In this puzzle I ignore the known dimensions of our square and work on the assumption that it is 13n by 13n. The value of n we can afterwards determine. Divide the square as shown (where the dotted lines indicate the original markings) into 169 squares. As 169 is the sum of the two squares 144 and 25, we will proceed to divide the veneer into two squares, measuring respectively 12 x 12 and 5 x 5; and as we know that two squares may be formed from one square by dissection in four pieces, we seek a solution in this number. The dark lines in the diagram show where the cuts are to be made. The square 5 x 5 is cut out whole, and the larger square is formed from the remaining three pieces, B, C, and D, which the reader can easily fit together. Now, n is clearly 5/13 of an inch. Consequently our larger square must be 60/13 in. x 60/13 in., and our smaller square 25/13 in. x 25/13 in. The square of 60/13 added to the square of 25/13 is 25. The square is thus divided into as few as four pieces that form two squares of known dimensions, and all the sixteen nails are avoided. Here is a general formula for finding two squares whose sum shall equal a given square, say a squared. In the case of the solution of our puzzle p = 3, q = 2, and a = 5. ________________________ 2pqa \/ a squared( p squared + q squared) squared - (2pqa) squared --------- = x; --------------------------- = y p squared + q squared p squared + q squared Here x squared + y squared = a squared. 160.--THE TWO HORSESHOES. The puzzle was to cut the two shoes (including the hoof contained within the outlines) into four pieces, two pieces each, that would fit together and form a perfect circle. It was also stipulated that all four pieces should be different in shape. As a matter of fact, it is a puzzle based on the principle contained in that curious Chinese symbol the Monad. (See No. 158.) [Illustration]