HENRY ERNEST DUDENEY urlous roblems & lee EDITED BY MARTIN GARDNER Avithmetical and Algebraical Problems SPECIAL VALUE SB Deming iil IMAL CL, ane 2496 Bat) 336 CURICUS PROBL Match Puzzles Curious Problems and Puzzles 536 Curious Problems and Puzzles HENRY ERNEST DUDENEY Originally published ae 536 Puzzles @ Curious Prablons. Copyright © 1967 by Charles Scribner's Sons. Allrights reserved, ‘This edition published by Harnes & Noble, Inc., hy arrangement with Scribner, an imprint of Simon 8 Schuster, 1995 Bares & Noble Books ISBN 1-56619-896-8 Printed and bound in the United States of America MOSTOH5452 Contents INTRODUCTION vil ARITHMBTICAL AND ALGEBBAICAL PROBLEMS 1 Money Puzzles 3 Age Puzzles un ‘Clock Puzzles 4 Speed and Distance Puzzles % ‘Weight Puzzles 2 Digital Puzzles 3a Skeleton Puzzles “a Ceypiarithm Puzzles a Miscellaneous Puzzles Py (GEOMETRICAL PROBLEMS a ‘Triangle, Square, and other Polygon Puzzles & Gisele Puzzles 6 Dividing-the-Flane Puzzles ror Plane Geometry Puzzles 104 Conienis Solid Geometry Puzzles Dissection Puzzles Paper Folding Puzzles Moving Counter Puzzles (COMBINATORIAL, AND TOPOLOGICAL PROBLEMS Magic Square Puzzles Magic Star Puzzles ‘Liquid Pouring Puzzles Route and Network Puzzles Point Alignment Puzzles ‘Map Coloting Puzzles GAME PUZZLES 1 ase 183 199 Introduction Henry Emest Dudeney (the last name is pronounced with a long “u” and a strong accent on the fist sylable, as in “scrutioy”) was England's greatest maker of puzzles. With respect to mathematical puzzles, especially problems ‘of move than trivial mathematical interest, the quantity and quality of bis out ‘Put surpassed that of any other puzzlist before or since, in or out of England. Dudeney was bom at Mayfield, in Sussex, on April 10, 1857, the son of a Toca schoolmaster. His father’s father, John Dudeney, was well known i Sussex as a shepherd who had taught himself mathematics and astronomy ‘while tending oheep on the downs above Lewes, a town fifly milee south of London. Later he became a schoolmaster in Lewes. Henry Dudeney, him- self a self-taught mathematician who never went to college, was understand ably proud to be the grandson of this famous shepherd-mathematician. Dudleney began his puzzle career by contributing short problems to news- papers and magazines. His earliest work, published under the pseudonym of “Sphinx,” scems to have been in cooperation with the American puzzlist, Sam Loyd. For a year aud a half, in the ete 1890's the two smen collaborated. on a series of articles in Zi-Bis, an English penny woekly. Later, using his own name, Dudeney contributed to a variety of publications including The Weokly Dispatch, The Queen, Blighty, and Cassels Magazine. For twenty years his puzzle page, “Perplexities” which he illustrated, tan in The Strand “Magazine. This was a popular monthly founded and edited by George Newnes, aan enthusiastic chess player who had also started and formerly edited 7i-Bix. ‘The Canierbury Przzles, Dudeney’s first book, was published in 1907. It was followed by Amusements in Mathematics (1917), The World's Best Word Puzzles (1925), and Modern Puzzles (1926). Two posthumous collections appewted: Puzzles and Curlous Problems (1991) und A Puzzle-Mine (undated). ‘The last book is a mixture of mathematical and word puzzles that Dudeney had at vii Introduction vomibuted to Blighty. With few exceptions, it repeats puzzles contained in is earlier books. The World's Best Word Puazies, published by the London Daily ‘News, contains nothing of mathematical interest. Dudeney’s frst two books have, since 1958, been available to American and British readers as paperback reprints, Modern Puztles and Puzzles and Curious Problems, inraany ways more interesting than the first two books becatise they contain less familiar puzzles, have long been out of print and are extremely hhard to obtain. The present volume includes almost the entire contents ‘of those two books. Readers familiar with the work of Sam Loyd will notice that many of the same purales appear, in different story forms, in the books of Loyd and Dudeney. Although the two men never met in person, they were in frequent votrespondeace, and they tad, Dudeaty once said in an interview. an informal agreement to exchange ideas. Who borrowed the most? This cannot be answered with finality umill someone makes a careful study of the newspaper and magazine contributions of both men, but it is say guess that most of the borrowing was done by Loyd. Dudeney never hesitated to give credits, He often gives the name of initials of someone who supplied him with a new des, and there are even ovcasional references to Loyd hitnself, But Loyd almost never mentioned anyone. Mrs. Macgery Fulleylove, Dudeney’s only child, recalla many oocasions on which her father fussed and funsed about the extent to which his ideas were being adapted by Loyd and presented in ‘America as the other puzzist’s own. Loyd was « dever and prolific creator of puzzles, especially in his ability to dramatize them as advertising novelties, ‘but when it came to problems of a more mathematically advanced nature, Dudeney was clearly his superior. There are even occasions on record whea, Loyd turned to Dudeney for help on difficult problems. ‘Geomettical dissections—cutting a polygon into the smallest number of pieces that can be refitted to make a different type of polygon—was a fleld in ‘which Dudeney was unusually skilful; the present volume contains many suprising, elegant dissections that Dudeney was the first to obtain, He was also an expert on magic squares and other problems of a combinatorial nature, ‘being the first to explore a variety of unorthodox types of magic oquares, such as prime-number squares and