# Four boxes, four questions, only one truthful answer

This is a harder variant of the following question: Three boxes, Four questions, Three times not true, Once the truth

The difference is that now there are four (not three) boxes, one of which contains a treasure. The rest is the same: You can ask exactly four yes/no questions, and the guardian will answer exactly one of them truthfully, and the other three falsely. Afterwards you open one box, the other three disappear, and you get the treasure if it was in the box you opened.

You can't ask about future events, the guardian's not psychic. Forcing the guardian to answer with a truth value ("Will you answer this question negatively?") will cause his head to explode, forfeiting your remaining questions.

How can you ensure you open the box with the treasure inside?

• The conditions aren't quite clear about the relationship between truthful guardians, lying guardians, and exploding heads. For example, if I had been standing in front of box #1 when asking the first question and box #2 while asking the second, and asked, while standing in front of the third, "When all is said and done, will it turn out that at least one question I asked while standing in front of the correct box had been answered truthfully?" Depending upon the treasure location and the truth of previous statements, that statement may or may not depend on the truth value of future events. Sep 16, 2017 at 18:48
• Not that I want to be nit-picky, but opening up a 3-state response (with the caveat that one of the response choices forfeits future questions) would open up interesting possibilities for this genre of puzzle. Sep 16, 2017 at 18:49
• @supercat The rule is simple, you can't ask about future events, so you're question about "will it turn out that" is not allowed. Sep 18, 2017 at 20:18
• Depending upon what has already happened, the answer to the question may or may not depend upon any future events. Whether or not you like that particular wording, the principle would be that blowing up the guardian's head would be an acceptable part of a solution but would forfeit the ability to ask any more questions. Asking N questions without that option would allow 2^N combinations of answers; adding the head-explode option would increase that to 2^(N+1)-1. Sep 18, 2017 at 21:36
• @supercat The rule isn't that you can't ask questions that depend on future events, it's that you can't ask about future events at all. "Will it turn out that X" breaks that rule regardless of X. Sep 19, 2017 at 18:53

You can:

Binary search!

So:

Is the treasure in the first two boxes? (x3)
Then, if there are 2 yeses and one no, it is in the last two boxes and the last answer will be false (the true answer having been used up); if there are 2 nos and one yes, it's in the first two boxes and the last answer will be false; three nos means it is in the first two boxes and the last answer will be true; and three yeses means it is in the last two boxes and the last answer will be true.

Then:

Ask about one of the two boxes you know the treasure's in, and since you know whether that answer will be true or false, you can precisely determine which box it's in.

• You ninja'd me! Sep 14, 2017 at 21:54
• @feelin Sorry! Great minds think alike! Sep 14, 2017 at 21:56

You can do this is JUST TWO QUESTIONS!

First, ask 'If I ask you if the treasure is in the first two bozes, will you say yes?'
This way, you can get the truth even when the guardian is lying and know which two boxes the treasure can be.

Then ask, 'If I ask you if the treasure is in the first (or third if he said no for the first question) box, would you say yes?
This is again getting the truth from a lie. Now you know where the treasure is.

• "will you say yes" is asking about the (hypothetical) future, so the guardian's head will explode. Jun 8, 2022 at 7:39
• @JaapScherphuis the wording could be changed to get around that - something like "what would you have said if I had asked ... instead of this?" Jun 8, 2022 at 17:35
• I don't quite understand how this would work, even if you can get around the future rule, since the guardian's truthfulness isn't consistent.. Suppose the guardian answers "yes". The guardian may have decided that he would lie to you if you asked if the treasure was in the first two boxes and be answering you truthfully, in which case it would be in the last two boxes. But he may also have decided he would tell you the truth if you asked about the first two boxes and is answering you truthfully, in which case the treasure's in the first two boxes. So how can you know which two boxes it's in? Jun 15, 2022 at 22:20