Ivars Peterson's MathTrek

January 13, 1997

# Hankies, Snarks, and Triangles

Charles Lutwidge Dodgson was a competent mathematician who taught at Christ Church, Oxford, during the nineteenth century. His creativity manifested itself not in proving important theorems but in the realm of recreational mathematics. His enduring fame rests on two books he wrote under the name Lewis Carroll.

Both Alice's Adventures in Wonderland and Through the Looking-Glass and What Alice Found There contain many examples of Dodgson's passion for mathematical games, puzzles, logic paradoxes, riddles, and all sorts of word play. Indeed, his fascination with card games and chess provided the background for his two Alice books.

Under the name of Lewis Carroll, Dodgson also published a variety of articles and leaflets devoted to puzzles, games, magic tricks, riddles, puns, anagrams, and ingeniously constructed verse. In the newly published book The Universe in a Handkerchief, Martin Gardner offers a varied selection of these little treasures, gleaned from Carroll's publications, diaries, and letters.

Gardner's title comes from a brief reference in a Carroll book called Sylvie and Bruno Concluded, in which a German professor explains how a Mobius band has only one side and one edge (see the MathLand column on Recycling Topology for more on Mobius bands). He then goes on to demonstrate how to sew together two handkerchiefs to make a three-dimensional one-sided surface known to topologists nowadays as a projective plane. Because this closed surface has neither an outside nor an inside, one can say it contains the entire universe.

In Carroll's lengthy nonsense ballad, The Hunting of the Snark, the Butcher tries to convince the Beaver that 2 + 1 = 3. He adopts the following procedure:

"Taking Three as the subject to reason about --
A convenient number to state --
We add Seven, and Ten, and then multiply out
By One Thousand diminished by Eight.

"The result we proceed to divide, as you see,
By Nine Hundred and Ninety Two:
Then subtract Seventeen, and the answer must be
Exactly and perfectly true."

Writing out and simplifying the algebraic expression for the operations, [(x + 7 + 10)(1,000 - 8)/992] - 17, one finds that the procedure always yields the same number that one starts with!

Dodgson's fascination with puzzles is evident in many diary entries. On December 19, 1898, he wrote: "Sat up last night till 4 a.m., over a tempting problem, sent me from New York, 'to find three equal rational-sided right-angled triangles.' I found two, whose sides are 20, 21, 29; 12, 35, 37; but could not find three."

Had Dodgson doubled the sides of the two triangles he had found, he would have obtained the first two triangles of the triple he was looking for. The smallest solution consists of triangles of sides 40, 42, 58; 24, 70, 74; and 15, 112, 113, all of which have the same area, 840.

There are actually infinitely many such right-angled triangle triples. Beyond the smallest triple, however, the integral sides of other triples are each at least six digits long.

Lewis Carroll certainly had a lot up his sleeve, and The Universe in a Handkerchief provides a welcome survey of Carroll's mathematical magic tricks.

Copyright © 1996 by Ivars Peterson. References: Carroll, Lewis (notes by Martin Gardner). 1990. More Annotated Alice: Alice's Adventures in Wonderland and Through the Looking-Glass and What Alice Found There. New York: Random House.

Gardner, Martin. 1996. The Universe in a Handkerchief: Lewis Carroll's Mathematical Recreations, Games, Puzzles, and Word Plays. New York: Copernicus.

Comments are welcome. Please send messages to Ivars Peterson at ipeterson@maa.org.