We study the efficient enumeration of fixed polycubes in dimensions three and higher, a central problem in enumerative combinatorics with exponential growth. Building on Redelmeier’s recursive framework and later refinements that replace a full recursion with combinatorial counting, we introduce an optimization for the terminal levels of the computation tree. The key insight is that the number of valid ways to extend a partial polycube depends only on its local adjacency structure, so the combinatorial counts for the remaining extensions can be updated incrementally as the search proceeds instead of being recomputed at each branch.
We implemented the resulting algorithm in a parallel and distributed setting, and computed new polycube counts in dimensions three, four, and five, including previously unknown values for three-dimensional polycubes of sizes 23 and 24. Experimental results show consistent speedups of 20–26 percent over prior methods.
