Tag: mathematics

Posted in History & Literature

Lo Stivale

Posted on by Jinavie

The Italian peninsula is nicknamed “Lo Stivale” (“the boot”) because of its iconic geography. Every child who has ever seen a world map will know this iconic boot-shaped country.

But hypothetically speaking, if Italy was actually a giant boot, what shoe size would you have to be to fit it?

Shoe sizing varies across the world. In Korea, Japan and Taiwan, the Mondopoint system is used where the foot length is measured in millimetres (the width is also considered).

But if you come from an English-speaking country, there is a good chance you are more familiar with the UK and US number system, typically ranging from 3 to 13.

The UK sizing system uses the length of the last that is used to make the shoe. A last is a model of a foot that can fill the entire cavity of the shoe. Because you typically need 1-1.5cm wiggle room for your toes, the last is bigger than the foot that would eventually wear the shoe. Instead of simply using the length of the last in millimetres, UK shoe sizes use a strange unit called the barleycorn.

The barleycorn originates from the 19th century when an inch was defined as the length of three barley corns (or grains). Hence, a barleycorn is ⅓ inch. For adult shoe sizes, a size 1 is 26 barleycorns, or 8 and 2/3 inches (220mm). For every size you go up, you add one barleycorn. This means a size 11 is 12 inches, while a size 10 is 11 and ⅔ inches.

Essentially, this means that your UK shoe size is:

(3 x heel-toe length of your foot in inches) – 23 (accounting for the toe wiggle room).

It is important to note that every manufacturer takes their own liberty with sizing, so this will often be inconsistent and can vary up to an inch, especially for women’s shoes. The US system starts counting at 1 instead of 0, meaning that you just add 1 to the equivalent UK size.

Now that we know how sizes work, let us size the Italian boot.

By rough estimate, the “sole” of the peninsula is approximately 360km long. This is accounting for the bend in the middle, as the heel height is tall. To use our formula, we must convert this into inches, which equals 14,173,200 inches.

Ergo, the shoe size calculates as follows:

= (3 x 14,173,200) – 23
= 42,519,577

Whether you use the UK or US sizing, the boot is roughly a size 42.5 million. Or, if you live in Korea, the shoe size would be recorded as 360 million. Either way, that is one big shoe to fill.

Posted in Science & Nature

Airplane Game

Posted on by Jinavie

You are cordially invited to a game that lets you earn money very easily. The game works like this:

  1. You pay $1000 to be recruited as a passenger to a plane.
  2. There are 8 passengers, managed by 4 crew members, who have 2 co-pilots above them, co-ordinated by a captain at the top.
  3. Everytime the “plane” is filled with 8 passengers, the captain retires and is paid out $8000.
  4. When the captain retires, the plane is split into two planes and everyone else is promoted one step higher (co-pilots each become a captain, crew become co-pilots, passengers become co-pilots).
  5. When each plane fills with 8 new patients, the captain of each plane gets paid out $8000 and retires.

This seems like a very easy way to earn money. Where else could you invest money and guarantee a 700% return, only needing to recruit 7 new people into the game?

The problem with the airplane game is that it is a classic example of a pyramid scheme. At first glance, it seems that the payout of $8000 is guaranteed because it seems that the promotions will keep coming.

But if you look at the mathematics, 8 people need to participate before the first player wins. 16 people have to participate for the second player to win. 80 people have to participate for the tenth person to win. If you are the one-thousandth person to join the game, you need a total number of 8000 people to be playing the game before you are paid out. At the end of the game, 87.5% of people playing will have lost money because they will never be paid out.

This is how simple exponential growth can result in a very real fraud, resulting in thousands of people losing their hard-earned money.

Posted in Science & Nature

Exponential Growth

Posted on by Jinavie

Imagine that you have won a strange lottery where they give you two options of payment: they can either pay you one million dollars up front, or they can pay you one cent on the first day, then double the amount you have every day for a month (i.e. 1 cent on day 1, 2 cents on day 2 etc.). Which would you choose?

