Missing Things

07 October 2010

A Most Ingenious Solution

Answers to last week's conundrum!

Let's start where you probably started - with the simple probability that, with one opportunity to make the choice and four options to choose from, you have a 25% chance of picking any given option and therefore a 25% chance of picking the right one.

This falls apart as soon as you realize that there are two options marked 25%. Two correct options out of four gives a probability of 50%, so we'll pick that one...

...wait. There's only one answer of 50%, and one of four is 25%. Again.

This is probably where most people give up.


Let's think about this a little more. Three out of our four answers are now in what amounts to a superposition of states - our argument proves that each of them is both true and false. This leaves us in a position of deciding what rule to use: either "all incompletely false statements are true" or "all incompletely true statements are false".

- If we go with the second, then we have eliminated three of the four options as acceptable answers and should default to the one remaining.
- If we go with the first, then since we have just judged that three of the four answers are true we have a 75% chance of selecting a correct answer at random.

In either case we ought to choose 75% as the correct answer!

If 75% is the correct answer, we have one chance out of four of choosing it correctly at random. Gotterdammerung!


In fact, however, this is the key to solving this problem. By this chain of reasoning, we have constructed an argument that places all four answers in a superposition of states; and, depending on whether we choose "all incompletely false statements are true" or "all incompletely true statements are false", the solution is either 100% or 0%, respectively. And neither 0% or 100% is presented as a solution to choose from. This means that the best possible answer does not appear as one of our four choices, leaving us with no chance at all of choosing the correct answer! No chance at all is 0% - and so, finally, we have an answer that is permitted to be true without also being false.

Since this choice does not appear, we simply do not select any of the answers, and in so doing get the problem right.


Final thought. Amidst all these convoluted circles of paradox in such a simple question, you may have forgotten that most people give up after discovering only the first circle. Having woven our way through the rest of the problem, we now learn that giving up is the best possible answer.

This means that, of all the people who ever encounter this question, most of them will answer it correctly - despite the fact that we just proved that each of them has no chance of choosing the right answer.

I'll leave you to think on that one for a while. :D

- Thursday