The Conway-Guy polyhedron, illustrated above, is the name given by Domokos and Kovács (2023) to the 19-face 34-vertex polyhedral solid independently found by Guy (1968;
Conway et al. 1969) and Knowlton (1969) which is stable on just its bottom
face, shown in yellow above. The dimensions of the polyhedral solid can be varied
by changing the lengths of the bottom rectangular face and the top ridge, though
only some combinations are unistable.

This solid remained the smallest known unistable polyhedron until the discovery of an 18-face 18-vertex polyhedral solid by Bezdek
(2011).

The Conway-Guy polyhedron illustrated above with parameters and in the parametrization of Hafner (2014) will be implemented
in a future version of the Wolfram Language
as PolyhedronData["ConwayGuyPolyhedron"].
Before scaling to unit minimum edge length, these correspond to ridge length and bottom rectangle length . For , this gives the smallest possible integer resulting in a unistable
polyhedron.

Bezdek, A. "Stability of Polyhedra." Workshop on Discrete Geometry, Sep 13-16,2011. Fields Institute, Toronto, Canada. pp. 2490-2491,
2011. http://www.fields.utoronto.ca/av/slides/11-12/wksp_geometry/bezdek/download.pdf.Conway,
J. H.; Goldberg, M.; and Guy, R. K. Problem 66-12 in SIAM Rev.11,
78-82, 1969.Croft, H. T.; Falconer, K. J.; and Guy, R. K.
Problem B12 in Unsolved
Problems in Geometry. New York: Springer-Verlag, p. 61, 1991.Domokos,
G. and Kovács, F. "Conway's Spiral and a Discrete Gömböc with
21 Point Masses." Amer. Math. Monthly130, 795-807, 2023.Guy,
R. K. "A Unistable Polyhedron." Calgary, Canada: University of Calgary
Department of Mathematics, 1968.Hafner, I. "Some Unistable Polyhedra."
https://demonstrations.wolfram.com/SomeUnistablePolyhedra/.
June 17, 2014.Knowlton, K. C. "A Unistable Polyhedron With
Only 19 Faces." Bell Telephone Laboratories, Report MM 69-1371-3, Jan. 3,
1969.Reshetov, A. "A Unistable Polyhedron With 14 Faces."
Int. J. Comput. Geom. Appl.24, 39-60, 2014.