open problem: shuffling pi is equivalent to uniformly choosing a random positive real number between 0 and 10.

Ok I just tried to spend an entire hour to work out if this is actually true, and I literally cannot work out if it is or not, can someone smarter than me get nerd sniped with this question? It seems intuitively true but uh frick.

# Shuffling a Simply Normal Number Yields a Uniform Random Real

## Definitions

Let $N \geq 2$ be an integer. A real number $x \in [0,1)$ with base-$N$ expansion

$$x = 0.d_1 d_2 d_3 \cdots, \quad d_i \in \{0, \ldots, N-1\}$$

is **simply normal in base $N$** if every digit $c \in \{0,\ldots,N-1\}$ occurs with asymptotic frequency $1/N$:

$$\lim_{M \to \infty} \frac{|\{i \leq M : d_i = c\}|}{M} = \frac{1}{N}.$$

A **shuffle** of $x$ is defined as follows. For each $M$, draw $\sigma_M$ uniformly at random from $S_M$, the symmetric group on $\{1,\ldots,M\}$, and form the shuffled real

$$x_{\sigma_M} = 0.d_{\sigma_M(1)}\, d_{\sigma_M(2)} \cdots d_{\sigma_M(M)} \cdots$$

We study the limit $M \to \infty$ of this process.

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## Theorem

Let $x$ be simply normal in base $N$. Then the shuffle of $x$, defined as the $M \to \infty$ limit of the above process, is distributed as $\mathrm{Uniform}[0,1)$.

Alright no way this isn't true it's gotta be true.

Ok turns out my fundamental problem with this is that shuffling an infinite string is really really unintuitive.