These puzzles, mathematical problems, and fallacies are treated more fully later on; but, as a few words of explanation at this point will be of assistance, we will take a look at the following display of shoes, wherein the turning over of the rhomboid piece has to be utilized. In the "tailor's gooses" we have a simple version of the paradoxical fallacies which will be introduced again in more difficult forms. The first, as may be seen, is constructed with seven pieces, while the other one, supposed to be of the same form and size in every respect, is built with only six pieces and has a superfluous triangle. It being self-evident that the same form and dimensions cannot be constructed with a different number of pieces, we are asked to determine which is correct.
The Chinese are wonderfully prolix in the treatment of a subject, and will ring the changes upon every possible variation, as if it was an all-important point to show that a certain figure can be constructed in many different ways. The designs of the six shoes are taken from a Chinese book published in Canton in 1690, which shows three styles of heels, six kinds of toes, and six different tops, all of which being interchangeable upon a permutative basis should produce 108 different styles, more or less, and we are coolly asked to guess the possible number of changes!
In many instances the designs are marked with a number, which is supposed to tell in how many different ways the same object can be constructed or varied.
Connecting Li Hung Chang's statement that he "knew all the figures of the seven books of Tan before he could talk," with his remarkable and hitherto unexplained reference to Tangrams as being "a progressive philosophy with seven interpretations," we get some idea of the scope of the work as described by Professor Challenor. He says that Confucius makes several allusions to Tangrams, which he likens to "a game where the babes learn the form of things; youths exercise their wits; men study mathematics; artists get designs; poets fire the imagination, and the wise ponder over the past, present, and future."
The philosopher Choofootze is said to have discoursed at length upon "the seven interpretations of Tangrams," although the same does not appear in any of the wise sayings ascribed to him; so we are compelled to accept the views of Confucius, which were so happily endorsed by li Hung Chang in his description of the pastime.
Many of the forms may readily be named, yet we detect a certain progressive degree of difficuty, suggestive of a puzzle feature combined whith a very instructive school of design. It may be of interest at the present stage of the game to challenge the reader to guess the following subjects, which will be referred to again as pertaining to an important feature of the work :
It can be proven that the seven volumes were written at different periods, and Professor Challenor, who was a life-long enthusiast upon the subject, attemps to give some idea of the date of their appearance, and quotes, in support of his argument, a well-known Chinese saying which speaks of "the fool who would write the eighth book of Tan." This does not harmonize altogether with his description of the seven volumes as one complete inseparable scheme wherein the first imperfect forms are to be reproduced afterward.
As a matter of fact, it would appear as if Tangrams might be studied to advantage from any number of different standpoints. Gustave Doré, whose power of imagery was almost akin to madness, first developed his love of art by designing figures with the seven magical pieces.
Lewis Carroll, who had the temerity and power of imagination to portray the vagaries of dreamland, was a votary of the pastime, and when he became the profound professor of mathematics at Oxford University was wont to employ the seven pieces to expound the problems of Euclid.
It may be mentioned that at a recent sale of Lewis Carroll's library, there was purchased by Henry C. Dudeney, the noted puzzlist, a little work, entitled "The fashionable Chinese Puzzle," containing 323 designs in tangrams. It was issued at a place called Sidmouth, England, and we get an approximate date of publication from the following interesting bit of information taken from the introduction: "This ingenious contrivance has for some time past been the favorite amusement of the ex-Emperor Napoleon, who, being now in a debilitated state, and living very retired, passes many hours a day in thus exercising his patience and ingenuity."
Just imagine the great fighter finding relaxation for his tired brain in illustrating the incidents of his eventful career. We wonder if it ever occured to him to portray that disastrous retreat from Moscow after the following manner :