How many possible closed hexagons? — A mathematical problem from the Tri‐Ominos game

Abstract

In this paper we discuss the interesting problem of finding the number of possible closed hexagons which can be formed by six Tri‐Ominos from the set supplied. Even though this problem is from a game there are some important mathematical facts and mathematical methods involved in solving it. First we establish two correspondences:

(1) Every closed hexagon formed by six Tri‐Ominos corresponds to a hexagon with the common number at the centre and six numbers at the vertices.

(2) Every hexagon corresponds to an ordered set of seven numbers which are the numbers at the centre and at the six vertices in the hexagon taken in clockwise order.

Then we prove nine properties about the seven numbers in the ordered set which corresponds to the hexagons formed by the Tri‐Ominos. As both the above‐mentioned correspondences are bijective, we can obtain the number of all possible closed hexagons formed by Tri‐Ominos by calculating the number of all the corresponding ordered sets which satisfy all the nine properties. Finally, according to these properties we classify these ordered sets, calculate their number using combinatorial theory, and obtain the number of all the ordered sets which satisfy all the nine properties. That special number, 666, is the answer to the problem.

Notes

∗Tang Fu‐Su, Associate Professor of the Department of Mathematics, Suzhou University, P.R. China, prepared this paper while working as a visiting research fellow at the Department of Mathematical Sciences, Loughborough University of Technology, UK.

Additional information

Notes on contributors

Tang Fu‐Su∗

∗Tang Fu‐Su, Associate Professor of the Department of Mathematics, Suzhou University, P.R. China, prepared this paper while working as a visiting research fellow at the Department of Mathematical Sciences, Loughborough University of Technology, UK.