The A, B, C, D Problem

The problem is “A affirmed that B denied that C declared that D was a liar. If A, B, C and D each speak the truth (independently) once in three times, what is the probability that D was speaking the truth ?” This problem, first propounded, I believe, by Sir A. Eddington, has aroused considerable interest, and a solution of it was given by Sir A. Eddington in the Gazette, October 1935. His method of attack was, however, somewhat unusual, and the object of this contribution is to show how the problem may be attacked on normal lines, and a perfectly general solution obtained. For this purpose let us suppose that the chance that A speaks the truth is a, that B speaks the truth is b, and similarly with C and D.