The 8th Book of Tan by Sam Loyd
When it dawns upon us that there is something in Tangrams more closely allied to the exact sciences than to a simple pastime we discover that there is a mystery in the dimensions of the pieces somewhat akin to the marvellous measurements of the Pyramids, whose ninth power proved the sun to be 91,840,000 miles distant, when for thousands of years it was supposed to be less than 20,000. The proportions of the Pyramid showed the ratio of the diameter of a circle to its circumference, and also proved that the Egyptians knew the distance to the center of the earth.
If you take an accurate rule and measure the dimensions of the following reproduction of the Tangram square it will prove to be 2 1/8 inches on its four sides. This being equivalent to 34/16 of an inch, we get 34x34 = 1,156 as the square. The short sides of the largest pieces appear to be 24/16 of an inch long, while the smaller pieces represent half of the dimensions, viz., 17/16 and 12/16. The four dimensions, therefore, are 34, 24, 17, and 12, and although it has been shown that the outside measurement 34x34 gives 1,156 as the area of the square, the measurement of the seven pieces computed separately makes it to be but 1,152 ! and the same is given as one of the unsolvable mysteries which Li Hung Chang said "cannot be explained."
The feature just described may throw some tangible light upon the Chinese philosophy quoted by Rev. Dr. Holt, which has been described by other writers as "pagan gibberish": "The illimitable produces the great extreme; the great extreme the two principles; the two principles produce the four dimensions, and the four dimensions develop what the Chinese call the eight diagrams of Feu-hi over 3,000 years ago."
The illimitable would thus be made to be the undiscovered pi of the square; the mysterious ratio of the sides of a square to its hypothenuse, just as in the diameter of the circle to its circumference.
The first of the above instalment of figures shows a mysterious change from one of the series of badges previously given, which paradoxical feat can only be appreciated by a careful study of the mathematics of Tangrams.
The second figure shows that a complete set of seven pieces forms a square, equal to sixteen triangles of the smaller size. The proportion, therefore, of the smallest piece to the whole set is what Mr. William Jennings Bryan would call the correct ratio of sixteen to one.
The small triangles, therefore, represent the unit value or one-sixteenth. The rhomboid piece, as well as the second size triangle and square, equal two-sixteenths each. The large triangles are equal to four-sixteenths.
In the puzzle of the two arrow-heads, given on page 32, the large one contains eleven-sixteenths and the smaller one five-sixteenths. Their relative proportions or sizes, therefore, are as 5 is to 11. A knowledge of this principle is absolutely necessary to solve some of the more difficult puzzles or to penetrate the mysterious features already discussed, as well as to determine the possibility of making some of the forms.