It may seem obvious that the $1 million up front is far better than accumulating a few cents every day. But by the end of the month (day 31), you would actually have accumulated $5.37 million. How did this happen?

The secret to this extraordinary increase is the power of exponential growth. If you double a number constantly at a regular interval, it grows at a staggering rate. Let us look at the above example again.

On day 1, you have 1 cent. By day 10, you already have 2(10-1) = $5.12. Now we can see that instead of mere cents, we are gaining $5 in one day.
By day 15, you have $163.84. Now the doubling nets you another $163.
By day 20, you suddenly have $10,485.76.
We pass $1 million at day 28 where we have $1.34 million.
Day 29 you have $2.68 million and you can see how we end up with $5.37 million – over five times the amount we would have received compared to the first option.

This shows the sheer power of doubling. It is an important principle to grasp as we see exponential growth all around us in life. Nuclear chain reactions undergo exponential growth to power nuclear reactors. Positive feedback in speakers undergoes doubling amplification, resulting in the sharp screeching sounds. Compound interest follows exponential growth, allowing investments to give substantial returns over time (or result in crushing debt). Bacteria divide in two each time, resulting in a rapid population boom.

Understanding exponential growth also helps us make sense of scary situations such as pandemics. Viral infections are spread from one person to multiple people, represented by a basic reproduction number (R0). In the case of the COVID-19 (2019 coronavirus) pandemic, the R0 was between 2 and 3, meaning that left unchecked, the number of infected individuals would essentially double every few days.

Although this seems obvious, if you didn’t know about exponential growth, it would be terrifying to hear that one day you have 8 cases in a country, but in a fortnight, there are over 1000 cases, with each day presenting increasing numbers of newly infected patients. The media preys on this effect by providing anxiety-inducing headlines. But in reality, the headlines might as well read: “virus continues spreading in predictable exponential fashion“.

Another strength of knowing about exponential growth in a pandemic is that it lets us predict what would happen without any intervention. The number of cases would explode in a matter of weeks, resulting in catastrophic numbers of unwell people taken off the workforce, accompanied by mass casualties. Hospitals would be completely overrun, crippling the nation’s healthcare system and resulting in even more deaths as the infection runs rampant.

Therefore, efforts to reduce the spread of the virus through social distancing and effective quarantining are vital to reduce the rate of exponential growth, flattening the curve and making the number of cases more manageable for the healthcare system to deal with.

File:Covid-19-curves-graphic-social-v3.gif
Posted in Science & Nature

Rainbow

Posted on by Jinavie

Rainbows have been associated with wonder and the heavens throughout the history of humanity. The Norse believed that the rainbow bridge, Bifröst, connects the realms of men and gods. The rainbow is mentioned in the Bible as a sign from God to signify to Noah that the flood had ended. Irish leprechauns are said to hide their pots of gold at the end of a rainbow. It is now adopted as a symbol for LGBT movements, symbolising diversity.

The massive scale and brilliant colours of a rainbow is awe-inspiring (famously captured in the Double Rainbow video). We now know that it is the result of sunlight interacting with water droplets: reflecting, refracting and dispersing.

Sunlight refracts (bends) as it enters the droplet. It then reflects off the inside wall of the droplet and refracts once more as it exits. Because each wavelength refracts slightly differently, light disperses and each colour can be seen separately, much like a prism breaking apart white light into colours.

Because of water’s refractive index being constant, the returning light is most intense at 42°, making the rainbow always form in a circle with an angular radius (angle of light compared to your eyes where a circle is seen as a specific diameter) of 42° surrounding the point opposite the sun. If you are standing exactly at this spot with the sun behind you, you will see a beautiful rainbow. Otherwise, the rainbow disappears.

Angular radius can sound like a complicated concept, but in this case, it results in something quite interesting. To capture a full rainbow with a camera, your camera lens must have a field of view (cone of light that the camera will photograph) of 84°. Most smartphone cameras have smaller fields of view than this (iPhone X has a 65° horizontal field of view for instance), meaning that it would be impossible to capture all of the rainbow in one photo.

