> Considering fibonacci - golden ratio relation, is it possible to devise a explict connection/translation between the puzzle 104's graph and the x²=1+x equation?

Here's my attempt -

Assume you want to compute the number of paths of length \$$n\$$, starting from \$$x\$$. You can start counting by taking one of two paths:

- Go along \$$f\$$, and end up at \$$x\$$. Then count the number paths of length \$$n - 1\$$
- Go along \$$h \circ g\$$, and end up at \$$x\$$ again. Then count the number of paths of length \$$n - 2\$$.

Let \$$f_n\$$ be the number of paths. Then

\$f_n = f_{n-1} + f_{n-2} \tag{★} \$

Now consider

\$\phi := \underset{n \to \infty}{\mathrm{lim}} \frac{f_{n}}{f_{n-1}} \$

With (★) we have

\\begin{align} \phi & = \underset{n \to \infty}{\mathrm{lim}} \frac{f_{n}}{f_{n-1}} \\\\ & = \underset{n \to \infty}{\mathrm{lim}} \frac{f_{n-1} + f_{n-2}}{f_{n-1}} \\\\ & = \underset{n \to \infty}{\mathrm{lim}} \left(1 + \frac{f_{n-2}}{f_{n-1}} \right) \\\\ & = 1 + \underset{n \to \infty}{\mathrm{lim}} \frac{f_{n-2}}{f_{n-1}} \\\\ & = 1 + \underset{n \to \infty}{\mathrm{lim}} \frac{f_{n-1}}{f_{n}} \\\\ & = 1 + \underset{n \to \infty}{\mathrm{lim}} \left(\frac{f_{n}}{f_{n-1}}\right)^{-1} \\\\ & = 1 + \left(\underset{n \to \infty}{\mathrm{lim}} \frac{f_{n}}{f_{n-1}}\right)^{-1} \\\\ & = 1 + \phi^{-1} \\\\ \end{align} \

Hence \$$\phi = 1 + \phi^{-1}\$$. Now multiply both sides by \$$\phi\$$. This gives:

\$\phi^2 = \phi + 1 \$

...as you wanted to see...