TITLE: How many unit simplices are needed to cover a unit $n$-cube?
QUESTION [18 upvotes]: The volume of an $n$-dimensional simplex of unit edge length is
$$V(n) = \frac{\sqrt{n+1}}{n! 2^{n/2}} \;,$$
so at least $\lceil 1/V(n) \rceil$ such simplices are needed to cover the unit $n$-cube.

Q1. What is an upperbound on the number of unit simplices needed to cover a unit $n$-cube?

I suspect that the $1/V(n)$ lowerbound is weak; 
perhaps there is no $c>0$ such that $c/V(n)$ is an upperbound?
For $n{=}2$, $1/V(2)=4/\sqrt{3} \approx 2.3$, but $4$ equilateral triangles are needed.
For $n{=}3$, $1/V(3)=6\sqrt{2} \approx 8.5$; 
I don't know how many tetrahedra are needed to cover the cube.

Q2. How many unit tetrahedra are needed to cover a unit cube?

REPLY [3 votes]: If I place a unit-edge tetrahedron at each face of a unit-edge icosahedron, so that they share a face and the tetrahedron is pointed into the icosahedron, I get 20 tetrahdera that together cover the icosahedron. I can now place a unit cube in such a way that nearly all of it is covered by the icosahedron. Only two corners will stick out, and we can easily cover them by adding two more tetrahedra. Therefore, a unit cube can be covered by 22 tetrahedra.
Coordinates for the cube:
{{-0.250201, -0.657675, -0.504839}, {-0.507631, -0.535251, 0.453671}, {-0.439869, 0.308569, -0.679191}, {-0.697299, 0.430994, 0.279319}, {0.697299, -0.430994, -0.279319}, {0.439869, -0.308569, 0.679191}, {0.507631, 0.535251, -0.453671}, {0.250201, 0.657675, 0.504839}}

Coordinates for the icosahedron:
{{0., 0., -0.951057}, {0., 0., 0.951057}, {-0.850651, 0., -0.425325}, {0.850651, 0., 0.425325}, {0.688191, -0.5, -0.425325}, {0.688191, 0.5, -0.425325}, {-0.688191, -0.5, 0.425325}, {-0.688191, 0.5, 0.425325}, {-0.262866, -0.809017, -0.425325}, {-0.262866, 0.809017, -0.425325}, {0.262866, -0.809017, 0.425325}, {0.262866, 0.809017, 0.425325}}