Another impossible thing when it comes to rainbows is finding the mythical pot of gold at the end of a rainbow. Because rainbows are the result of optics, they are different to every observer and how they are positioned to the sun and water droplets. This means that no two people observe a rainbow in the same way and a rainbow is not static.

You can also never approach the rainbow as it will disappear given the angular radius mentioned above.

Furthermore, there is no end to a rainbow because it is actually a full circle that extends through the horizon. We cannot see it as there is ground between us and the rainbow, but you can sometimes see a ring rainbow from a plane.

However, because the rainbow is technically just light from the sun bouncing off water and into your eyes, we can imagine it not as a circle, but a double-ended cone that ends in your eyes. By this logic, your retinas that sense the rainbow (and by extension, you) are the pot of gold at the end of the rainbow.

The End of the Rainbow
(Image source: https://xkcd.com/1944/)

Posted in Science & Nature

Periodical Cicada

Posted on by Jinavie

In certain parts of eastern North America, it has been noted for centuries that some summers seem to bring a massive swarm of cicadas. Observant naturalists such as Pehr Kalm noted in the mid-1700’s that this mass emergence of adult cicadas happened every 17 years. Since then, a similar pattern has been observed with many different broods of cicadas, with precisely 17 or 13 years between emergences of mature cicadas.

What could possibly explain such a specific, long gap between these spikes?

This phenomenon has been well-researched and the species of cicadas (Magicicada) are known as periodical cicadas. They can be distinguished by their striking black bodies and red eyes. Like most cicadas, periodical cicadas start their lives as nymphs living underground, feeding on tree roots. They take 13 or 17 years (depending on the genus) until they emerge all at once in the summer as mature adults – far longer than the 1-9 years seen in other cicadas. After such a long period of growth, they emerge for a few glorious weeks in the sun to mate, before laying eggs and disappearing.

The astute reader would notice that both 13 and 17 are prime numbers (a number divisible only by itself or 1). Is this a sheer coincidence or a beautiful example of mathematics in nature?

This curious, specifically long period of maturation has been a great point of interest for scientists. The phenomenon of mass, synchronised maturation is a well-documented survival strategy known as predator satiation. Essentially, if the entire population emerges at the same time, predators feast on the large numbers, get full and stop hunting as much. The surviving proportion (still a great number), carry on to reproduce and the species survives.

One theory holds that the prime numbers are so that predators cannot synchronise their population booms with the cicadas. If the cicadas all emerged every 4 years, a predator who matures every 4 or 2 years could exploit this by having a reliable source of food in a cyclical pattern. 13 and 17 are large enough prime numbers that it would be very difficult for a predator to synchronise its maturation cycles with.

Another possible theory is that it is a remnant of a survival strategy from the Ice Age. Mathematical models have shown that staying as a nymph for a longer period increased the chances of adults emerging during a warm summer, rather than when it is too cold for reproduction. This resulted in broods of varying, lengthy cycles, but this created another problem: hybridisation. When broods of different cycle lengths intermingled, hybridisation could occur and disrupt the precise timing of maturation cycles, decreasing the brood’s survival rate. Prime number cycles such as 13 or 17 years have a much less chance of hybridisation, increasing the survival rate.

As Galileo Galilei said, mathematics is the language in which the universe is written. It is fascinating to see examples of how maths can influence natural phenomena, even the life cycles of insects.

Posted in Science & Nature

Compound Interest

Posted on by Jinavie

When is the best time to invest? Is it when you have sufficient income and savings that you feel that you have a surplus to invest with?

The correct answer is much simpler: yesterday, with the second best time being today. Because of the magic of compound interest, investing early is the best strategy possible.

Thanks to a simple mathematic rule, compound interest rewards early, small investments more than late, large investments.

