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.