This page is linked from here; go there for some background.

Take a tetrahedron and snap a new one onto three of its sides and you'll get one of these:

There are two ways to add a tetrahedron to the triple augmented tetrahedron (7V3) to get a new (8-vertex) deltahedron. Which ones among the following 14 are they? (Source for image: this paper by N. Tsuruta, J. Mitani, Y. Kanamori, Y. Fukui (2015), Random Realization of Polyhedral Graphs as Deltahedra, Journal for Geometry and Graphics, Volume 19 (2015), No. 2, 227–236.)