If you are familiar with SingingBanana on youtube, he posted the following question:
There is a 10 digit number where the first digit tells me how many 0 there are in the number, the second digit tells me how many 1 there are in the number, and so on. What is this number?
I formulated the problem as a discrete dynamical system.
Let $\mathbf{x^0}\in \mathbb{R}^{10}$ be an initial condition such that $0\leq\mathbf{x}_i\leq9$ for all $i$ between 0 and 9.
Then, the next iteration will produce
$$ \mathbf{x^{n+1}} = f(\mathbf{x^n})$$
where
$$\mathbf{x^{n+1}}_i = \mbox{How many $i$ occur in $\mathbf{x^n}$}$$
Easy to compute, or so I thought. I'm getting stuck in oscillations between 7 1 0 1 0 0 1 0 0 0 and
6 3 0 0 0 0 0 1 0 0.
So, I am stuck in a two cycle and need to find the actual steady state.
I've reasoned that the answer is 621000100. Any ideas on how to find this analytically or computationally?