 104348/33215 – which is 99.99999998944% accurate. But, since it is an irrational number, it can never be written in the form of a fraction.
Exact Value
The exact value of pi is impossible to write in digits because the number of digits needed to write it would be infinite and could never be fit inside the known universe.
To think of it in another way, if you divided the whole universe into the smallest possible volumes (plank volume), you’d end up with a mind bogglingly large number of volumes. Suppose you started writing the digits of pi inside these little volumes, you’d finish up the universe and would be still left with infinite more digits to write.
The Digits of Pi
The latest record for the maximum number of known digits of pi is 12.1 Trillion digits (December 28, 2013), as calculated by Alexander J. Yee & Shigeru Kondo. They have run out of disk space to store more numbers. Here you can have a look at the first 100,000 digits of pi. And One million digits, if you need more than that.
In these first one million digits, the sequence 12345 occurs 8 times!
The Feynman Point: If you’d like to hear what pie would sound like if you mapped a couple of pleasant sounding notes to each of the digits of pi, try listening to this. If you kept listening for a while and made it till the 762th digit, you’d hear a series of (6 of them) high frequency notes (the ones mapped to the digit nine) that get played continuously. This place in the digits of pi is called the Feynman point where six 9s occur one after another. Isn’t it incredible for six same numbers to be there consecutively in a random irrational number!
The Feynman point – series of six consecutive 9s highlighted in red.
Practically useful pi
Pi can be used in real life to make a couple of things easier. For instance, if you were to find the size of your hat (usually measured in diameters), you’d have a hard time measuring the diameter of your head. This is what you can do to get a good approximation:
Measure the circumference of your head and divide it by π.
Another one trick is used by forest guards: To estimate the height of an elephant the Diameter of an elephant’s foot is multiplied by 2 π.
Pi Jokes, facts and Coincidences
It is an impressive coincidence that 3.14 if horizontally flipped, looks like the word “Pie”. You can check this in the mirror.
Pi looks lie pie in the mirror
Another one is that, the 16th Greek letter is ‘Pi’ and the 16th letter in the English alphabet is ‘P’.
The famous comedian John Evans once made a joke: “What do you get if you divide the circumference of a jack-o’-lantern by its diameter? Pumpkin π.
There is a cologne named pi and is sold with the following marketing mantra: “highlighting the sexual appeal of intelligent and visionary men.”
The height of the Great Pyramid of Giza multiplied by 2 π is equal to the perimeter of its base.
The 90841th place in pi is 122189 – which is also my birth date in the mmddyy format. Find yours here and tell me in the comments below.
Do share with me in the comments, other facts about pi you know and I haven’t covered them here.
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The logical engineer in me has always paid a lot of attention to how well my money is being put to use, or if something I bought was well worth it. So, before buying anything, I usually love to add in a basic mathematical calculation that would ensure the best logical use of my money. I used to do the same when I was studying engineering and had come up with a handful of tricks, which enabled me to eat the best food, in best quantities at the lowest prices.
Optimized Pizza Buying
Till date, I had relied on calculations for individual joints to buy the pizza that gave me the best value for money (irrespective of what my stomach could fit). In other words, I had never used statistical methods, as I always went to only 2 or 3 pizza places and never felt a need to do it statistically.
So yesterday, while skimming through blogs on NPR, I came across a post by Quoctrun Bui, where he had calculated the best valued pizza size using statistical methods.
The final findings of his study based on 74,476 prices from 3,678 pizza places were condensed into a graph which depicted data as follows (here is the link to the article for an interactive version of the graph).
Conclusion
The above graph plots 74,476 data points to find the pizza size that gives you the best value for money. The y-axis plots price-per-square-inch – the lesser price-per-square-inch you pay, the better deal you score.
This basically means that buying the largest pizza gets the most value out of your money. As the size increases the value for money increases or the price you pay per-square-inch of pizza decreases. – Statistically speaking.
