This is the fourth part of the Pitless Pit series. For your convenience you can find other parts in the table of contents in Part 1 — Furmanek Test for consciousness

We’re often encouraged to think outside the box and apply our creativity to solve problems. While it’s hard to disagree with the statement, it’s barely constructive. What does it mean exactly? What techniques should we use and how can we develop this skill?

Thinking outside the box often starts with finding the box. It’s somewhat hard to find “new solutions” or “different perspectives” if we don’t know what is already inside the box. Finding the boundaries of the problem if crucial as this is where most of our models collapse. We already mentioned the Furmanek Test for consciousness which relies on identifying paradoxes. In this post, we’re going to solve few popular paradoxes. We’ll see that paradoxes are one of the best way to find the box and understand the hidden assumptions in our models.

Missing dollar riddle

Before solving the paradoxes per se, we need to understand how paradoxes even emerge. Let’s start with the popular missing dollar riddle:

Three guests check into a hotel room. The manager says the bill is 30 USD, so each guest pays 10 USD. Later the manager realizes the bill should only have been 25 USD. To rectify this, he gives the bellhop 5 USD as five one-dollar bills to return to the guests.

On the way to the guests’ room to refund the money, the bellhop realizes that he cannot equally divide the five one-dollar bills among the three guests. As the guests are not aware of the total of the revised bill, the bellhop decides to just give each guest 1 USD back and keep 2 USD as a tip for himself, and proceeds to do so.

As each guest got 1 USD back, each guest only paid 9 USD, bringing the total paid to 27 USD. The bellhop kept 2 USD, which when added to the 27 USD, comes to 29 USD. So if the guests originally handed over 30 USD, what happened to the remaining 1 USD?

If it’s the first time you encounter this riddle, I encourage you to solve it. It’s crucial to understand the solution before moving on. And if (when) you already know the solution, then think for a second why this is a riddle after all.

Why is this a riddle? Is there any riddle here?

We understand there is a misdirection in the riddle. As Wikipedia puts it:

The misdirection in this riddle is in the second half of the description, where unrelated amounts are added together and the person to whom the riddle is posed assumes those amounts should add up to 30, and is then surprised when they do not

Once we realize that “unrelated amounts are added together”, then it’s all clear and the riddle is solved. This gives us two most important things when looking for the box.

First, correct calculations do not prove the correctness of the model. Notice that there is no error in adding “9 + 9 + 9 + 2 = 29”. This sentence is entirely valid in the mathematical framework we commonly accept. It all adds up. The real problem is in “why do we even add 9 + 9 + 9 + 2”? Like, what does it really mean?

When we think twice about these calculations, we notice that they do not represent anything related to the situation in the riddle. Yes, they use the same values (hence the misdirection), but they pose no meaning. The formula, reasoning, and calculations are entirely correct, but they do not carry any meaning that would bring us closer to anything meaningful outside of the calculations.

The importance of doing calculations for something more than just the sake of the calculations is something we often miss. For instance, Occam’s razor (also known as principle of parsimony) suggests that we should look for solutions that use as few assumptions or moving parts as possible. That’s a useful technique (given that we’ve historically seen it leading to “better” results), but it doesn’t prove anything about the model itself.

Let’s think about the principle of parsimony for a little longer. Imagine moving back to 1800s. If you tried telling a layman about quantum physics, they would look at you like you’re crazy. “Why would we need anything like that? We should apply the principle of parsimony and use the simplest explanation”. And that would be the very right reasoning. Why would we introduce quantum physics with unclear interpretation and complex math when we have Newton’s laws of dynamics that explain everything sufficiently?

The same argument can be applied to other things. Do we need “god” per se to explain most of the nature now? Probably not, so if we apply the Occam’s razor, then we should not assume “god” exists. But the principle doesn’t prove that assuming non-existence of the god is correct. A god (or gods) may still exist. Similarly, do we believe in the afterlife (especially given that post-bereavement hallucinations are common), or do we apply Occam’s razor and claim that those “hallucinations” can be explained via other means (by the way, notice that many of these papers in fact assume those are “hallucinations” instead of trying to prove that)?

To summarize, a perfectly valid inference doesn’t prove anything about the correctness of the model.

Second, using wrong models leads to meaningless conclusions, and this can be exploited. Paradoxes and riddles often come from hidden “misdirection” that rely on our hidden assumptions about the world. It’s hard to ignore these assumptions or even identify them because they are so common and have been with us for so long. However, once an observation falsifies the model, we can find the box.

