Constructing Constrained-Version of Magic Squares Using Selection Hyper-heuristics

Constructing Constrained-Version of Magic Squares Using Selection Hyper-heuristics

A square matrix of distinct numbers in which every row, column and both diagonals has the same total is referred to as a magic square. Constructing a magic square of a given order is considered as a difficult computational problem, particularly when additional constraints are imposed. Hyper-heuristics are emerging high level search methodologies that explore the space of heuristics for solving ...

Corrigendum: Constructing Constrained-Version of Magic Squares Using Selection Hyper-heuristics

Markov Chain Selection Hyper-heuristic for the Optimisation of Constrained Magic Squares

A square matrix of size n × n, containing each of the numbers (1, . . . , n) in which every row, column and both diagonals has the same total is referred to as a magic square. The problem can be formulated as an optimisation problem where the task is to minimise the deviation from the magic square constraints and is tackled here by using hyperheuristics. Hyper-heuristics have recently attracted...

Constructing all magic squares of order three

A magic square of order n is an n by n matrix with distinct nonnegative integer entries such that every row sum, column sum, and (two) diagonal sums equals to the same number m, the magic number. Adding 1 to every entry will give us a traditional magic square of positive integers. A magic square is pure if the entries are the consecutive numbers from 0 to n − 1, and hence it has magic number 3 ...

Solving high school timetabling problems worldwide using selection hyper-heuristics

High school timetabling is one of those recurring NP-hard real-world combinatorial optimisation problems that has to be dealt with by many educational institutions periodically, and so has been of interest to practitioners and researchers. Solving a high school timetabling problem requires scheduling of resources and events into time slots subject to a set of constraints. Recently, an internati...