Continuing our analysis of the square, rhomboid, or other geometrical shapes which can be constructed by the seven pieces, we find a paradoxical demonstration that the same forms may be built from either six, seven, or eight pieces!
The first three figures show imperfect designs containing seven pieces each, which might be completed by the addition of an eighth piece, and yet we know that they can be made perfect with only seven pieces!
Strange to relate, the first of the figures is a famous puzzle, known as the Japanese Problem, the authorship of which has been claimed by several puzzlists and mathematicians. It is given in form of a square with one-ninth removed, and the conditions are to "discover how to devide the same into three pieces which will fit together and form a perfect square." It was a beautiful and difficult puzzle 3,000 years ago!
In Fig. 4 is given the answer to Fig. 1, showing how the same may be built with seven pieces, which would therefore require an eighth piece to complete, whereas we know that by a different arrangement the square can be made complete with only the seven!
The heavy white lines are given to show how to cut it into three pieces which will fit together and solve the Japanese problem.
In the same way in Fig. 5 is shown a perfect rhomboid constructed from the seven pieces, and yet in the next four designs are shown similar forms, each one containing the same seven pieces, although one of them appears to be missing, so that, as in Fig. 8, it would require nine pieces to make the form perfect.
We will now pry into the mysteries of the pyramids and ask the puzzlists and mathematicians to explain how it is possible to construct a perfect pyramid with seven pieces, or the imperfect ones, with one piece lacking, with the same number! it reminds one of the well-known story of the king who, in attempting to destroy one of the pyramids, removed one top stone, which the entire ability and resources of his nation could not restore. The encyclopædias say: "The Pyramids were built by the respective kings for their own tombs, and were begun at the beginning of each reign, and their different sizes therefore correspond with the length of the reign." In which case they must have been constructed by the Irishman who knew how to put up the chimney first and then build down to the ground!
Two of the designs are marked off with white lines so as to show how to make a complete as well as an imperfect pyramid with the same seven pieces.
An examination of the second book of Tan shows that every figure there given is susceptible of being represented in two other forms; one a contraction and the other an extension, and that the expended or extreme form produces the mysterious paradoxical principle of the missing piece.
Take, for example, the following shapes, to be found in all modern collections, and note the different rendering of the same subject, as well as the trick of the vanishing piece shown in the second figure. By this paradoxical way of building it can be shown that some figures which contain seven pieces might be made with eight, or might also be built just as perfect with only six, so as to leave a piece to spare of any desired shape.