The 8th Book of Tan by Sam Loyd
The Chinese are past masters in the matter of symmetry, and with the aid of Tangrams prove the truth of their old saying that "two ugly make a pretty." Any crooked, irregular line creates a beautiful and artistic design when duplicated in reverse, as if reflected in a mirror. Observe the lines on one side of any of the following figures and the result of duplicating the same.
Scattered through the original books are many little groupings of a disconnected character which do not appear to represent any particular point or narrative, fable or proverb, such as can be readily detected in other parts. These sketches undoubtedly conform harmoniously with the progressive development of the work, in that they introduce new costumes, pastimes, etc. It is believed, however, that they show the origin of an ancient but still popular style of Chinese picture-book for the young, which presents a number of illustrations for which the parents or some of the more clever children must invent appropriate stories:
Here is a slippery little scene, suggestive of wintry out-door sports on the ice and snow, with a fairly good representation of a Canadian tobogganing mischap. Space will not permit of placing the sketch properly bias-ways on the page, as occurs not unfrequently in the Chinese books, when, by a slight effort of the imagination, you might readily see them slide:
Despite of the careless or inaccurate manner of printing the designs in the original works, which is equivalent to giving imperfect data for a problem, there appears to be an over punctilious requirement for exact answers. So many changes are rung upon the slightest possible variation in a pattern, as if there were great merit in showing different ways of construction, that the same becomes monotonous. We can readily see how four designs can be produced by the slightest modification in the following baptismal fonts, but it is not so easy to master the insignificant change necessary to alter the style of the Jockey's caps.
With the figures of the three cabinet organs we reach that borderland of mystery in the black art which can only be solved mathematically. The second and third organ in the original Chinese works are exactly alike; each is built from the same seven pieces, and yet the last one shows a folding lid which calls for an extra piece! Is this a fallacy or an optical illusion?
This paradoxical feature of Tangrams, whereby almost any of the geometrical forms appear to be susceptible of being constructed at pleasure with one piece more or less, is the great mystery referred to by ancient Chinese writers.
It has never been touched upon or even discovered by any writer within 2,000 years, although an important principle of Tangrams. Take the following magic dice-cup trick: Fig. 1 represents the cup, built with the seven pieces. Fig. 2 represents the same cup, of the same size, with a vacant space, although all the pieces are used! Observe that the third figure is also built with the same pieces, but has a still smaller vacancy to fill. Of course, it is a fallacy, a paradox, or an optical illusion, for you will say the feat is impossible! But look at the answers and see if any light is thrown on the subject!
Count the pieces and measure the dimensions carefully! The seventh and eighth figures represent the mysterious square, built with seven pieces; then with one corner clipped off, and still the same seven pieces employed. Explain this also, if you can, for there are greater mysteries yet to investigate!