Adding value
I felt a need to add value to the study by finding how well ‘buying a large pizza’ to get the best value works in India. So, I selected a popular joint Dominos (where I go all the time) for the test. I dug out their menu (probably an old one) to see if buying the large pizza always works in India. Here is a record of price you pay per square inch at Dominos for various sizes and categories of pizzas. (Click the image to see a better version)
I did not pay much attention. Please point if there is a mistake.
Conclusion (Dominos India)
No, always buying the large pizza clearly is not the best option at Dominos in India.
A small pizza is the best option (economically) if you are buying from the categories: Simply veg, Veg I or Simply Non-veg.
A large pizza is the best option (economically) if you are buying from the categories: Veg II, Non-Veg I or Non-Veg II
Never go for the small Non-veg II pizza. It is the worst choice you can make.
Never ever get a medium pizza from any category!
I’d love to see someone doing the same thing for other popular pizza joints. Do get back to me if you have done it. I’ll add it to the article as an update.
Till the year 2003, there were seven mathematical problems that had not been solved. Then came in Grigori Perelman, a Russian mathematician, who solved The Poincare Conjecture, a problem which was the first one of those seven unsolved problems.
To Grigori Perelman the prize was completely irrelevant. Sir John Ball, president of the International Mathematical Union tried persuading him for 10 long hours to accept the prize. But, he did not attend the ceremony, and declined to accept the medal, making him the first and only person to decline this prestigious prize.
6 problems yet to be solved
One down. Today, six of them still remain unsolved. Each one of those six problems carries a $ 1 Million for whoever solves it. A total of $ 6 Million to be won! For more than a century the solutions to these six problems have eluded mathematicians.
P versus NP
The Hodge conjecture
The Riemann hypothesis
Yang–Mills existence and mass gap
Navier–Stokes existence and smoothness
The Birch and Swinnerton-Dyer conjecture
Today, I’m going to talk about the first and probably the most popular problem among the six millennium prize problems.
P versus NP
The first one and one of the most vexing questions in computer science and mathematics is the P versus NP problem – polynomial versus non-deterministic polynomial. It is quite a popular one and has made appearances in TV shows like The Simpsons and Numbers and in a video game, SIMS 3.
The reason this one interests me more than the other 5 problems is because P versus NP is a problem which is the most likely, among all of them, to be solved by an amateur.
Presently it is not known if P equals NP. The problem if solved could figure which problems can or cannot be solved by a computer. Seems abstract, but if solved it could have great implications. It could dramatically affect our everyday lives.
Although mathematicians expect it to go the other way, but if it is proved that P = NP, it would make our current definitions of security obsolete. Public-key cryptography could become impossible. We could face problems with online security if wrong people get proper resources to break public key – That means it would become possible for people to break into your bank accounts, communications, emails, encrypted data etc…
Dealing with optimization problems would become easier. That means everything will be much more efficient. Transportation of will be scheduled optimally. Moving people and goods would become quicker and cheaper. Manufacturing units would be able to improve their production speed and make less waste etc…
Weather, earthquakes and other natural phenomenon would get easier to predict. We might even find the perfect cancer cure.
Until today, I had no idea that non-spherical objects could have the same diameter at every point! Don’t believe me? Have a look at these wonderful little metal objects that aren’t anything even close to a sphere and are still able to roll a flat surface on them, as spheres would. [Video]
Like Gombocs, these shapes are a Mathematician’s fantasy. They have a generic name – constant width objects. In fact, these carefully machined metal objects could be perfect gifts for your mathematician friend. I know, I would some day, if you think you need these too, you can buy them here. (I’m in no way related to Grad-Illusions, nor am I an affiliate marketer)
Note: Though in some manner they look like Gombocs, they are not Gombocs. To know more about what Gombocs are, read this – [Gomboc – An Object That Never Falls]
How are they made?
Theoretically, to understand how they are made, you need to understand that the 3D objects of constant width are usually* spun out of a 2D object – Just like a sphere can be made by spinning a circle. Though there is a kind of constant width 3D object that is not a spun version of any 2D curve.
