There are at least $\ 2^{n-1}\ $ triangulations of $\ [0;1]^n\ $ into nice $n$-simplexes. I am quite sure that there are exactly $\ 2^{n-1}\ $ of them.

**CONSTRUCTION**   First I'll present one *nice triangulation* of $\ [0;1]^n$.

>**NOTE**   Włodzimerz Kuperberg and myself have obtained this *nice triangulation* (see below; but of course we didn't use term *nice*) independently. I did it during the first half of 1996 (I don't know Włodek K's exact date). Looking back, it is closely related to the old homological/combinatorial triangulation of a prism (one may check for example a Pontryagin's small monograph on Combinatorial Topology; see Лев Семёнович Понтря́гин); small but great.

Let $\ \pi:\{1 \ldots n\}\rightarrow \{1 \ldots n \}\ $ be an arbitrary permutation. Let

$$\Delta_{\pi}\ :=\ \{(x_1\ldots x_n)\in [0;1]^n\ :\ \forall_{k=2}^n\ x_{\pi(k-1)}\le x_{\pi(x_k)}\} $$

The family of $n$-simplexes $\ \Delta_{\pi} : \pi\in S_n,\ $ together with their simplicial faces, forms a nice triangulation (which has $\ n!\ $ simplexes of dimension $\ n.\ $ We get the $\ 2^{n-1}\ $ such triangulations due to the $\ \mathbb Z_2^n\ $ action of isometries on $\ [0;1]^n$.

I feel that there are no other nice triangulations in $\ [0;1]^n$.

>**COUNTEREXAMPLE(?) in dimension 3**   If you see a counterexample in dimension $\ 3\ $ then here is a place to write down a $5$-th nice triangulation (just replace each star by a real number)--here simplexes are specified by their vertices:

 1. simplex:   $((*\ *\ *)\ \ (*\ *\ *)\ \ (*\ *\ *)\ \ (*\ *\ *))$
 2. simplex:   $((*\ *\ *)\ \ (*\ *\ *)\ \ (*\ *\ *)\ \ (*\ *\ *))$
 3. simplex:   $((*\ *\ *)\ \ (*\ *\ *)\ \ (*\ *\ *)\ \ (*\ *\ *))$
 4. simplex:   $((*\ *\ *)\ \ (*\ *\ *)\ \ (*\ *\ *)\ \ (*\ *\ *))$
 5. simplex:   $((*\ *\ *)\ \ (*\ *\ *)\ \ (*\ *\ *)\ \ (*\ *\ *))$
 6. simplex:   $((*\ *\ *)\ \ (*\ *\ *)\ \ (*\ *\ *)\ \ (*\ *\ *))$

Of course a picture can be nicer and (almost :-) as good.