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.
2. First we ask how probable it is that Alice utters a true statement and Bob makes a true statement of support. Since they both tell the truth 1/4 of the time, these events will both turn out to be true 1/16 of the time (1/4 x 1/4). Now we ask how probable it is that Bob will make a statement of support. Since Bob will utter his support when either both he and Alice tell the truth or when they both lie, the probability of this is 10/16 (1/4 x 1/4 + 3/4 x 3/4). Thus the probability that Alice is telling the truth given that Bob supports her is 1/10 ( the ratio of 1/16 to 10/16). The moral: Confirmation of a very dishonest person's unreliable statement by another very dishonest person makes the statement even less reliable.
3. Consider all possible sequences of 10 statements, no 2 consecutive ones of which are false. Some of these sequences end with a T and some with an F. There are exactly as many 10-element sequences ending in T as there are 9-element sequences of T's and F's, since any sequence of either of these types can be turned into one of the other type by adding or subtracting a T at the end. Furthermore, there are as many 10-element sequences ending in F as there are 8-element sequences of T's and F's, since any sequence of either of these types can be turned into one of the other type by adding or subtracting a TF at the end.
Putting these two facts together shows us that there are just as many 10-element sequences of T's and F's as there 8-element sequences and 9-element sequences put together. In particular, there are as many 3-element sequences as there are 2-element and 1-element sequences combined, as many 4-element sequences as there are 3-element and 2-element sequences combined, as many 5-element sequences as there are 4-element and 3-element sequences combined, and so on. But this is the definition of the famous Fibonacci sequence, each of whose terms after the second is the sum of its two predecessors. So the answer to the question is that there are 144 possible 10-element sequences of T's and F's having no 2 consecutive F's. Note that the "Fib" in "Fibonacci" acquires a new resonance.
Maybe to the polemical books listed above I should add Lying Politicians and the Convoluted Mathematical Truths They Reveal.
Professor of mathematics at Temple University and adjunct professor of journalism at Columbia University, John Allen Paulos is the author of several best-selling books, including Innumeracy, and the just released A Mathematician Plays the Stock Market. His Who’s Counting? column on ABCNEWS.com appears the first weekend of every month.