We take a smaller instance for the same problem (see Figure at the right). Now one can see immediately where the problem is: the diagonal in the 3 × 8 rectangle ist not a straight line. Hence, none of the forms is a triangle. A quick calculation also shows that they cannot be triangles. The proportion of the height and the length of these triangles is 3 8. At the position 5 at the length, the height of this “triangle” is 2. But we know that 3 8 ⁄= 2 5. Hence, these forms are not triangles.
How did we discover these figures? They are “special numbers” 3 × 8 is almost 5 × 5, and 8 × 21 is almost 13 × 13. Sounds familiar? 1, 1, 2, 3, 5, 8, 13, 21, 34,... These are consecutive Fibonacci numbers. These numbers are defined by:
The Fibonacci numbers have many interesting properties, one of them is
Proof (by induction):
The property is true for n = 1, since 1 ⋅ 0 - 1^{2} = (-1)^{1}.
Supposing the property is valid for n, then it is valid for n + 1, namely we substitute F_{n-1} with F_{n+1} -F_{n} in the property and we get:
By multiplying with -1 and transforming it we finally get:
That proves the property.
In particular, for n = 6 we have: 13 ⋅ 5 - 8^{2} = (-1)^{6} producing the initial graph of the problem. We also have: 8 ⋅ 3 - 5^{2} = (-1)^{5}.