Logic and Algebra in Ulam's Searching Game with Lies

What I tell you three times is true LEWIS CARROLL Introduction In the familiar two-person game "Guess a Number", the first player chooses a number c in a certain set S, and the second player tries to find the secret number, by asking a few questions, to which the first player can only answer yes or no. Here, one might correctly object that since 1 is the only odd number in S that is also smaller than 3, there is no need of a third question. However, if we allow the possibility that some of the answers may be wrong, (as would be the case if the first player were a liar, or—more generally—if some of his answers were subject to distortion during the travel from his mind to our ears) then more questions and answers are necessary to find c. So let's go on. We shall naturally identify the first player with Pinocchio, and ask the reader to join the author to impersonate the second player—the Investigator. —Q 3 : Is c=1 ? —A 3 : no. Here we meet an experimental fact, well known to readers of detective stories: liars tend to contradict themselves. Accordingly, in our logic investigation we must learn from Pinocchio's contradictions. —Q 4 : is c odd ? —A 4 : yes. Here again we are experiencing another useful principle, well known to teachers: repetitions are helpful, or, repetita juvant. If, for instance, the first player is allowed to lie at most once, then the repeated information that c is odd must be taken for granted.