The Diagonal Paradox

Shown here is a square with edges of length 1. The two orange edges will sum up to 2. Additionally, the length of the diagonal can be calculated using the Pythagorean Theorem. You know, A² + B² = C². That one. This gives √2 ≈ 1.41 for the diagonal.

Start carving out a stairway as shown in the pictures below. Realise that the total length of those orange lines will remain 2 no matter how many steps you chose to make!

From left to right, there are more (smaller) steps each time. But hang on a second: that orange line is quickly starting to look like the diagonal. Would it also approach 1.41 in length? If you made infinitely many steps, would its length turn out to be exactly √2?
No, not so, on both accounts.

The Necktie Paradox

Tom and Michael are both given a cheap necktie by their respective wives for Christmas. At the office Christmas party they start arguing over which (who) is the cheapest. Having had a few drinks, they agree to have a silly bet. They will ask their wives how much their neckties cost. The guy with the more expensive necktie has to give it to the other as the prize.

Tom eagerly accepts the bet. He reasons that winning and losing are equally likely. “If I lose, then I lose the value of my necktie. But if I win, then I win more than the value of my necktie. Therefore this bet is ultimately to my advantage!”
Michael is eager to accept the bet as well... since he reasons in exactly the same way.

Either guy's reasoning seems sound, yet they cannot both have the advantage in the bet! Where is the fault?

Details: For a fair bet you'd expect both guys to have a 50% chance of winning it (100% together). In the extreme case where one guy would have a 100% chance of winning, it can only follow that the other guy has 0% chance, i.e. no chance at all. So that's why they can't both have the advantage (= a chance bigger than 50%).

The Fuses Riddle

You have a bunch of fuses, each of which burns for exactly one minute. But, they burn unevenly. That is to say, half a length of fuse does not necessarily burn for half a minute. You have as many fuses and matches as you want.

How can you use these to measure 45 seconds?

The Coastline Paradox

What is the length of the coastline of Great Britain?

If you'd ask the Ordnance Survey (the mapping authority for the United Kingdom), they might give you a number of 11,073 miles (17,820 km). That's all well and good, but what does this number actually mean?

If I said the coastline had an infinite length, would I be wrong?

The Town's New Roads

A small town has four important places: a factory, school, soccer stadium and the shops. These four places are located on the corners of a square exactly 1 kilometer in size (see image on the left).
Now the municipality is planning the construction of roads in between. Each location should be directly or indirectly reachable from any other location. Extra intersections can be placed wherever. However, because of cutbacks, the designers are instructed to plan as little road as possible.

Here are some attempts to do that.

1. The first image connects all places directly to every other place. While this makes for fast travelling, it doesn't quite result in a small amount of road. The total length here is 4×1 + 2×√2 ≈ 6.83 km.
2. Since indirect connections were fine, the second image does away with the diagonals giving it a total road length of 4 kilometers.
3. Looking for even shorter solutions, what happens if the road is built in the shape of a circle, touching every location? With π×√2 ≈ 4.44 km, that third image is worse.
4. So in the last image, one of the roads is removed from the square, making a total of 3 km. While moving from the factory to the shops will be a pain, it is the shortest solution so far. Although an H-shape would be more practical, this is not relevant to the problem (that's still 3 km).

But 3 kilometers is not the shortest solution. What is?

The Uncountable Hotel

This story is a follow-up to The Infinite Hotel.

David Hilbert had been getting quite some attention with his Infinite Hotel. Georg Cantor, a fellow mathematician who was always in for an impossible challenge, wondered if he could design an even bigger hotel. So that when it was completely full, there would be no way all those guests could ever fit in Hilbert's hotel. No matter what clever trick Hilbert would come up with (and as we know, he had quite a few of those).

So Cantor started to think. Hilbert's hotel had an infinite amount of rooms, each denoted by its own room number. The first was number 1, the next number 2, and so on — all the way to infinity. Basically, there was a room for every possible positive (whole) number. How could he design more rooms than that? Hmmm... what if, maybe, his hotel was to include rooms for all negative numbers as well? A list of all rooms would then go on indefinitely in both the positive and the negative direction! Surely he would then have twice as many rooms as Hilbert did.

Cantor soon realized it was not going to be that easy. Infinity times two? That's still infinity. Just arrange the rooms so that you can match them, and you'll see:

Cantor's Room Number	1	-1	2	-2	3	-3	4	-4	...	∞
Hilbert's Room Number	1	2	3	4	5	6	7	8	...	∞

The idea sounded good at first, but the above shows that Hilbert's hotel would still be able to house every potential guest in Cantor's hotel. It doesn't matter if you change the labels on the doors; the number of rooms is still infinite.

