The Hexacoto

The above image made its rounds on Reddit the other day. The question asks “If you choose an answer to this question at random, what is the chance you will be correct?” The options are:

a) 25%
b) 50%
c) 60%
d) 25%

Since the randomly choosing one out of four answers is a 25% chance, so it’s a)… and d)? So since there are two correct answers, out of four choices, that is 50%, which is b). But there’s only one b), it’s 25%, so it’s a) and d)… ad nauseam.

STOP. You’re doing this wrong. Let semantics easily (and hopefully painlessly) tell you how to solve this question.

Let’s look at the question again.

“If you choose an answer to this question at random”

Let’s break it down:

IF [You] [choose 1 answer randomly] to [this] question, [percentage answer=TRUE?]

The secret is in the word, “IF”. It summons a counterfactual version of you, that you are able to discuss things in an “if” world, while not being constrained to answer by “if” rules. Thus, [counterfactual You] is supposed to pick 1 answer to [this], where [this] is self-referential to a world that has 2 correct answers out of 4. The answer is 50% for you in this world, not the world [counterfactual You] inhabits.

Hence, in your reality, not the [counterfactual You] in the question, just answer the question that they asked about counterfactual you, simple as that. An equivalent question, substituting counterfactual you with a third person, is:

Kevin has to randomly pick 1 answer out of four. However, 2 of the answers are identical and correct. What is the percentage that Kevin will pick a right answer?

Don’t sweat the counterfactuals, just stick with this reality. The right answer is B.

(No need to read the below if you don’t want technical explanations)

If you want a really convoluted discussion about semantics and counterfactuals and why we can discuss counterfactuals without being constrained by counterfactual rules, it’s simple. In counterfactual semantics we often discuss the death of Aristotle (or was it Plato?), such as “Aristotle might not have been a philosopher if he had died as a kid.” This relates to the topic of indices and what names refer to, largely researched and discussed by many linguists and philosophers, such as Kripke.

A quick answer, without going too in-depth, is that if we are bound by the indices of the counterfactuals we refer to, we will be unable to talk or respond because the counterfactuals are in an infinite loop. Thus, we can talk about Aristotle’s death without having to go back in time to kill him, or talk about what would happen at the end of the world without destroying the world to be able to talk about it. Take the following multiple self-indexed sentence.

If I were you, I would kill me

There are two people involved in the conversation, “you” and “me”, yet to our minds there seems to be a conventional understanding of what the sentence means. It means that “I am such a terrible person that if there were another person, and that person were talking to me, he would hate me so much that he would kill me.” For such a short sentence, it takes such a long sentence to elaborate. Thank goodness for indices! This is how the above sentence works with indices:

IF [counterfactual I][sees]me, [counterfactual I][wants][kill] me.

There you go.