The way compound interest works is that after a given time interval (e.g. year), interest (as a percentage of the original investment) is paid out. The next year, interest is paid out again but as a percentage of the new amount. As an example:
1000 x 1.08 = 1080 (end of year 1)
1080 x 1.08 = 1166.40 (end of year 2)
1166.40 x 1.08 = 1259.71 (end of year 3)
…until end of year 10

If we use mathematical shortcuts and convert this into a formula, we can express it as:

(A = future value, P = present value, r = interest rate as decimal, n = number of periods/years)

For example, if we invest $1000 (PV) at an interest rate of 5% (r=0.05) for 10 years, then:

$1000 x 1.08^10 = $2158.92,

meaning we have earned $1158.92 over 10 years. Taking it further, in 30 years our investment would have grown to $10062.66 – ten times our original investment.

Because the formula uses exponents (or powers) for the time, your investments grow exponentially with time. This means that the earlier you invest, the greater your returns become disproportionately. This is why within 10 years, we have more than doubled our initial investment despite a reasonable interest rate and not doing anything else.

A rule of thumb for calculating how long it will take your investment to double is to divide 72 by the interest rate in % (e.g. 7). This is the number of years it will take for your investment to double (e.g. 72/7 = 10.3 years).

On top of this, if we invest small amounts every year, then we can benefit even more from the exponential growth of our investment. For example, just by adding in $100 every year, we end up with an additional $564.55 of investment earnings at the end of 10 years – a 50% increase in returns.

Unfortunately, mathematics works both ways and compound interest also applies to certain loans, such as credit cards. This means that your debt will grow exponentially unless you aggressively pay it back, making it seem impossible to pay off your credit card debt sometimes.

(This graph shows that investing early and consistently is the best strategy to maximise your eventual earnings. Compare the grey and purple line and you will see that despite investing a third of what Lyla invests total, Quincy ends up with a higher portfolio by retirement.)
Posted in Science & Nature

Sudoku

Posted on by Jinavie

Sudoku is a mathematic puzzle that has gained considerable popularity in the 21st century, rivalling the classic puzzle that is the crossword. You are given a 9×9 table divided into 9 equal squares, filled with a certain number of digits. Your goal is to fill in the table so that each row, column and subsquare (of 9 small squares) contains every digit from 1 to 9. You are not allowed to have the same number appear on the same row, column or subsquare, as there are not enough spaces for spare digits.

The more digits (“clues”) that you are given at the start of the puzzle, the easier it is to solve it. This begs the question: what is the minimum number of clues that you need to solve a sudoku puzzle?

Sudoku puzzles with 17 clues have been completed traditionally. We know that 7 clues is not enough as the last 2 digits can be interchanged, creating puzzles with more than one solution. Using mathematics, we know that if we can solve a puzzle with n clues, then a puzzle with n+1 clues can be solved as well. Ergo, the answer lies somewhere between 8 and 16.

In 2012, Gary McGuire, Bastian Tugemann and Gilles Civario tackled this problem using one of the oldest tricks in mathematical analysis: brute force. The total number of possible sudoku puzzles that can be generated is 6,670,903,752,021,072,936,960, or 6.67 x 10²¹. After accounting for symmetry arguments (meaning that two puzzles may be essentially identical, but just rotated or flipped), we are left with 5,472,730,538 possible unique solutions.

The team used supercomputers to analyse all of these possibilities to see if any puzzle can be solved with just 16 clues, as the conventional thought was that 17 was the minimum number of clues possible from traditional methods. After a year of calculations, the computer found no sudoku puzzle could be solved with only 16 clues. This was confirmed by another team from Taiwan a year later, proving that the minimum number of clues required for sudoku is indeed 17.

Posted in Science & Nature

The Dangerous Number

Posted on by Jinavie

Everyone has learned of the Pythagorean theorem in maths class:

In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (a² + b² = c²).

image

A lesser known fact is that Pythagoras, the Greek mathematician who came up with the theorem, had a school where numbers were essentially worshipped. The school of Pythagoras were obsessed with whole numbers and their ratios, believing the universe was built around whole numbers. Their motto was “All is number”.

In 520BC, a mathematician named Hippasus was murdered by members of the school of Pythagoras, by being thrown off the side of a ship. Why did a group of scholars go as far as killing a fellow mathematician? The reason lies in a special number.