The Ant On An Elastic Rope

An ant starts to walk along an elastic rope which is 1 km long, at a speed of 1 cm per second (relative to the rope it is crawling on). While this happens the rope is uniformly stretched by 1 km per second (i.e. after 1 second it is 2 km long, after 2 seconds it is 3 km long, etc.). Assume the rope will never break and we have an infinite amount of time on our hands. Will the ant ever reach the end of the rope?

Here's a hint: Yes, yes it will.

The Cube Dovetailing Puzzle

A cube consists of two halves, interlocked together as pictured above. The cube looks like that on all of its four sides. How would you separate the two halves without cutting, breaking or distorting them in any way? The answer is inside...

The Missing Euro

“Three guests check into a hotel room. The clerk says the bill is €30, so each guest pays €10. Later the clerk realizes the bill should only be €25. To rectify this, he gives the bellhop €5 to return to the guests. On the way to the room, the bellhop realizes that he cannot divide the money equally. As the guests didn't know the total of the revised bill, the bellhop decides to just give each guest €1 and keep €2 for himself.

Now that each of the guests has been given €1 back, each has paid €9, bringing the total paid to €27. The bellhop has €2. If the guests originally handed over €30, what happened to the remaining €1?”   ^source

The Mixing Problem

Say you have two barrels, one containing wine and the other containing an equal amount of water. You take a cup of wine from the wine barrel and add it to the water. If you want, you can stir it some — or not, it doesn't matter. Then take a cup of that wine/water mixture and return it into the wine barrel. The cup should hold the exact same amount so that both barrels once again contain equal volumes (but now mixed). The question is: which one of the mixtures is purer?

Thought about it? The answer is that both mixtures are equally pure! (← click to spoil)

The Supertask

In Greek mythology, Achilles was a hero of the Trojan War. Some 2500 years ago, philosopher Zeno of Elea included him into one of his paradoxes. Achilles was quite the runner... or maybe the paradox shows otherwise.

Suppose Achilles was in a 100 meter foot race with a tortoise. The tortoise is given a 25 meter head start. It would seem obvious that when Achilles eventually starts running, he will easily overtake the tortoise and then comfortably reach the finish line (with time to spare). Looking at it logically though, one could wonder how Achilles can ever beat the tortoise.

The Survey

A census taker rings the doorbell and a woman opens. She informs the census taker that she lives in the house with her three sons. “What are the ages of your boys, please?” the census taker asks.
“When you multiply all their ages, the result is 72,” the woman cryptically informs him.

The census taker has a confused look on his face, so the woman adds: “The sum of their ages is the same as the housenumber of the house next door.”
He finds this rather odd, but walks to the house next door, only to return shortly after. “I still don't know, can you give me another hint?”

“Sure,” the woman says, “my oldest son likes strawberries.”
The census taker nods and writes down the ages. What are they?

The Infinite Hotel

This story is a follow-up to The Highest Number.

David Hilbert had worked hard for it, but after what seemed like an eternity, his Grand Hotel was finally completed. So many people had been involved in building it, it was impossible to keep count. But there it was, up and running, with more guests coming in every day. And Hilbert's Grand Hotel was grand alright. You see, this hypothetical hotel had an infinite amount of rooms. A feature Hilbert was keen to advertise: “The hotel that always has a room available!”

Business was good. Hilbert enjoyed his tasks as manager, as quirky situations tend to occur a lot in a hotel with infinitely many rooms. Such a situation often required a mind-bending solution, just the kind of challenge Hilbert liked. And with the hotel quickly filling up, things were about to get crazy.

The Coin Flip

Here's a quick riddle. Setup: you are blindfolded, wearing thick gloves. There are twenty coins on the table in front of you: ten are heads and ten are tails. You do not know which; you cannot see them because of the blindfold and you cannot feel them because of the gloves. The only thing you can do is flip coins upside down or move them around.

You are to divide the coins into two equal groups (thus ten coins each). The assignment is to get the same number of heads and the same number of tails in both groups. How can you do it? No peeking!

Hint: The solution is easy. But reasoning to that point... less so!

The 20 Prisoners And The Hats

The warden from The 100 Prisoners And The Light Bulb is up to another one of his sick games! He puts 20 death sentenced prisoners together in a cell and explains the rules of their upcoming ordeal.

“Tomorrow, at the execution, I will give you a chance to go free. I will queue you up randomly and put a hat on your head. The hat is either red or blue. You cannot see the color of your own hat, only those of the prisoners in front of you. You will not be allowed to look behind you, nor are you allowed to touch, or talk to the other prisoners in any way. To be clear, the last prisoner will only see the 19 prisoners in front of him. The second-to-last prisoner will only see the 18 prisoners in front of him, and so on.”

