A bag contains 2 counters, as to which nothing is known except that each is either black or white. Ascertain their colours without taking them out of the bag.I thought this must be a trick question. My answer was along the lines of "Ask someone else to take the counters out of the bag for you." But turning to the solution, I found this calculation:
"Whaaaat? That's absurd," I thought. "Both of the original counters might have been white, or both might have been black—a calculation can't retroactively change the counters' colors!"
I thought this must be a jest on Carroll's part. But then again, Carroll took seriously his role as a public communicator of mathematics, and so he would never leave a misleading calculation on the page for people to absorb as if it were true. At the same time, Carroll himself couldn't be mistaken about such a simple problem, especially since some other problems in the book show Carroll creating probability paradoxes and resolving them correctly.
Then I noticed the words "THE END." I thought, It would be just like him to end with a laugh. But still I didn't quite believe that he would commit a fallacy to paper, even for a joke. On the other hand, fallacious proofs like 0 = 1 are a commonplace in mathematical recreations....
I returned to the introduction, which I hadn't read in several months, and there I found this paragraph:
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 in which, I believe, very little has yet been done by even the most enterprising of mathematical explorers. To the casual reader it may seem abnormal, and even paradoxical; but I would have such a reader ask himself, candidly, the question "Is not Life itself a Paradox?"OK, there we go—found the wink-wink! With problem number 72, Carroll was taking tongue in cheek and showing the reader a tempting piece of sophistry.
Next morning, I searched online for discussions of the problem. One is here, but the content is not available for free and I didn't read it. The problem was also mentioned by the great Martin Gardner (1914–2010) in his book The Universe in a Handkerchief: Lewis Carroll's Mathematical Recreations, Puzzles, Games, and Word Plays. (I don't own this book, but in case anybody reading this post is responsible for putting together a Christmas list....)
The book's last problem, number 72, has been the subject of much controversy. ... The proof is so obviously false that it is hard to comprehend how several top mathematicians could have taken it seriously and cited it as an example of how little Carroll understood probability theory!Gardner goes on to explain how in the book's Introduction Carroll "gives the hoax away." Gardner however doesn't seem to spend any time exploring the paradox. If I come up with any additional thoughts about it, I'll update this post with them. In the meantime, I leave you in the capable hands of another admirer of Lewis Carroll.