This seemingly simple problem had been open for many years : triangulate a cube into tetrahedra such that each triangle has acute angles and all angles between triangles are acute. It was not even known if such a triangulation exists ! Our paper gives the solution of this problem with the help of algorithms and software we developed for well-centered triangulation (WCT). WCT is described in many papers available on my publications page. This is work of my student Evan VanderZee (co-advised with Vadim Zharnitsky) in collaboration with Damrong Guoy. Edgar Ramos is another collaborator in the WCT project. The vertex data and program for computing the angles is available. Paper appears in journal Computational Geometry: Theory and Applications. Also available as a preprint.