# How would one make a good liars puzzle?

I’ve seen and made a few liars puzzles, and a recurring theme with liars puzzles is that they can be too easy, and can be dull, perhaps a bit repetitive. How can you come up with a Liars puzzle that is not too easy, but has interest and appeal?

In your answer you can, and are encouraged to make reference to previous liars puzzle, or even create one in order to support your reasoning.

If you're referring to Knights and Knaves type puzzles, the Wikipedia article on them suggests some variants that might produce some interesting complexity.

I find that a Liars mechanic can be very interesting when incorporated into other types of logic deduction puzzles, such as this Logic Grid puzzle I created earlier this year, in which a liars mechanic greatly increases the difficulty.

Liars mechanics can also increase complexity in Grid Deduction puzzles such as Sudoku. For example, here is a Slitherlink puzzle by Prasanna Seshadri, in which one clue from each row and column is lying, and must be different from its stated value. Similarly, here is a collection of Area 51 puzzles by Dave Millar, which is basically the same idea, but with several additional types of clues. A couple of Nikoli grid deduction puzzles have Liar mechanics built into their standard rules, including Yajisan-Kazusan and Usowan.

In fact, it's not uncommon for the World Puzzle and World Sudoku competitions to have an entire section devoted to this mechanic.

In any case, one way I might start to make a puzzle of this type would be to create a set of clues in which certain combinations produce a contradiction, constructing those combinations so that one can deduce previously uncertain clues or statements to be true.

For example, in the Logic Grid puzzle linked above, five people have lists of five clues each, one clue being false on each list. I created a circular chain of potential contradictions, which can be abstracted as follows:

A1-5, B1-5, C1-5 are all clues. Exactly one each of the A, B, and C clues is false. A1 conflicts with B2, B1 conflicts with C2, and C1 conflicts with A2. There are only two possibilities, all of the _1 clues are false or all of the _2 clues are false. Therefore all of the remaining A, B, and C clues must be true. Trenin finds this chain in his walkthrough in the section called Mary, Jeff, and Katie.

Instead of doing the traditional 'A is lying' or 'B is telling the truth', add other elements - such as last names, etc. and make it more like a logic grid, for example:

You are a police officer investigating a crime and have narrowed the criminal down to four suspects. You also know that exactly one of the suspects is guilty and will always lie, whereas the other three will always tell the truth. The suspects are A,B,C,D with last names W,X,Y,Z in some order, and their preferred weapons are a knife, dagger, sword and gun. It is quite obvious that the criminal used a gun but you have no idea who the criminal is. Here are their four accounts:

A:

• B and D have last names X and Y in some order
• My last name is Z
• I use a sword

B:

• The person with last name X does not use a knife or sword
• I do not use a gun

etc.

To my mind, puzzles where the goal is to formulate questions are more interesting than those which merely require finding a distribution of liars and truth-tellers that satisfies a given set of conditions. Further, puzzles where one can't ask enough questions to find out everything (but must find some particular piece of information with certainty) are often more interesting than those where one can ask more questions.

A critical requirement for a good puzzle of this form is that it clearly delineate what questions may be asked, and how different kinds of subjects will respond. While the rules may be expressed "in-universe" [e.g. if you ask the Neila an ambiguous question, he will get annoyed and shoot you, thereby denying you a chance to rescue the princess, convince her to marry you, and live happily ever on Tenalp], good rules should generally also be easy to express mathematically.

If S would be the set of possible solutions, and an entity which is asked a question may respond N ways, then for a typical puzzle is should be possible to represent any question as a collection of N subsets of S. For example, if there are three people, one of whom is a liar, a truth teller, and a devil [who may arbitrarily answer yes or no without restriction], and I ask the first if the second is a truth teller, then if the person says No the possibilities would be LTD, TDL, TLD, DLT, and DTL, and if the person says Yes the possibilities would be DLT, DTL, and LDT. Such a question would thus be [[LTD, TDL, TLD, DLT, DTL],[DLT, DTL, LDT]].

Even in puzzles with a relatively small number of possible arrangements, the number of possible questions may be quite large. There will often be restrictions on the questions one may ask [e.g. if one is asking the first person a question, any possible arrangement in which the first person is a devil must be present in both the 'yes' and 'no' sets] and some rule sets may require keeping track of state, but formulating questions as sets avoids quibbles over how to handle potentially-ambiguous questions, since for any question and arrangement, it would be clear what responses the subject would be allowed to give.