Inexpensive advertising premiums are popular in all countries where businesses compete for consumer attention, and frequently such premiums are based on mathematical puzzles. Many premiums of this kind have been discussed in columns that are reprinted in my earlier book collections, and one involving a "map fold" will be found in this book in the chapter on paper folding. Now I shall consider some classic puzzle premiums that I have not previously discussed.

One of the oldest and best is the T-puzzle shown in Figure 70. The reader is urged to trace or photocopy the four pieces, paste them on cardboard, cut them out and try to fit them together to make a capital T. I know of no polygon-dissection puzzle with as few pieces that is so intractable. The number of giveaway premiums based on this puzzle, particularly in the early decades of the century, runs into the hundreds.

Figure 70

Figure 70

The classic T-puzzle

Less well known, although equally ancient and charming, is the square puzzle shown in Figure 71. It is best shown to a friend by first giving him only the four nonsquare pieces and asking him to make a square. After he succeeds, hand him the square piece and see how much longer it takes him to make a square that uses all five pieces.

The Pythagorean-square puzzle

I have not seen this as a die-cut premium in recent years, but at least two plastic versions are currently on sale in the United States. Milton Bradley's One Way was designed by Henry Adams, and another version, designed by Frank Armbruster, is called Madagascar Madness. Armbruster's instructions point out how the puzzle illustrates the Pythagorean theorem. If the big and little squares shown in the illustration are on the sides of a right triangle, the square formed by all five pieces will, of course, be the square on the hypotenuse.

In this country the most prolific creator of mathematical premiums unquestionably was Sam Lloyd (1841-1911), the famous Philadelphia-born puzzlist and chess-problem inventor. In his cluttered, musty office in a decaying Manhattan building occupied by The Evening Globe, Loyd concocted hundreds of puzzles of fantastic variety and ingenuity. As described in a 1911 magazine article, his small office "would be dark even if the one window were washed, a cataclysm of which there seems no immediate prospect. There are two desks, a typewriter and a printing-press in it, and countless shelves loaded with papers, pictures, magazines, stereotype plates and one thousand other things which have spilled out upon the floor and risen like strange, dirty snowdrifts breast high in the corners. Sam Loyd says he does all his business on a cash basis and keeps no books. The reason probably is that he couldn't find the books. Th-at would be too much of a puzzle even for him."

.Loyd's first big success with a premium came with his invention, at the age of 17, of the Trick Donkeys. The task is to ar-ran ge three cardboard rectangles so that two riders are astride two- donkeys. The puzzle is still widely used as a giveaway item throughout the world. Loyd's original version, which P. T. Bam urn distributed by the millions to publicize his circus, is reproduced in the chapter on Loyd in The Scientific American Book of Mathematical Puzzles & Diversions. Modernized versions can be found in the article "Problem-solving" in Scientific American for April 1963; on page 124 of The Mind (a Life Science Library book), and in an American Can Company advertisement in Time for March 22, 1968. Loyd once related in an interview that Barnum used to make periodic treks to his office saying, "Hang- it all, Sam, show me how to do my puzzle. I've forgotten again.

Another of Loyd's early premium hits, even more widely used today than then, is nothing more than a pencil with a short loop of string on its eraser end. Loyd designed the trick for agents of the New York Life Insurance Company, who would attach the pencil to prospective customers' coats with the promise to remove it if a sale was consummated. The loop is placed around a lapel buttonhole, then the cloth is pulled forward through the loop until the pencil goes back far enough for its point to enter the buttonhole from behind. When the pencil is pulled foward through the hole, it is fastened in such a way that it seems impossible to remove the pencil without cutting the string.

Loyd produced numerous geometric puzzles, but none with a more unexpected solution than his Pony Puzzle, shown in Figure 72 exactly as he himself originally drew it. The problem is to rearrange the six pieces to make the best possible picture of a trotting horse. In his Cyclopedia of Puzzles Loyd claimed that over one billion copies of the Pony Puzzle had been sold.

