Imagine, if you will, a very long piece of ropethat loops around the Earth, fitting it tightly around the equator like a belt. If you wanted to raise this rope off the surface by one metre all around, how much more rope will you need?
The length of rope is the same as the circumference of the Earth which is 40,075km (24,901 miles). Ergo, it is easy to think that you would need kilometres of rope to extend it enough to float a metre off the Earth’s surface. However, in reality you need a mere 6.28m of extra rope to achieve this.
The reason is extremely simple, mathematically speaking. The circumference of any given circle is given by the equation 2πr, where r is the radius of the circle. Therefore, if you increase r by 1 unit (e.g. 1m), then the circumference increases by 2π x 1 = 2π = 6.28. No matter how large the circle may be, this rule does not change.
(This is a famous maths riddle, but here’s a much more interesting application of the concept in this What If? article. God I love that blog! http://what-if.xkcd.com/67/)
I have two coins in my pocket that add up to the value of 30 cents. One of them is not a nickel. What two coins do I have in my pocket? (This riddle uses US currency, which means that the only coins available for the riddle are: penny (1c), nickel (5c), dime (10c) and quarter (25c))
The answer is a quarter and a nickel. Some might angrily retort that one of them should not be a nickel and that is correct. One of them is not a nickel, the other one is. We commonly place such simple yet seemingly unbreakable barriers in our minds. If you cannot solve a problem, take a step back and try to look past the barriers.
You are walking along the road when you reach a fork, where you see two guards chatting with each other. You approach them to ask which way you should go, but you notice a peculiar sign that says: “One of us always tells the truth, one of us always lies”. There is no way for you to know who tells the truth and who tells lies. If you could only ask one question to help you walk down the right path, what would you ask the guards?
The solution is simple: all you have to do is ask “If I asked the other guard which road is the right one, which road would he tell me to take?” then go to the opposite road. The reason being, no matter who you ask – truth-teller or liar – they will tell you the wrong answer. If you asked the truth-teller, he would honestly reply with what the liar would say, which is the wrong answer. If you asked the liar, he would tell you the opposite of what the truth-teller would say, which would ultimately be the wrong answer.
A plague struck the ancient Greek island of Delos. As the disease ravaged the island, the people went to the oracle at Apollo’s temple for help. This is what the oracle said:
Double the volume of the cube-shaped altar in Apollo’s temple
People considered this a simple task and made a new altar where each side was double the original length. However, instead of disappearing, the plague worsened and people were confused.
Reason being, given that the length of one side of a cube is a, the volume is a³; if one side is 2a, the volume becomes 8a³, or eight times the original volume. Therefore, to double the volume of a cube, the number ³√2 is required. The problem is, whether ³√2 can be found using only compass and straightedge construction (where only the two tools are used to solve a geometric problem).
This problem, also known as the Doubling the cube problem, is one of three geometric problems known to be unsolvable by compass and straightedge construction. In other words, without the help of other mathematical methods, the answer cannot be found.
However, the solution to the above story is very simple.
A Cretan named Epimenides once said: “All Cretans are liars.”
So if Epimenides is a Cretan and he is a liar, then the statement is false. But that means that Cretans tell the truth, and Cretans are in fact liars. So what is the truth? This paradox continues ad infinitum due to the self-referencing nature of the statement.
This is a well-known example of a logical fallacy, or a flaw in a logic. It is also referred to as the Liar Paradox, seen in: “This sentence is false”.
The power of a paradox is best portrayed in the following parable.
A wise woman who worked as a fortune teller was tried for being a witch. In her trial, the king demanded she tell a fortune. If the fortune was correct, she would be drowned. If the fortune was wrong, she would be burnt at the stake. The woman smiled, and replied: “I will be burnt at the stake”.
A man asked how old a man’s three daughters were. The father replied with the following statement.
“The product of their ages is 36.”
“It’s hard to determine their ages from just that.” the man asking replied.
“The sum of their ages is same as the number of my house.”
“I still can’t figure out the answer!” the man replied again.
“My eldest daughter is blonde.” the father said, and the man, now smiling, replied.
“Oh, is that so? Then I can figure out how old your daughters are.”
How old is each daughter? And how did the man figure it out?
A computer cannot solve this problem, as it can only be solved using human logic.
This scenario psychoanalyses how you prioritise your life and what you place the most importance in.
You are out camping, but end up lost in the woods. It has become dark but luckily you find an empty cabin. Under the dim light, you discover the cabin is equipped with a fireplace, a stove, a lamp and a candlestick. In your backpack you only have one match.