Solving a binary puzzle

A Binary puzzle is a Sudoku-like puzzle with values in each cell taken from the set (Formula presented.). Let (Formula presented.) be an even integer, a solved binary puzzle is an (Formula presented.) binary array that satisfies the following conditions: (1) no three consecutive ones and no three consecutive zeros in each row and each column; (2) the number of ones and zeros must be equal in each row and in each column; (3) there can be no repeated row and no repeated column. This paper proposes three approaches to solve the puzzle. The first method is based on a complete backtrack-based search algorithm. The idea is to propagate and fill an unsolved binary puzzle according to the three constraints, followed by a random guess if the puzzle remains unsolved. The second method of solving a binary puzzle is by representing it as an instance of a Boolean satisfiability problem which allows the solution for a binary puzzle to be obtained using SAT solvers. The third approach is based on expressing a binary puzzle as a system of polynomial equations over the binary field (Formula presented.). The set of solutions for the equation system implies the solutions for the binary puzzle and it is obtained by computing a Gröbner basis of the ideal generated by the polynomials. We experimentally compare the three approaches with binary puzzles of various sizes and different numbers of empty cells using a computer algebra system

Quantum-accelerated constraint programming

Constraint programming (CP) is a paradigm used to model and solve constraint satisfaction and combinatorial optimization problems. In CP, problems are modeled with constraints that describe acceptable solutions and solved with backtracking tree search augmented with logical inference. In this paper, we show how quantum algorithms can accelerate CP, at both the levels of inference and search. Leveraging existing quantum algorithms, we introduce a quantum-accelerated filtering algorithm for the $\texttt{alldifferent}$ global constraint and discuss its applicability to a broader family of global constraints with similar structure. We propose frameworks for the integration of quantum filtering algorithms within both classical and quantum backtracking search schemes, including a novel hybrid classical-quantum backtracking search method. This work suggests that CP is a promising candidate application for early fault-tolerant quantum computers and beyond.Comment: published in Quantu

MATHEMATICS OF HUNG-PING TSAO

Tsao, Hung-ping (2020). Mathematics of Hung-ping Tsao. In: "Evolutionary Progress in Science, Technology, Engineering, Arts, and Mathematics (STEAM)", Wang, Lawrence K. and Tsao, Hung-ping (editors). Volume 2, Number 11, November 2020; 336 pages. Lenox Institute Press, Newtonville, NY, 12128-0405, USA. No. STEAM-VOL2-NUM11-NOV2020; ISBN 978-0-9890870-3-2.............ABSTRACT: I would like to share some of my ideas in Number Theory, Actuarial Mathematics, Sudoku Solving and Optimization Teaching with college students and colleagues. ............KEYWORDS: Natural sequence, AP-sequence, Power-sum, Product-sum, Sorting, Combination, Permutation, Cycle, Subset, Binomial coefficient, Stirling number, Pascal triangle, Bernoulli coefficient, Eulerian number, Bell number, Ordered Bell polynomial, Eulerian Bell polynomial, Recursive formula, q-Gaussian coefficient, Life insurance, Life annuity, Interest, Mortality, Contingency, Premium, Reserve, Sudoku, Puzzle, Row, Column, Box, Unique solution, Flipflops chain, Residue

Global Constraint Catalog, 2nd Edition

This report presents a catalogue of global constraints where each constraint is explicitly described in terms of graph properties and/or automata and/or first order logical formulae with arithmetic. When available, it also presents some typical usage as well as some pointers to existing filtering algorithms