gnomons
The diagram shows gnomons whose areas (labelled in black) are Fibonacci numbers. It shows the lighter and darker yellow gnomons $F_{2n-1}$ and $F_{2n}$ fitting together to make $F_{2n+1}$ with the total orange and yellow area given by $F_{2n-1}+ F_{2n} = F_{2n+1}$. The gnomon representing $F_{2n+1}$ fits together with the blue gnomon representing $F_{2n}$ to make the gnomon for $F_{2n+2}$. The red labelling shows the lengths of the edges.

Thank you Andrei from School No. 205 Bucharest, Romania for the following solution.

I drew separately the gnomons for F r 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 F 2n and F 2n+1 respectively.

For Fr with r even, I obtained the following table, with l j for j = 1, 2,..., 6, the lengths of the sides of the gnomon:

l 1 l 2 l 3 l 4 l 5 l 6
F 2 F0 F 0 F 1 F 1 F 2 F 2
F 4 F 4 F 4 F 2 F 2 F 3 F 3
F 6 F 2 F 2 F 3 F 3 F 4 F 4
F 8 F 3 F 3 F 4 F 4 F 5 F 5
F 10 F 4 F 4 F 5 F 5 F 6 F 6
F 12 F 5 F 5 F 6 F 6 F 7 F 7

So, for the gnomon F2n I observed the following rule:

F 2n F n-1 F n-1 F n F n F n+1 F n+1

This is an inductive process and at the end I have to verify it.

For F r with r odd, I obtained the following table, with lj for j = 1, 2,..., 6, the lengths of the sides of the gnomon:

l 1 l 2 l 3 l 4 l 5 l 6
F 5 F 4 F 3 F 3 F 2 F 2 F 1
F 7 F 5 F 4 F 4 F 3 F 3 F 2
F 9 F 6 F 5 F 5 F 4 F 4 F 3
F 11 F 7 F 6 F 6 F 5 F 5 F 4

For the gnomon F2n+1 the following rule could be observed:

F 2n+1 F n+2 F n+1 F n+1 F n F n F n-1

Using these rules, the gnomons for F20 and F21 have the following lengths of the sides:

F 20 F 9 F 9 F 10 F 10 F 11 F 11
F 21 F 12 F 11 F 11 F 10 F 10 F 9

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:

F 2n = F n (F n-1 + F n+1 ) F 2n+1 = F n+2 F n + F n-1 F n+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

F 2n = (F n+1 ) 2 - (F n-1 ) 2

and

F 2n+1 =(F n+1 ) 2 + (F n ) 2 .

These four relations are illustrated in the gnomon diagram above.