Brain Teasers on Lying Politicians

Many recent bestselling books have titles stating directly or indirectly that politicians and political partisans in general are flat-out liars; they fabricate, spin, deceive, and prevaricate.

On the left these books include Al Franken's Lies and the Lying Liars Who Tell Them, Joe Conason's Big Lies: The Right-Wing Propaganda Machine and How It Distorts the Truth, Eric Alterman's What Liberal Media? The Truth About Bias and the News. The latter do battle with Ann Coulter's Slander: Liberal Lies About the American Right, Dick Morris' Off with Their Heads: Traitors, Crooks & Obstructionists in American Politics, Media & Business, and Bernard Goldberg's Bias: A CBS Insider Exposes How the Media Distort the News on the right.

These book scuffles bring to mind three tricky puzzles having to do with lies and lying, which, although not very realistic, lead to some important ideas in logic, probability, and number theory.

The Three Puzzles

1. The first sort of puzzle was made popular by the logician Raymond Smullyan and it concerns, if I may adapt it for my purposes here, a very unusual state, each of whose politicians either always tells the truth or always lies. One of these politicians is standing at a fork in the road and you wish to know which of the two roads leads to the state capital. The politician's public relations person will allow him to answer only one question, however. Not knowing which of the two types of politician he is, you try to phrase your question carefully to determine the correct road to take. What question should you ask him?

2. The politicians in a different state are a bit more nuanced. Each of them tells the truth 1/4th of the time, lying at random 3/4th of the time. Alice, one of these very dishonest politicians, makes a statement. The probability that it is true is, by assumption, 1/4. Then Bob, another very dishonest politician, backs her up, saying Alice's statement is true. Given that Bob supports it, what is the probability that Alice's statement is true now?

3. In a third unusual state, there is another type of politician. This type lies at times, but then becomes conscience-stricken and makes it a point never to tell two lies in succession. Note that there are 2 possible single statements such a politician can make. They may be denoted simply as T and F, "T" standing for a true statement and "F" for a false one. There are 3 possible sequences of 2 statements, no 2 consecutive ones of which are false — TT, TF, and FT — and there are 5 sequences in which to make 3 such statements — TTT, FTT, TFT, TTF, and FTF. How many different sequences of 10 statements, no 2 consecutive ones of which are false, may a politician in this state utter?

The Three Solutions

The following are more than hints, but less than full explanations. For full understanding, time and a quiet corner may be necessary.

1. You could ask him, "Is it the case that you are a truth-teller if and only if the left road leads to the capital?" Another question that would work is, "If I were to ask you if the left road leads to the capital, would you say Yes?" The virtue of these questions is that both truth-tellers and liars give the same true answer to them, albeit for different reasons.

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