The question we ask if “What would the other guard say if I asked him which door is the good one?”

Liar says Bad Door

Truther says Bad Door

Now, for their answers to update, they would have to ne answering the question, “Which door would the other guard say if I asked him ‘Which guard would the other guard say is the good door?’”

We want a guard to answer “What would the other guard say is the good door?” Regardless of how they answer our “outer” question, the answer to the “inner” question (“which is the good door?”) doesn’t change.

Liar doesn’t care that Truther would say that “Liar would say the right door is the good one,” Liar is being asked how Truther would answer “Which door is the good one”.

I know I basically just said the same thing three times. My brain isn’t working to break this out the elegant way I can’t quite assemble. But hopefully some part of all this helps. The crux is that the question that they are imagining the other guard’s answer to is not the same question they themselves are being asked.

The liar, knowing the truth-teller will point to the good door, points to the bad door.

The truth-teller, knowing the liar would point to the bad door, points to the bad door.

Either way, you take the one your guard doesn’t point to.