## Friday, 15 March 2019

### Answer to Fermat's Room puzzle

A very diffiicult puzzle this week!

Three people guard two doors. You know that:
• one person always tells the truth;
• one person always lies;
• one person randomly decides whether to tell the truth or lie (assume lies and truth are equally likely);
• the three know amongst themselves who they are.
You can ask two questions to the people. The answer to which must either be yes or no. What question do you ask and who do you ask?

This is an extension of the famous two person puzzle. Normally, you only have two guards, one tells the truth and one lies. You have to choose and open one of the doors, but you can only ask a single question to one of the guards.

What do you ask so you can pick the door to freedom?

In this case the solution is:
If I asked what door would lead to freedom, what door would the other guard point to?

This works by considering the two possible outcomes. Namely:

• If you asked the truth-guard, the truth-guard would tell you that the liar-guard would point to the door that leads to death.
• If you asked the liar-guard, the liar-guard would tell you that the truth-guard would point to the door that leads to death.
Therefore, no matter who you ask, the guards tell you which door leads to death, and therefore you can pick the other door.

This puzzle is so famous it's appeared many times in media

The inclusion of the trickster guard, however, changes the puzzle dramatically. Specifically, you questions have to work no matter who is being asked (truth-teller, liar, or trickster). Further, no matter what you ask, you always have to worry about the trickster screwing up your logic.

Thus, one strategy is to identify one person is NOT the trickster. We don't have to identify whether they are truth-teller, or liar.

Call the three gaurds A, B and C. You ask A:
"Is exactly one of these statements true:
1. You are the truth-teller
2. B is the trickster
If you get back the answer yes, then the possibilities are:
• A is the truth-teller and B is the liar (1. true, 2. false, so one statement true, so answer is yes which truth-teller truthfully gives)
• A is the trickster
• A is the liar and B is the truth-teller (both statements false so answer is no which liar lies about)
In all three cases, B is not the trickster.

If you get back the answer no, then the possibilities are:
• A is the truth-teller and B is the trickster (both statements true, so answer is no which truth-teller truthfully gives)
• A is the trickster
• A is the liar and B is the trickster (1. false, 2. true so one statement true so answer is yes which liar lies about)
In all three cases, C is not the trickster.

Once you have found a person who is not the trickster, just point to a door and ask the person:

Thus, reducing the problem to the previous case.