Posted in Science & Nature

## Airplane Game

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

## Compound Interest

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.

Posted in History & Literature

## Ponzi Scheme

Money is a human invention that acts as a medium of exchange and a store of value. It lets us easily carry around our assets in the form of paper, or nowadays, on a card. But because of its characteristics, money allows for some very intricate, complicated con schemes. One famous example is the Ponzi scheme – a type of investment scam.

In 1920, Charles Ponzi had a great idea. Back then, people would send an international reply coupon along with international mail so the receiver could send a reply using the coupon. Ponzi noted that since the value of these coupons were constant across nations, buying it cheap in one country then selling them in the United States would lead to profit. To kickstart his business idea, he began gathering investors, to whom he promised large returns that the investors could not refuse. However, when Ponzi began trading the coupons, he quickly found that it was not effective and he did not make the profit he expected. But instead of telling the investors that his plan failed and that they would not get the returns he promised, he decided to try something different.

Having not heard of Ponzi’s failure, new excited investors asked Ponzi to invest their savings too. Ponzi used the invested funds to pay off the original investors by redistributing the money. Happy that they got their promised returns, the original investors told their friends and family about the incredible opportunity. This brought more new investors to Ponzi, whose money he used to pay off the previous investors. Because Ponzi took a commission from each investment, he quickly raked in a massive amount of money – just by redistributing money around and not actually investing a single cent.

But Ponzi schemes do not last. Eventually, the amount of new investments were not enough to pay off the previous investors and people began investigating, only to discover that Ponzi was scamming them all along.

The Ponzi scheme relies on three things: enticing investors with the promise of high returns, intricate redistribution of money to feed the previous group of investors and the good reputation built through word of mouth. Because the scheme relies on paying old investors with money taken from new investors, a larger number of new investors is needed compared to the old investors. This leads to the formation of a pyramid. When the number of new investors is not enough, for example during a recession, the pyramid’s base becomes weak and the whole scheme collapses, with everyone losing money except for the schemer and the few original investors.

(Click Read More for diagram explaining a Ponzi scheme)

(Image source: http://browse.deviantart.com/art/Support-83476199)

Posted in Philosophy

## Achilles And The Tortoise

In 450 BC, a Greek philosopher named Zeno thought of the following paradox. Let us imagine that Achilles and a tortoise were to have a footrace. Achilles, obvious being faster than the tortoise, allows the tortoise to have a head start of 100 metres. Once the race starts, Achilles will quickly catch up to the tortoise. However, within the time he took to cover the distance, the tortoise would have travelled some distance as well (say 10 metres). When Achilles runs the 10m to catch up again, the tortoise has once again toddled on another metre. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Because there are an infinite number of points Achilles must reach where the tortoise has already been, theoretically the tortoise will be ahead of Achilles for eternity.

According to this thought experiment, motion is paradoxical and theoretically impossible. However, we know for a fact that motion happens. So how can we break Zeno’s paradox?

The main flaw of Zeno’s paradox is that he uses the concept of “eternity”. If we record the story mathematically, the time taken for Achilles to run the footrace is (if it took him 10 seconds to run 100m): 10 + 1 + 0.1 + 0.01 + 0.001… = 11.111… Ergo, the tortoise is only ahead of Achilles for less than 11.2 seconds (rounded). After 11.2 seconds pass, the time passed exceeds the sum of the infinite series and the paradox no longer applies.

Although it is a flawed paradox, the story of Achilles and the tortoise teaches the concept of geometric series – that something finite can be divided an infinite amount of times. For example, 1 = ½ + ¼ + 1/8 + 1/16… ad infinitum. This principle is a crucial part of mathematics and has significant implications in the field of economics. For example, it can be used to calculate the value of money in the future, which is necessary for working out mortgage payments and investment returns. Perhaps it is because of this mathematical principle that it seemingly takes an infinite amount of time to pay off a mortgage.

Zeno’s paradox teaches us that one should not take the concept of infinity for granted.

Posted in Science & Nature

## Marriageable Age

When is the right time to get married? According to Professor Tony Dooley, you can use an equation to find the right age for proposing. To do this, take “the youngest age you want to marry” and minus it from “the oldest age you want to marry” then times 0.368. Add this number to the youngest age. For example, if you would consider getting married from age 21 onwards and at the latest 30, your ideal age to marry is: (30 – 21) x 0.368 = 3.312 + 21 = 24.312, thus about 24 years and 4 months old.

This equation is very practical as it is a modified version of equations used in financial and medical fields. This equation is used to maximise profit while minimising loss using mathematics. It may not sound romantic, but according to Professor Dooley, after you reach the calculated age you should not waste time and ask the hand of the next person you date in marriage.