Geometric Approach

For this geometric picture, we use a state space diagram with one axis along the $|w⟩$ direction, the target state we want to find, and the other perpendicular to $|w⟩$. That perpendicular direction represents a superposition of all the other basis states, each of which individually is perpendicular to $|w⟩$. Let’s call it $|ψ_{other}⟩$. In the following figure, we will also include the equal-amplitude superposition state

$$\begin{aligned}\left|s_0\right\rangle & =\frac{1}{\sqrt{2^n}}(|00 \ldots 0\rangle+|00 \ldots 1\rangle+\ldots+|11 \ldots 1\rangle) \\ & =\frac{1}{\sqrt{2^n}} \sum_{x=0}^{2^n-1}|x\rangle=\left\{\frac{1}{\sqrt{2^n}}|x\rangle\right\}, \end{aligned}$$

where we made use of the curly-brace state notation to avoid the summation symbol.