The 2D form is called the Reuleaux triangle and it looks like this [image]. The one shown in the link is a constant width curve based on an equilateral triangle (triangle with equal sides). It turns out, you can construct a constant width curve out of any triangle, and a polygon too (like the 50 pence coin shown above made out of a regular heptagon). To make a constant width curve using an equilateral triangle, all you need is a compass, a paper and a pencil:
Draw an equilateral triangle.
Put the point of your compass on one vertex.
Trace out an arc that starts from one of the other vertex and ends at the third one.
Repeat the same for rest of the two vertices. There! You have your constant width curve. Cut it out of paper.
Now around an axis dividing the shape into half, spin it. You have a theoretical 3D shape that resembles the one shown in the video above.
Vehicle tyres and square hole drills
So, since these shapes can roll things around like circles can, wheels could be made of these shapes too! Then, why aren’t wheels made that way? That is because when these shapes roll, they don’t have their centers at one place. If vehicles had tyres like these, engineers would have had a hard time designing axle systems.
They are in fact used in a Wankel engine. And since the center traces a square when Reuleaux triangle rolls, they have been used in drills that can drill out square holes.
Random constant width facts:
There are a few pencils which are manufactured in an extruded-reuleaux triangle shape. These pencils can roll around smoothly like circular pencils.
For some reason, even guitar picks are often manufactured in these shapes.
Instead of spinning around a constant width 2D object, a 3D constant width object can be made by modifying flat tetrahedron faces using intersections of sphere faces. As it can’t be done on a lathe, these are particularly hard to machine. It is called the Meissner’s tetrahedron or the Reuleaux Tetrahedron.
Benford’s law is a fairly simple law to grasp and it will blow your mind. It deals with the leading digits of numbers.
So, for example, you have the number 28 – The leading digit for it would be 2. Similarly, the leading digit for 934 would be 9. Just pick the first digit. Now…
In a data set you’d say – it is common sense to assume that the probability of leading digit one (1) appearing would be more or less equal to that of leading digit nine (9).
As there are 9 possible leading digits, you’d think that the probability of each leading digit would compute to something around 0.11
You’d imagine that it would be normal to assume a nearly straight graph of probability vs. leading digit. But this isn’t true.
Benford’s law says
Your common sense fails. What actually happens is that the likelihood of 1 appearing as the first digit in a data set is around 0.3
For the following digits, the probability keeps decreasing. And the following graph appears. You’ll see that the numbers rarely start with nine!
When does it work?
This counter-intuitive result applies to a wide variety of natural data sets. It works the best if your set spans quite a few orders of magnitude. Natural set of data like stock prices, electricity bills, populations, which could range from few single digit values to several digits work the best. Other data like the heights of people doesn’t work because it does not span “quite a few orders of magnitude”. Also, artificially tampered data fails to comply because the person who tampers does the same mistake everyone does. Therefore, Benford’s law is also used to detect frauds in data.
Example:
Count the number of data points in a data set which have the leading digit 1 and write the number next to the number 1 in a table.
Then, keep repeating it for all the numbers 2, 3, 4 and so on.
Calculate the probabilities for each. In the end you’ll be left with a table that would look something like this. (Probability = Number of Data Points for that digit / Total Data Points)
There is no astrologer in the world that can tell you for sure if you’ll die this year or not. But, thanks to Gompertz Law, if you ask me, there is one thing I can tell you for sure – Whatever may be the odds of you dying this year, in 8 years, the likelihood of you dying will double.
This dreadful law of death was named after the first person who noted it – Benjamin Gompertz, in the year 1825. The law rests on a general assumption that a person’s resistance to death decreases as he ages. The Gompertz Law of mortality, put simply in a sentence would compute to this:
Your probability of dying during a given year doubles every 8 years.
It is amazing, and no one knows how it works exactly. Why does nature pick the number 8, to double our likelihood of death? We’ll probably never know.
There is a whole table which relies on census data, and statistically notes the probabilities of people dying at different ages. And when it is plotted on a Probability of death vs. Age graph, you get an exponentially increasing mortality rate with age. That is death coming faster as you get older.