The most spectacular of all Loyd premiums, by all odds, was his mind-bending "Get off the Earth" paradox. He patented the device in 1896 and first sold it as a premium to advertise Bergen Beach, a resort that had just opened in New Jersey. Copies of the original are now rare collector's items. The art, based on Loyd's sketches, was done by Anthony Fiala, then a cartoonist on The Brooklyn Daily Eagle. (Later Fiala was commander of the Ziegler Polar Expedition of 1903-1905 and wrote a book about it, Fighting the Polar Ice.) The puzzle consisted of a cardboard disk fastened by a central rivet to a card-

Sam Loyd's Pony Puzzle board rectangle. A tab attached to the disk projected through a curved slot in the backing so that by moving the tab up or down the disk could be rotated to two positions [see Figures 73, 74]. In one of the positions you can count 13 Chinese warriors. When the disk is turned to the other position, there are only 12 warriors. Which man vanishes, the premium asked, and where does he go?

For more than a year Loyd filled his weekly puzzle column in The Brooklyn Daily Eagle with letters from readers attempting to explain this astonishing phenomenon. In Loyd's own lengthy, mock-serious explanation (January 3, 1897, page 22) he called attention to a curious feature easily overlooked by a person unless he has tried the difficult task of drawing human figures properly around the rim of a disk. "The grotesqueness of the figures and a necessary legerdemain feat of changing a right leg for a left one between the fourth and fifth men does the trick. If it were not for that particular acrobatic feat, all of the men on the left side would come down head end first. Some pirates, who brought out the puzzle in different parts of Europe, with different figures, found it absolutely necessary to retain that flop over of the legs."

At that time Americans were aroused over the "yellow peril," a fact that explains the premium's unpleasant racist connotations. As if not to be partial to either China or Japan (the two nations had been at war in 1894), Loyd provided the Metropolitan Life Insurance Company in 1897 with a more elaborate Japanese version of his paradox. Nine Japanese men alternate around the circle with eight lanterns. When the disk is turned, one man vanishes and a ninth lantern appears, giving the

Loyd's greatest puzzle starts with 13 Chinese warriors

Figure 74

Figure 74

impression that a man has turned into a lantern. The premium announced a contest with 20 prizes, from \$5 to \$100, for the best explanation. Although the names of the winners were printed, none of the prize-winning letters were published. Perhaps the reason is to be found in a typical "explanation" that was quoted: "When the handle is down I find nine Japanese, but when the handle is up there are only eight, as one has disappeared." In 1909 Loyd issued a third version of the paradox called Teddy (Roosevelt) and the Lions, in which an African native seems to turn into a lion. It too is reproduced in the chapter on Loyd in The Scientific American Book of Mathematical Puzzles & Diversions.

The basic principle behind Loyd's three versions was not original with him. He simply took earlier linear forms of the paradox and bent them into circular shape. I have seen in a private collection of advertising cards an 1880 premium, copyrighted by Wemple and Company of New York, called "The Magical Eggs." A rectangular card is cut into four smaller rectangles. Different arrangements of the pieces produce eight, nine or 10 eggs. Scores of variations on this paradox have since been used in the United States and abroad. The latest and funniest version, in three pieces, is "The Vanishing Leprechaun," skillfully drawn by Pat Patterson, a Toronto graphic designer, and issued in Canada by William Elliott, a producer of puzzles and magic tricks. The paradox is repro-

Figure 75

Figure 75

Which leprechaun vanishes?

duced in Figure 75. An eight-by- 19-inch two-color print on pa-perboard can be obtained from W. A. Elliott Company, 212 Adelaide Street West, Toronto 1, Canada.

From hundreds of other mathematical premiums I select as a final specimen a card that advertises a brand of Scoth whisky [see Figure 76]. This seemingly trivial addition problem trips most people whether they have had a drink or not. To obtain the correct sum the use of a pocket or desk calculator is advised.

### Figure 76

Can you add this column of figures? Place your hand over all but the top number and move it down the column, revealing one number at a time. Add all the numbers, as you go along. When you get the total, turn over for correct answer.

1000 40 1000 30 1000 20 1000 10

An advertising giveaway card

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