Here we have two books, bound as one, by two authors. However, both were born in 1832 and died in 1898.

*A Tangled Tale*, by Lewis Carroll, originally appeared in installments starting in 1880 in a British periodical whose full title was *The Monthly Packet of Evening Readings for Members of the English Church*. It was edited by Charlotte M. Yonge, a well-known novelist of the time, and was intended for young female members of the middle and upper-middle classes.

Carroll, famous because of the *Alice* books (1865 and 1871), was no doubt asked to contribute to the magazine and responded with a series of ten “Knots” containing problems, mostly of a recreational sort, for readers to solve. For example, one of the easier ones is in Knot IV: sacks A, B, C, D, and E are such that A and B together weigh 12 pounds, B and C 13½ pounds, C and D 11 pounds, D and E 8 pounds, and C, D, and E 16 pounds; how much does each sack weigh? There are also jokes:

A and *B* began the year with only 1000*l*. a-piece. They borrowed nought; they stole nought. On the next New-Year’s day they had 60,000*l*. between them. How did they do it?

Of course, one stood in front of the Bank of England and other stood behind it. The joke may have been old enough to elicit groans even in Carroll’s day.

The *Tale* is not a story, but a collection of excuses to present problems, though there are characters who appear in several Knots. One, “Balbus”, Carroll may have modeled after himself. (Balbus Blaesius was an ancient Roman who stuttered, as did Carroll.)

After a Knot had appeared, Carroll would give solutions to the problems therein. Also, readers submitted solutions on which Carroll would comment with his customary gentleness, grace, and humor. Judging from the large proportion of wrong answers (for Knot VI, “Twenty-nine answers have been received, of which five are right, and twenty-four wrong”) readers of *The Monthly Packet* were not highly skilled in recreational mathematics. One, who used the pseudonym “Tympanum”, caused Carroll to write “Oh, Tympanum! My tympa is exhausted: my brain is num. I can say no more.”

The Knots were collected into a book published in 1885. They are entertaining reading still and provide a window through which we can see part of Victorian England.

*Pillow-Problems*, “By Charles L. Dodgson, M. A., Student and late Mathematical Lecturer of Christ Church, Oxford”, was published in 1895. Its subtitle is “Thought Out During Wakeful Hours” and it contains seventy-two problems, followed by answers and solutions, that Dodgson solved mentally while in bed at night.

Dodgson did not, he said, suffer from insomnia. His mental problem-solving was “a remedy for the *harassing thoughts* that are apt to invade a wholly-unoccupied mind.” He said that when his

brain is in so wakeful a condition that, do what I will, I am *certain* to remain awake for the next hour or so, I must choose between two courses, viz. either to submit to the fruitless self-torture of going through some worrying topic, over and over again, or else to dictate to myself some topic sufficiently absorbing to keep the worry at bay.

There are, he said,

sceptical thoughts, which seem for the moment to uproot the firmest faith; there are blasphemous thoughts, which dart unbidden into the most reverent souls; there are unholy thoughts, which torture, with their hateful presence, the fancy that fain would be pure.

Poor Dodgson!

The problems are manly drawn from geometry, number theory, and probability. All are elementary but are by no means easy. Though Dodgson said that he had no special skill in mental work and that anyone could do as well with a little practice, I doubt that very much. Here is problem 61:

Prove that, if any 3 Numbers be taken, which cannot be arranged in *A. P*., and whose sum is a multiple of 3, the sum of their squares is also the sum of another set of 3 squares, the 2 sets having no common term.

I was trained in number theory, but I could never do that mentally, and probably not with paper and pencil. Dodgson’s solution is more than a page long and contains several complicated algebraic expressions. Some of his geometrical diagrams are so complicated that I marvel at his ability to retain them in his head.

The problems are good, though, worth reading and within the grasp of undergraduate students.

Dodgson could not keep Carroll from showing through. His last problem, he says, is one in “transcendental probabilities”:

A bag contains 2 counters, as to which nothing is known except that each is either black or white. Ascertain their colors without taking them out of the bag.

The answer is that one is black and one is white, and the solution provides a good example of the slipperiness that reasoning with probabilities can sometimes have.

Dover Publications is to be congratulated for keeping this book (any many others) in print for many years. The Dover edition was first published in 1958 and is still here more than fifty years later. Its price, a monotone step function, has gone from $1.50 to $15.95, but its content is unchanged.

Woody Dudley still has his copy of the $1.50 edition. It has aged better than he has.