Gompertz Law can be verified for real-life data – the 2005 US census data. The following graph and the probability vs. age plotted using the law match almost perfectly. Amazingly, the law holds true for several other countries too.
That means, the probability of me, a 25-year-old dying during the next year is very small — about 1 in 3,000. When I become 33, this probability will grow to something around 1 in 1,500. In the next 8 years, the probability of me dying will be 1 in 750, and so on…At the age 100, the probability a person’s death will be about 1 out of 2 – fat chance of successfully moving on to 101!
Theoretically, using this data, it can be said with 99.999999% certainty that no human will ever live to the age of 130 (of course only if medicine doesn’t start tampering with human genes, or some other artificial factor). There is one thing for sure – there is almost no chance that you are going to beat Mr. Ming.
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To appreciate the beauty of mathematics and nature there is no escaping without learning about a Klein Bottle. A three-dimensional representation of a Klein bottle looks like this – [image]
There are number of phrases you can use to describe (not exhaustively) it. A few of them are as follows:
An object with no boundaries.
An object with no inside or outside.
One sided surface.
Non-orientable surface
Wikipedia describes it as:
The Klein bottle is a non-orientable surface; informally, it is a surface in which notions of left and right cannot be consistently defined.
Simplifying things: A Möbius strip is a simpler example of a non-orientable object. That means it has no inside or outside. Add another aspect – having no boundaries – to it, it gets more complex and you end up with a Klein bottle.
If you haven’t heard of Möbius strips, to understand such surfaces, you can make one for yourself now.
Tear off a strip of paper.
Hold it horizontally, straight with both of the short edges in your hands.
Now, twist one of the edges by 180 degrees and join the two short edges. You’ll have something like this in your hands – [image]
Test the surface and edges: On this object you just created, move your finger along the surface. You’ll find that your finger comes back to the same place eventually. There is no inside or outside for this object, there is just one surface.
The same thing happens with its edge (try moving your finger along the edge). Here is a Music box playing a Harry Potter theme continuous – forward, inverted, forward and so on – manner; Relevant video: [video]
Now spin it (the Möbius Strip) fast. You can NOT practically do it. I mean, spinning it like you spin a circle and get a sphere. There! You have a Klein bottle. It is better than a Möbius strip in a way that it (Klein Bottle) has no boundaries.
Klein bottles cannot actually exist in our three-dimensional worlds, the ones that look like them (Klein Bottles) are just 3D representations of a 4D object. Like a two-dimensional drawing of a 3D cube. These models are available for you to buy. Interestingly, in spite of having no inside or outside, they can be filled with a liquid. But, given the opposing force of air, they are pretty tough to fill. It is important to note that the 3D representation of a 4D Klein bottle has an intersection of material, this doesn’t happen in 4D. It is like the intersecting edges of a 3D cube in the 2D representation.
If you are having a tough time imagining this 4D object, the following 4D animation might help (or leave you perplexed) – [video]
[Extra reading for math geeks] as if they already didn’t know about Klein bottles.
There’d be hardly anyone among us who hasn’t played with a roly-poly toy during their childhoods. If you know it by some other name, you could think of it as a toy that never falls, no matter how hard you hit it, and sells in variants which look like this. That isn’t exactly what a Gomboc is, but you get an idea about what it does – It does not fall. For more, read on.
What is a Gomboc?
A Gomboc (Gömböc) is a mathematical 3-D shape which has only one position in which it can stand and is made up of a single material of uniform density. If you try to make it stand in some other way, or try to knock it down, it moves back to that single stable position, gradually. When placed on its side, it starts rocking magically, gains momentum, straightens itself and gradually comes to rest in that single position. Here is a video of a Gomboc doing its thing.
A Gomboc is an object surrounded by a number of complex curves, it takes an immense amount of accuracy to get the surfaces right. An accuracy of the orders of around 1/10th of a human hair’s thickness is required for it to work properly. For better, people have started 3D printing these complex shapes.