Every time you spot a paradox, stop and think twice. The paradox typically shows that either the data point is incorrect (e.g., we incorrectly observed something or have wrong measurements), or the model is incorrect (or incomplete). Paradoxes may show that we did something wrong (as in we measured something incorrectly) or that our reasoning started from the very wrong assumptions. Paradoxes are not “wrong”. It’s exactly the opposite – paradoxes clearly show us that we cross the boundary of the box.

Solving paradoxes to find the box

Let’s now take a look at few paradoxes that in my opinion are “easy to solve” once we realize our mental model is incorrect. I obviously don’t think I can convince anyone this way (otherwise the paradoxes would be considered “solved” long time ago) but I want to show that the paradoxes are not about “calculations within a model” but more about “choosing the right model”.

Kelly criterion

Let’s start with this game: you start with some money X and toss a fair coin many times. When it goes heads, you get 50% more of what you have, so you end up with 1.5X. When the coin goes tails, you lose 40%, so you end up with 0.6X. Should you play this game?

You might calculate the expected value of a single toss. You get 0.5 * 1.5X + 0.5 * 0.6X = 1.05 X Therefore, you should earn in a single toss, so you should take the game, right? If you earn in a single toss, then you’ll earn in many tosses, correct?

Wrong. The expected value is calculated correctly, but the expected value doesn’t represent what happens after many consecutive tosses. It shows what happens on average when MANY people take ONE toss. It doesn’t show what happens when ONE person takes MANY tosses.

The correct way to calculate that is to apply the Kelly criterion to find the long-term expected value.

So what is the problem here? As mentioned above, the calculations are correct. The problem is with using the wrong model. If you simulate the game many times, you’ll notice that observations (“most players are losing”) are not consistent with the expectations (“the expected value of a single toss is greater than one, so we reason that most players should be winning”). Once you realize observations disagree with the model, then either you have a bug in your simulation, or your model is incorrect.

Two envelopes problem

The two envelopes problem goes like this:

Imagine you are given two identical envelopes, each containing money. One contains twice as much as the other. You may pick one envelope and keep the money it contains. Having chosen an envelope at will, but before inspecting it, you are given the chance to switch envelopes. Should you switch?

Again, expected value tells us that switching should give us advantage. But this leads to the “paradox” that you should switch again after switching which would lead to the initial situation but should give you more money.

What’s wrong here? Again, we applied the incorrect model again.

First, by calculating the expected value of a single switch, we change the game. The expected value would give us the desired result if the game was like this: “once you pick an envelope, we toss a coin to decide if the other envelope has double the value or half of the value of your envelope”. If that was the case, we should switch because it’s generally beneficial on average. What’s more, if the game was “once you switch, we toss a coin again and replace the money in your initial envelope with either double or half of the value in the envelope you’re holding now”, then we should switch again.

However, that’s not the game we’re playing here. The game goes “we first put money in the envelopes, then you can pick one envelope, and then you can decide to switch”. We can’t just ignore the fact how the envelopes were created. Once we keep in mind the problem setup, the paradox disappears. Once you pick the right model, the math adds up.

Boy or girl paradox

The boy or girl paradox is another example of choosing the right model. You are asked two questions:

• Mr. Jones has two children. The older child is a girl. What is the probability that both children are girls?
• Mr. Smith has two children. At least one of them is a boy. What is the probability that both children are boys?

Once you do the math, you will realize that something is wrong. Zach Star has two great videos on the topic: video and video

Again, the issue here is not with the calculations. The problem is with the model you pick. If you disregard the setup of the problem (i.e., how you get the information about the sex), then you get different results.

Sleeping Beauty problem

The Sleeping Beauty problem goes like this:

Some researchers are going to put you to sleep. During the two days that your sleep will last, they will briefly wake you up either once or twice, depending on the toss of a fair coin (Heads: once; Tails: twice). After each waking, they will put you back to sleep with a drug that makes you forget that waking. When you are first awakened, to what degree ought you believe that the outcome of the coin toss is Heads?

Again, the problem comes from picking the wrong model. If you disregard the problem setting, then you may end up with calculations disagreeing with the assumption about the fairness of the coin. But this shows again that you need to pick the right model. If you include the setting in your model (i.e., you know there was a fair coin used), then you get the right result. But if you disregard the coin existence, then you again get the “right” result but you can’t draw a conclusion that the coin was unfair because there was no coin at all in your model.

Summary

Paradoxes often show that there is something wrong with our models. Once we realize that the results are suspicious (either because we have two “correct” methods giving different results or because something is off), we effectively find the box. At the very end of the day, either the coin was fair or not, there was no other way regardless of how “correct” the calculations are.