Problem 21

Lewis Carroll's Pillow Problem No. 72 (1894)

Problem. A bag contains two counters that are each equally likely to be either black or white. Can you ascertain their colors without taking them out of the bag?

Solution. The problem is ill-posed: it is impossible to ascertain the colors of the counters from the given information.

# 21.1 Discussion

This problem was adduced by the famous English writer, logician, and mathematician Lewis Carroll^{1} (1832–1898) in his book Pillow Problems^{2} (Dodgson, 1894) (Figs. 21.1 and 21.2). The latter contains a total of 72 problems of which 13 are of a probabilistic nature. Of all these problems, Problem 72^{3} has attracted the most attention. It appears under the rather mysterious heading Transcendental Probabilities. In the preface of his book, Carroll says (p. xiv)

If any of my readers should feel inclined to reproach me with having worked too uniformly in the region of Common-place, and with never having ventured to wander out of the beaten tracks, I can proudly point to my one problem in Transcendental Probabilities - a subject on which very little has yet been done by even the most enterprising of mathematical explorers. To the casual reader it may seem abnormal or even paradoxical, but I would have such a reader ask himself candidly the question, ‘Is not life itself a paradox'?

On p. 27 of his book, Carroll gives the wrong answer ...