The efficiency of A* searching depends on the quality of the lower bound estimates of the solution cost. Pattern databases enumerate all possible subgoals required by any solution, subject to constraints on the subgoal size. Each subgoal in the database provides a tight lower bound on the cost of achieving it. For a given state in the search space, all possible subgoals are looked up in the pat...

We investigate spin effects in four-quark systems consisting of two heavy anti-bottom quarks and two light up/down quarks. To this end we use the Born-Oppenheimer approximation. We utilize potentials of two static antiquarks in the presence of two quarks of finite mass computed via lattice QCD and solve a coupled-channel Schrödinger equation for the anti-bottom-anti-bottom separation. Without t...

The 8-puzzle is the largest puzzle of its type that can be completely solved. It is simple, and yet obeys a combinatorially large problem space of 9!/2 states. The N x N extension of the 8-puzzle is NP-hard. In the first part of this paper, we present complete statistical data based on an exhaustive evaluation of all possible tile configurations. Our results include data on the expected solutio...

Published papers rarely disclose implementation details. In this paper we show how such details can account for speedups of up to a factor of 28 for different implementations of the same algorithm. We perform an in-depth analysis of the most popular benchmark in heuristic search: the 15-puzzle. We study implementation choices in C++ for both IDA* and A* using the Manhattan distance heuristic. R...

The tile packing problem is a challenging combinatorial puzzle based on tiles with colored edges or colored corners. In its different incarnations, the puzzle gives rise to a number of interesting problems. In this paper, we sketch the background of the tile packing problem and present solutions to the puzzle. We hope that this work will stimulate further interest in this puzzle amongst readers...

Sudoku’ is the Japanese abbreviation of a longer phrase, ‘Suuji wa dokushin ni kagiru’, meaning ‘the digits must remain single’. It is a very popular puzzle that trains our logical mind. There are several approaches to solve this well-liked puzzle. In any case, the problem of solving a given Sudoku puzzle finds numerous applications in practice. In this paper, an exhaustive study has been made ...

We investigate a class of "shape allophiles" that fit together like puzzle pieces as a method to access and stabilize desired structures by controlling directional entropic forces. Squares are cut into rectangular halves, which are shaped in an allophilic manner with the goal of re-assembling the squares while self-assembling the square lattice. We examine the assembly characteristics of this s...