Hippasus raised an interesting question regarding the Pythagorean theorem. Imagine a square where each side is 1 unit long. What is the length of the diagonal?

image

Using the theorem a² + b² = c²: 1² + 1² = 2 = c². Ergo, c = √2. This does not appear to be so controversial. The Pythagoreans would reason that it was simply a ratio between two whole numbers, much like ½ or ¾.

But as they tried to quantify what this ratio was, a horrifying truth emerged – no ratio between whole numbers could produce √2. It is what we now call an irrational number.

This was heresy – how can such a number exist in a universe built around whole numbers? The Pythagoreans would not allow this. Hippasus tried to argue that √2 was just as real a number as any other, but his attempts to propagate the knowledge of irrational numbers was quashed through murder.

Knowledge is power, but knowledge can also lead to tragedy.

image
Posted in History & Literature

Tic Tac Toe

Posted on by Jinavie

Tic Tac Toe is a simple game where you and an opponent make a mark (X or O) on a 3×3 grid once per turn, until one person has made a line of three marks in a row (horizontal, vertical or diagonal).

However, it is so simple that there are only a certain number of permutations, meaning that if you know the algorithm, you can win most of your games (assuming your opponent does not also know the algorithm). This is called a solved game – unlike chess, where there is a near infinite number of ways the game can play out.

First, let’s take the case of you starting first. Put a X in a corner. If your opponent does not put an O in the centre, you automatically win. Your next move is to put an X in any corner away from the O. Your opponent will have to put an O between the two X’s to prevent a loss. Once they do this, you can either put an X in the centre or another corner to create two possible winning moves and your opponent can only block one. You win.

image

If your opponent puts their O in the centre, things get more complicated. Now you can only win if your opponent makes a mistake – otherwise the game is guaranteed to end in a draw. You can take one of two options:

– Place an X in the corner diagonally opposite to your first X. If your opponent puts an O in a corner, you win by putting an X on the last corner to block their attack and create your own double-attack.
– Place an X on an edge square that is not next to your X. You can win if your opponent puts an O in a corner not next to an X by blocking their attack and creating a double-attack.

image

If your opponent plays first, then you can never lose. If an opponent starts in the corner, put your X in the centre. All you have to do now is block your opponent’s attacks and you will force a draw.
The same strategy applies if your opponent starts in the centre – put an X in any corner then block every attack. The game will end in a draw unless your opponent slips up.

As you can see, there are only so many ways a game can play out, meaning it is very easy to force a draw.
A more interesting game is omok (오목 in Korean, gomoku in Japanese) – where you put white or black stones on a 15×15 board to try and connect five stones in a row.

Posted in Science & Nature

Birthday Problem

Posted on by Jinavie

How many people do you need in a room until there are two people with the same birthdays? The pigeonhole principle dictates that (excluding February 29) since there are 365 birthdays, 366 people in a room would guarantee two people sharing birthdays. However, this is only the number needed to absolutely guarantee a pairing. Using a neat statistical trick known as the birthday problem (or birthday paradox), we can find that a much smaller number is needed to solve the problem.

Let us assume that every birthday is equally possible (in real life, some birthdays are more common than others). If there are 30 people in the room, Person 1 has a chance of sharing a birthday with each of the other 29 people (possible pairs). Person 2 can be paired with 28 people (since they have already been “paired” to 1), Person 3 with 27 people and so forth. Therefore, the number of chances are: 29 + 28 + 27+ … + 1. Using Gauss’s handy addition trick, the total number is (29 + 1) x 29/2 = 435. We can see already that although the total number of individuals is only 30, the total number of pairs already exceeds 365. Since the probability of having a certain birthday is 1/365, it is likely that it would occur when you have so many possible chances.

Using statistical analysis, it can be found that when there are 23 people, the odds of there being a match surpasses 50%, making it more likely that two people share a birthday than not. By 70 people, the probability of a match grows to 99.9%. Therefore, with only 19% of the number required by the pigeonhole principle, the birthday problem can say with 99.9% certainty that there will be two people sharing a birthday.