A Comparison of Various Extensions of Strong Truthteller and Strong Liar Puzzles (Mutes and Crazies)

Abstract

Truthteller liar puzzles are popular in science and also in recreational mathematics. In this paper, we compare five different types of puzzles. In each of our puzzles, the persons of the puzzle may state some statements about their types. In strong truthteller-strong liar puzzles (SS puzzles), each statement of a truthteller must be true and each statement of a liar must be false, and there is no third type of person in these puzzles. It is known that there is no good SS puzzle, where a puzzle is good if it has exactly one solution. In fact, because of symmetry, by flipping the type of person in a solution, another (dual) solution is obtained. Therefore, to break this symmetry, there are various ways to introduce a third type of person, e.g., Mutes and crazies. In SSS puzzles, crazy people may appear, each of whom can tell only a self-contradicting statement. In SSW puzzles, a crazy person may say some additional statements apart from his or her self-contradicting statement. In SSM puzzles, that we investigate here, there can also be some Mute people (as the third type together with truthtellers and liars). We differentiate two types of SSM puzzles. In SSMW puzzles a mute person may be a Mute (type), but he or she could also be either a truthteller or a liar (type). In SSMS puzzles, each person who did not say any statement must be a Mute in the solution. Various examples are presented and it is also highlighted how a puzzle changes from unsolvable to solvable or to a good puzzle when the interpretation, the type of the puzzle changes, i.e., shifted from one to other, and symmetry breaks. Among other data, the number of solvable and good puzzles are counted and compared for the five types when few people appear.