Season 4 Episode 3

Two guards stand before two doors. One always tells the truth and the other always lies. In this episode, we work out how to get the guards to tell us which door has a tiger behind it.

Further Reading

Variants

Here are some variants we discussed briefly on the livestream.

• Two doors are guarded by three guards. One door has a tiger behind it. One of the guards always tells the truth, one guard always lies, and the other answers questions at random, deciding to lie with probability 50% or to tell the truth with probability 50%. You do not know which door is which or which guard is which. The guards are all aware of each other’s behaviours and, importantly, are aware of the tiger. You may choose any guard and ask them a yes/no question. How many questions do you need to work out which door has a tiger?
• Two doors are guarded by two guards. One door has a tiger behind it. Both guards speak a language that is unfamiliar to you, but you know that the words “da” and “ya” mean “yes” and “no” in some order. One of the guards always tells the truth, and the other always lies. You do not know which door is which or which guard is which. You may choose any guard and ask them a yes/no question. Or in this case, I suppose it’s a da/ya question. Or possibly a ya/da question. Can you work out which door has the tiger?

This puzzle is famous enough to have a Wikipedia page, where it’s known as the Knights and Knaves puzzle. Perhaps unusually, it also has a TV tropes page.

A variant with a third guard who stabs people who ask tricky questions is from the webcomic xkcd.

John Finnemore’s comedy sketch show had a version of this puzzle with more conversation between the guards.

Raymond Smullyan did a lot to popularise this problem and came up with many variants in his book “What is the Name of This Book?” (originally published 1978, but there are more recent printings available on, for example, Amazon).

Other logic puzzles

This one is a classic that also appears in the movie Fermat’s Room:

Three boxes are labelled “lemons”, “limes”, and “mixture of lemons and limes”. You are told that all the labels are wrong. You may pull one random fruit from one box of your choosing. You must determine which box is which. From which box would you like a fruit?

And this one was used at a Maths Festival we ran recently:

Three bags are labelled with statements. You are told that at least one of the statements is true and at least one of the statements is false. The bags and statements are
Bag 1: “the silver is in this bag”
Bag 2: “the silver is not in this bag”
Bag 3: “the silver is in Bag 1”
Which bag is the silver in?

Exercise: design your own version of this puzzle, with different statements (and perhaps with more bags).

MAT logical reasoning

For other similar MAT questions, see MAT 2011 Q6, MAT 2012 Q6, MAT 2013 Q6. All three of those questions are available to download together on this worksheet from the 2020 MAT livestream, or separately from the table of past papers on the MAT webpage. The 2022 MAT livestream starts 04 August 2022 at www.maths.ox.ac.uk/r/matlive.

If you want to get in touch with us about any of the mathematics in the video or the further reading, feel free to email us on oomc [at] maths.ox.ac.uk.