The world’s largest Gomboc was displayed in China in the year 2010 which measured around 3 meters in all directions.
Terrestrial tortoises, who use a similarly shaped shell to get on their feet when turned upside down, were using it long before humans had found a way to construct it. The first time we made it, was in the year 2006. Evolution got there first!
How is it different than a Roly-Poly toy?
A roly-poly toy usually has an internal counter weight made up of a heavier material. But a Gomboc is made up of a single material.
Uses: Use it as a paper weight or to gift it to your friend who is a math geek. Tortoises use it to save their own lives.
Where can I buy one?
You can get one for yourself from an official website of the inventors – Here.
There exist a few unusual units of measurement which you must have never heard of, or would have never thought of them as units until now. Here is a list of 10 of the many fancy units of measurement.
Note: These units are not official. They’re often used for their humor value or for simplicity’s sake):
1. Car length – It is not a very unusual unit of measurement and is used normally to mention the braking distance of a vehicle. Deriving its length from a typical car’s length, 4 meters is referred as one “car length”. You must have heard one spy advising another spy to keep a 2 car length distance from a vehicle to avoid detection.
2. Nanoacres – A measure of area which is equal to about 4 sq.mm (4.0468564224 sq.mm exactly). It is the area of a single VLSI chip which is square in shape and measures 2 mm on each side. This unit is widely popular as a joke among electronic engineers – who often are known to make quips about VLSI nanoacres having costs in the same range as real acres.
3. Grave – It is a unit that measures mass and equals 1 kilogram or 1000 grams. Grave was set to be the standard unit of mass for the metric system, but it was replaced by kilogram in 1799. [read more about it]
4. Moment – Moment is actually something that was used to measure 90 seconds during the Medieval times. But for modern times, the Hebrew calendar’s definition of moment makes more sense. According to it, a moment is equal to 5/114 of a second or around 0.0438 seconds. [read more]
5. Jiffy – Jiffy is used popularly as an informal time in English. Think of someone saying “I’ll be back in a jiffy”. But, we’ve never thought of it as a unit. Also, every field has a different definition of Jiffy.
Early usage – 33.35 picoseconds or the time take by light to travel 1 cm.
Electronics – 1/50th or 1/60th of a second, depending on the AC power supply frequency.
Computing – Typically anything between 1 millisecond to 10 millisecond. Commonly: 10 ms.
Animation – The time interval between each frame of a dot GIF file or 1/100th of a second or 10 ms.
Physics/Chemistry – Time taken by light to travel 1 Fermi or 3X10^-24 seconds.
6. Dog Year – Based on a popular myth that dog’s age can be calculated in human years by multiplying it with 7. So, a single Dog year comes to around 52 days (365/7 – Days in a human year divided by 7)
7. A Bible – Used as measure of digital data volumes. It is like measuring the size of a disk in number of movies it can fit which I used in this article. A single Bible in uncompressed 8-bits, has around 4.5 million characters and 150 of them can be stored in a single CD. Hence, a bible can be measured to be approximately equal to 4.67 Megabytes. Similarly, Encyclopedia Britannica and Library of Congress are used to represent much larger data volumes.
8. Kardashian – Yes, it is named after the 72 day marriage of Kim Kardashian to Kris Humphries. Of course, it measures 72 days of marriage. So, a 25 year marriage would amount to around 126.7 Kardashians.
9. Wheaton – Used to measure the number of twitter followers relative to the popular celebrity Will Wheaton. This became a standard when he had 0.5 million Twitter followers. Today, Will Wheaton himself has 4.88 Wheatons. I, for instance, with 210 followers, have about 0.00042 Wheatons.
10. Warhol – Derived from the widely used expression coined by Andy Warhol – “15 minutes of fame” – 1 Warhol measures exactly what you’d expect it to – 15 minutes of fame. Yes, it measures the amount of fame.
Consequently, 1 kilowarhol is equal to 15,000 minutes of fame or 10.42 days and 1 megawarhol measures 15 million minutes of fame or about 28.5 years.
