## [ tt x [ n

3. The large polygon in Figure 15 can be cut into five congruent polygons as shown. The method obviously enables one to dissect the polygon into any desired number of congruent shapes. L. Vosburgh Lyons first published this in The Pallbearers Review for July, 1969, page 268.

Figure 15

Figure 15

Solution to the dissection problem

4. The 32 chess pieces can be placed so that 36 "moves" are needed to transfer the pieces to a correct starting position with black at the top and white at the bottom [see Figure 16].

It was stated in the problem that it was not necessary for black to be at the top. However, if the final position is black at the bottom, then 37 moves are required to produce a starting pattern with the queens on the right color. If it is required that black be at the top, then a standard starting position, with white at the bottom, requires 38 moves to effect the change.

5. The Texas oilman's bank deposit problem reduces to the Diophantine equation 2,584x + 4,18131= 1,000,000. It can be solved by Diophantine techniques such as the continued-frac-

A solution to the chess problem tion method explained in Chapter 2. The first two deposits are \$154 and \$144.

The shortcut, given that x and y start the longest possible Fibonacci chain terminating in 1,000,000, rests on the fact that the longer a generalized Fibonacci series continues, the closer the ratio of two adjacent terms approaches the golden ratio. To find the longest generalized Fibonacci chain that ends with a given number, place the number over x and let it equal the golden ratio. In this case the equation is

0 0