squares magic with respect to operations other thud addition, (There is an excellent article by Dudeney, on magic squares, in the fourteenth edition of the Encyclopacdia Britannica.) tn recteational numn- ber theory he was the first to apply “digital roots"—the term was probably coined by him—to numerous problems in which their application had tot Introduction ix been previously recognized as relevant. (For typical exemple of how digital roots furnish a shott cut to an answer otherwise difficult to obtain, soe the answer to Problem 131 ia this volume.) Dudeney was tall end handsome, with brown hair and brown eyes, a slightly aquitine nose, and, in his later years, a pray moustache and short chin whiskers, As onc would expect, he was a man of many hobbies, “He was naturally fond of, and skilled at games.” his wife Alice wrote in # preface 4o Puzzies and Curious Problems, “although he cared comparatively tte for cards, He was a good chess player, and a better problemist. As a young man he was fond of billiards, and also played croquet well” In his elderly days he enjoyed bowling every evening on the old bowling green within the Castle Precincts, an atea sutrounding the ruins of an old caste in Lewes. The Dudeneys owned a two inmdred-year-old house in this area, where they were Living at the time of Dudeney's death on April 24, 1930, (In Alice Dudeney’s ‘preface this date erronsoualy appears as 1931.) ‘Mrs, Fulleylove recalls, ina private communication, that her father’s croquet lawn, “ino matter how it was rolled and fused over, was always full of natural hazards. Father applied bis mathematical and logical skill to the game, with special reference to the surface of our lawn. He would infuriate some of our Visitors, who were uot familiar with tbe terran, by striking a ball ia what ap- peated tobe the wrong direction. The ball would gop, down, around the hills and through valleys, then roll gaily through the boop...” Alice Dudeney speaks of her husband as a “brilliant pianist and organist.” adding that, at different times, he was honotary organist of more than one church. He was deeply interested in ancient church music, especially plain song, which he studied intensively and taught to a choir at Woodham Church, Surrey. Mr, Fulleylove tells te that her futher, as a small boy, played the organ every Sunday at a fashionable chorch in Taunton, Somerset, He was a faithiul Anglican throughout his lite, attending High Church services, keenly Interested in theology, and occasionally writing vigorous tracts in defense of this ot that postion of the Anglican church, ‘Asa little girl, Mrs. Fulleylove sometimes accompanied bet father to bis London club for dinner. She remeinbers one oocasion oa which she felt very proud and grown-up, hoping the waiter and other guests would notice her sophistication atid good manners. To her hortor, het father, preoccupied with some geometrical puzzle, began peociling diagrams om the fine damask tablecloth, In his Inter life, Mrs, Fulleylove weites, her father lost interest in all x Eatraduction ‘composers except Richard Wagner. “He had complete transpositions for the ‘Fiano of all Wagner's works, and played them uuceasingly—to the great grief ‘of my mother and myself, who preferred the gentler chamber music. “The house at Littlewick, in Surey,” Mrs. Folleylove continues, “where we lived from 1899 to 1911, was always filled with weekend guests, mostly pub- fishers, woiters, editors, artists, mathematicians, musicians, and freethinkers.” One of Dudeney's friends was Cyril Arthur Pearson, founder of the Daily Press and of C. Arthur Pearson, Lid. a publishing house that bromght out Dudeney’s Modern Puzzles. Other fiends included Newnes and Alfred Harmsworth (later Lord Northelifte), another prominent newspaper publisher. “Father provided me, by degrees, with a marvelous collection of puzie toys, morly Chinese, in ebony, ivory, and wood... .,” Mrs. Folleylove recalls. “He was a huge success at children’s partes, entertaining them with feats of Jegerdemain, charades, and other party gamacs and stunts... “we hati a mongrel etree that I adored. His name, for some obscure reason, ‘was Chanoe, One day father fel over the dog’s leash and broke his arm, His comment, made without anger, was a quolation: ‘Chance is but direction ‘which thou canst not see."” In am interview in The Strand (Apri, 1926) Dudeney tells an amusing story bout a code message that had appeared in the “agony column” of a London newspaper. A man was asking 2 girl to meet bim but not to let her partnts mow about it. Dudeney cracked the code, then placed in the column a ‘message to the girl, written in the same cipher, that said: “Do not trust him. He means no good. Well Wisher.” This was soon followed by a code mestage from the girl fo "Well Wisher.” thanking him for his good advice. ‘Alice Dudeney, it should be added, was much better known in het time than her husband, She was the author of more than thirty popular, romantic novels. ‘A.good photograph af her provides the frontispiece of her 1909 hook, A Sense Of Scarlet and Other Stortes, and hee biographical sketch will be found in the British Who Was Who. “A Sussex Novelist at Home.” aa interview with her that appeared in The Sussex County Magazine (Vol. 1, No. 1, December 1925, PP. 6-5), includes her picture and photographs of the “quaint and curious” ‘Castle Precincts House where she and her hushand thea lived. Dudeney himself tried his hand on at least one short story, “Dr. Bemart's Patlent,” (The Strand Vol, 13, 1897, pp, 50-55). Aside from his puzde ‘eatures, he also wrote occasional nonfiction pieces, of which I shall mention only two: “The Antiquity of Modem Inventions” (The Strand, Vol. 45, 1913,