Jeu du Carré Solution
Mathematically, you'll see that the solution (Example 1) does not give a perfect square. Using the box as a tray for the puzzle helps to hide the irregularity and assure that the solution is correct, but it is not a very satisfying solution. In fact, if you found the solution, you probably didn't think it was correct! This solution seems just too far off to even be loosely acceptable by sight as a square (especially when the pieces are not being fit into a tray).
The puzzle is based on the dissection of one square into five equal smaller squares (Example 2). There are two extra B pieces in Example 2, so the area of two B pieces is actually missing from the solution above. This missing area is in fact equal to half of a square, or 10% of the area of the puzzle in Example 1! This is extreme fudging, and that solution is not very satisfying mathematically or aesthetically.
To try to keep only 8 pieces, one could fill in the missing area from the solution by dissecting a square as in Example 3. This gives though two new larger sizes of pieces, piece C and piece D. For the pieces in my copy of the French puzzle, one of the tetragons seemed to be a bit larger than the others by about 4% (39.5mm compared to 38mm on the longest side). The difference between the sizes of piece A and piece D in Example 3 (a mathematically correct square) is about 12.6% and is easy to notice, especially if the pieces of Example 3 are rearranged as in Example 3b.
Rather than just one larger trapezoid, the solution in Example 3 shows that two trapezoids (pieces C) should have been slightly larger. So I don't know if this piece was only accidentally larger in my set (because of irregularity of handcut pieces) and there should have been two larger pieces, or if one larger trapezoid is correct for the puzzle and there is a different answer than I came up with.
The point of all this, though, is that the puzzle is made so that the solution doesn't fit very well in the box because the mathematics of the pieces is "fudged", the solution doesn't really make a square. In fact, it doesn't come close! This caused me to become very interested in trying to find out what the best set of pieces would be in order to give the best puzzle using only eight pieces.
Now the challenge for all the mathematicians out there:
Example 1 shows how five of piece A and three of piece B can be arranged into a pretty poor approximation of a square. Can you redesign the Jeu du Carré puzzle with new dimensions for piece A and piece B, so that 5 identical piece A's and 3 identical piece B's make the best approximation of a square?
The final solution should retain the relative configuration of pieces in Example 1. Your solution does not need to be a square, but it should be the best "eye fooling" square possible-- the solution that most looks like a square within a reasonable doubt.
I am not trained as a mathematician, but I intuitively worked my way using geometry to what I believe to be the best solution of this problem. Send me your solution at email@example.com and I'll send you mine!