Gnomon II Thank you Andrei Lazanu, age 12, School No. 205 Bucharest, Romania for this solution. I drew separately the gnomons for Fr with r even and r odd, up to r = 12. I replaced the lengths of the sides with the terms in the Fibonacci sequence, and then I tried to find a rule for the gnomons corresponding to F2n and F2n+1 respectively. For Fr with r even, I obtained the following table, with lj for j = 1, 2,..., 6, the lengths of the sides of the gnomon: l1 l2 l3 l4 l5 l6 F2 F0 F0 F1 F1 F2 F2 F4 F1 F1 F2 F2 F3 F3 F6 F2 F2 F3 F3 F4 F4 F8 F3 F3 F4 F4 F5 F5 F10 F4 F4 F5 F5 F6 F6 F12 F5 F5 F6 F6 F7 F7 So, for the gnomon F2n I observed the following rule: F2n Fn-1 Fn-1 Fn Fn Fn+1 Fn+1 This is an inductive process and at the end I have to verify it. For Fr with r odd, I obtained the following table, with lj for j = 1, 2,..., 6, the lengths of the sides of the gnomon: l1 l2 l3 l4 l5 l6 F5 F4 F3 F3 F2 F2 F1 F7 F5 F4 F4 F3 F3 F2 F9 F6 F5 F5 F4 F4 F3 F11 F7 F6 F6 F5 F5 F4 For the gnomon F2n+1 the following rule could be observed: F2n+1 Fn+2 Fn+1 Fn+1 Fn Fn Fn-1 Using these rules, the gnomons for F20 and F21 have the following lengths of the sides: F20 F9 F9 F10 F10 F11 F11 F21 F12 F11 F11 F10 F10 F9 Keeping into account that the area of the gnomon equals the corresponding number in the Fibonacci sequence, and also the shape of each kind of the gnomon, I found the following recursive relations: F2n = Fn (Fn-1 + Fn+1) F2n+1 = Fn+2 Fn + Fn-1 Fn+1 I hope that both relations are satisfied, but I don't know so much algebra to verify them. As discovered in the solution to Gnomon I from May 2001 we also have the relations F2n= (Fn+1)2 - (Fn-1)2 and F2n+1=(Fn+1)2 + (Fn)2. These four relations are illustrated in the gnomon diagram below. The diagram shows gnomons whose areas (labelled in black) are Fibonacci numbers. It shows the lighter and darker yellow gnomons F2n-1 and F2n fitting together to make F2n+1 with the total yellow area given by F2n-1+F2n=F2n+1. The yellow gnomon representing F2n+1 fits together with the blue gnomon representing F2n to make the gnomon for F2n+2. The red labelling shows the lengths of the edges. Fn+1 Fn Fn F2n Fn Fn-1 Fn-1 Fn-1 Fn-1 Fn-1 Fn-2 Fn-1 Fn+1 F2n F2n-1 Fn Fn+1 Fn