The warden grins. “Starting with the last person in the row, I will ask a simple question: What is the color of your hat?”

“If he answers correctly, I will set him free. But you can guess what happens when the answer is wrong. He will be put to death immediately. Regardless of the outcome, I will then move to the prisoner in front of him, and ask the same question. From the last one in the row to the first one, you will all be asked.”

The warden lowers his voice. “And another thing. I will tolerate no cheating. If anyone answers anything besides ‘red’ or ‘blue’, I will execute you all right then and there!”

As he leaves, the warden sarcastically shouts: “Good luck tomorrow!”

The twenty prisoners can still talk freely during the night, so they are having heated discussions about how to free as many prisoners as possible. This turns out to be a difficult task. What is the most prisoners that can definitely be saved, and how?

The Exponential Folding

How many times do you need to fold a piece of paper (A4 size) in half, to make it reach the moon? Assume that the paper is 0.1 mm thick and that the distance to the moon is 384,403 kilometers (238,857 miles) — the accepted average.

When you calculate this, you should find that there are only 42 folds necessary. Welcome to the world of exponentials!

The Counterfeit Coin

You are given 12 golden coins, though you know that one of them is a counterfeit. The counterfeit coin can be detected because it does not weigh completely the same the others. But at the moment you don't know which coin it is, nor do you know whether it will be lighter or heavier.

Unfortunately, all you have is a scale of the classical type. It is the kind where there are two balanced dishes. By putting objects in, the scale will indicate that one side is heavier than the other (the dish will hang lower) or that each side weighs the same.

How can you find the counterfeit coin by only weighing three times? And is the coin lighter or heavier than the others?

The Road To Safety

A lone traveler comes upon a fork in the road. One path leads to safety, the other will lead to certain doom. At the fork there are two twin brothers and each guards one of the paths. One always lies and the other always tells the truth — you do not know who is which.

You can only ask one yes/no question to one of the brothers. What should your question be to find the safe path?

This is one form of a well known logical puzzle known as ‘Knights and Knaves’. This particular format was used in the 1986 film Labyrinth.

The puzzle has many variations and in the future I will post one that is annoyingly more complex. For now, let's see what the answer should be for this one.

The Rickety Bridge

Andrew, Beth, Carol and Daniel are on vacation and have spent the day exploring the mountain ranges in Yellowstone National Park. Having lost track of time, darkness is setting in and they are in a hurry to get back. But then there is a rickety old wooden bridge on their path, suspended high over a deep ravine. There's a warning-sign stating that the bridge will only be able to carry the weight of two persons at a time.

No-one is willing to cross the dangerous bridge without the light of a flashlight... unfortunately the group only has one of those with them. They can't risk throwing it, thus it needs to be carried back and forth.

Because of their different ages and fitness levels they will all cross at different speeds. Andrew can cross in 1 minute, Beth in 2 minutes, Carol in 4 minutes and Daniel in 5 minutes. For each duo, the slowest will of course determine the duration of crossing.

Soon it will be night and pitch black. Therefore the group wants to cross the bridge in the minimum time possible. Andrew thinks for a moment and then announces it can be done in 12 minutes. No trick. How?

The Highest Number

What is the highest number there is? Let's try and find out by working our way up. To save myself from having to write down a lot of digits, let me first explain the scientific notation:

• 1 × 10² = 1 × 10 × 10 = 100
• 1 × 10³ = 1 × 10 × 10 × 10 = 1000
• 1 × 10⁴ = 1 × 10 × 10 × 10 × 10 = 10000

Basically, powers of 10 can be used to define the number of zeros that follow the ‘1’. So here's some large numbers you might know: a million, a billion, a trillion, a quadrillion... These can be written respectively as 10⁶, 10⁹, 10¹² and 10¹⁵. You might knew these first few, but that list actually goes on for a while. Take for example a vigintillion, which is 10⁶³. Are there any higher?

Of course! How about the googol? It is notated as 10¹⁰⁰. That's a ‘1’ with a hundred zeros! Is this the largest number? Nah, we can put at least the centillion (10³⁰³), septuagintacentillion (10⁵¹³) and the millinillion (10³⁰⁰³) on the table.

We need to do better than this, so let's push it into crazy territory. A googolplex is 10^googol — yes, that's 10 to the power of a googol. A ‘1’ with a googol zeros. 10¹⁰¹⁰⁰. Fine! It is a 1 with 10 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 zeros. So as you can see I just wrote out a googol. I cannot write out a googolplex: there is not enough room in our universe to do so. See, the number of atoms in the observable universe is estimated to be around 10⁸⁰. This means that writing down a googolplex requires far more zeros than there are atoms in the universe!