This Demonstration shows Bezdek's unistable polyhedron with 18 faces. A face is stable if, and only if, the orthogonal projection (red point) of the center of mass (black point) onto the face lies inside the face. A unistable polyhedron has only one stable face. The polyhedron is a skew pyramid placed along the axis and can be changed using the sliders.

Guy constructed unistable 19-faces solid in 1968 [2, 3, 4]. Bezdek found unistable 18-faces solid in 2011 [1]. In [5] Reshetov constructed unistable polyhedra with 14, 15, 16, and 17 faces.

References:

[1] A. Bezdek, On stability of polyhedra, Workshop on Discrete Geometry, 13-16 September 2011, Fields Institute, Canada (2011), pp. 2490-2491.

[2] J. Bryant and C. Sangwin, How Round Is Your Circle?: Where Engineering and Mathematics Meet, Princeton: Princeton University Press, 2008 pp. 273–276.

[3] R. K. Guy, A Unistable Polyhedron, Calgary: University of Calgary Department of Mathematics, 1968.

[4] J. H. Conway, M. Goldberg, and R. K. Guy, Problem 66-12, SIAM Review 11, 1969 pp. 78–82.

[5] A. Reshetov, A Unistable Polyhedron with 14 Faces, International Journal of Computational Geometry & Applications, Vol. 24, No. 1 (2014) pp. 39-59.