New Results on Tripod Packings

Abstract

Consider an \(n \times n \times n\) cube Q consisting of \(n^3\) unit cubes. A tripod of order n is obtained by taking the \(3n-2\) unit cubes along three mutually adjacent edges of Q. The unit cube corresponding to the vertex of Q where the edges meet is called the center cube of the tripod. The function f(n) is defined as the largest number of integral translates of such a tripod that have disjoint interiors and whose center cubes coincide with unit cubes of Q. The value of f(n) has earlier been determined for \(n \le 9.\) The function f(n) is here studied in the framework of the maximum clique problem, and the values \(f(10) = 32\) and \(f(11)=38\) are obtained computationally. Moreover, by prescribing symmetries, constructive lower bounds on f(n) are obtained for \(n \le 26.\) A conjecture that f(n) is always attained by a packing with a symmetry of order 3 that rotates Q around the axis through two opposite vertices is disproved.