5 - The Wisdom of Lewis CarrollThe wisdom of the English mathematician and writer Lewis Carroll lay in his uncanny ability to intelligently yet delightfully fuse semantics and mathematics—the world of words and the world of numbers. He wove wordplay and logic intricately into his writings, notably in
Alice’s Adventures in Wonderland and
Through the Looking Glass as well as in
Symbolic Logic, his instruction book on elementary logic. Reading these books ourselves—and encouraging our children to do the same—certainly can enhance the quality of our language and thinking both as individuals and as a people.
Carroll passionately believed that mental recreation through riddles and mathematical puzzles was necessary for mental health. He also saw that only through logical thinking could one clearly see through the veneer of fallacious and irrational thinking that permeates many societies. To limber up the mind for rational thinking, he developed mathematical games of increasing complexity, the most famous of which was this puzzle that I posed to readers of
The Manila Times the week before:
“If 6 cats kill 6 rats in 6 minutes, how many will be needed to kill 100 rats in 50 minutes?” IMAGE CREDIT: NEWSCIENTIST.COM Physicists are turning to Lewis Carroll for help with their maths
Responding to my challenge, 11
Manila Times readers from various parts of the world sent in solutions to the puzzle in exchange for the solution of Lewis Carroll himself. One, Gil Fernandez of Springfield, Illinois, USA, presented two solutions that yielded the same probable answers as those of Carroll’s—12 and 13 cats, using two different assumptions.
Here, verbatim, are Fernandez’s two solutions:
“1st solution: 6 cats’ rate of killing is 6 rats/6 min., or 1 rat/min. Now for 100 rats/50 min., that will be 2 rats/min., which will require 2 times 6 cats = 12 cats to achieve the 2 rats/min. rate of killing.
“To check: For 12 cats with a rate of 2 rats/min. x 50 min. = 100 rats. Answer therefore is: 12 cats.
“2nd solution: You can also say that it takes 1 cat 6 minutes to kill 1 rat. Therefore, it can kill 50 min./6 min. or 8 rats in 50 minutes. But it has 2 minutes left. What do we do with it? Since there are 100 rats to be killed, you will need 100 rats/8 rats or 12.5 cats.
“12 cats can kill 96 rats in 48 minutes. So what can be done with the idle 2 minutes? Can those be used to kill another 4 rats? If not, then maybe you should have 13 cats, considering that half a cat could be as messy as all those rats killed and may not be in a position to kill any.
“Answer therefore is: 13 cats.”
Rhegeon Abes of the Subic Bay Freeport Zone in the Philippines and Douglas Maliszewski of New Jersey, USA, sent in solutions that yielded 12 cats. Theirs were particularly elegant solutions, incisively annotated, so I’ll be happy to send copies to interested readers.
The eight others who solved the problem gave answers ranging from 2 to 200 cats, and one of them—“Nom DePlume” in the U.S.—offered a purely semantic, non-mathematical solution: “If I were to encounter 100 rats, logic tells me to immediately call the exterminator.”
Now let’s see Lewis Carroll’s solution, which first appeared in the February 1880 issue of
The Monthly Packet in England: “If 6 cats kill 6 rats in 6 minutes, how many will be needed to kill 100 rats in 50 minutes? This is a good example of a phenomenon that often occurs in working problems in double proportion; the answer looks all right at first, but, when we come to test it, we find that, owing to peculiar circumstances in the case, the solution is either impossible or else indefinite, and needing further data. The ‘peculiar circumstance’ here is that fractional cats or rats are excluded from consideration, and in consequence of this the solution is, as we shall see, indefinite.”
Carroll used the ordinary rules of double proportion—in the same way that Gil Fernandez did in our own time over a century later—to show that either 12 or 13 cats would have to do the killing job. Carroll backtracked, though: “But when we come to trace the history of this sanguinary scene through all its horrid details, we find that at the end of 48 minutes 96 rats are dead, and that there remain 4 live rats and 2 minutes to kill them in: the question is, can this be done?” He then showed four ways of how 6 cats might kill 6 rats in 6 minutes, but finally conceded that a solution to the problem had been “made ‘indefinite’ by the circumstances of the case.”
Then Carroll ended with this touch of whimsy: “If a cat can kill a rat in a minute, how long would it be killing 60,000 rats? Ah, how long, indeed! My private opinion is that the rats would kill the cat.”
(October 21, 2003)
