One side of this, there is a standard picture. In $\mathbb R^n,$ take the $n$ points
$$ (1,0,0,\ldots,0), $$
$$ (0,1,0,\ldots,0), $$
$$ \cdots $$
$$ (0,0,0,\ldots,1). $$
These are all at pairwise distance $\sqrt 2$ apart.
At the same time, they lie in the $(n-1)$-dimensional plane
$$ x_1 + x_2 + \cdots + x_n = 1. $$ If you wish to work at it, you can rotate this into $\mathbb R^{n-1};$ in any case, $n$ points in $\mathbb R^{n-1}.$
If you prefer, you can keep $\mathbb R^n$ and place a point numbered $(n+1)$ at
$$ (-t,-t,-t, \ldots, -t) $$
for a special value of $t > 0$ that makes all the distances $\sqrt 2.$ I get
$$ n t^2 + 2 t - 1 = 0 $$ or, with $t>0,$
$$ t = \frac{\sqrt {n+1} - 1}{n } = \frac{1}{\sqrt {n+1} + 1} $$