Magic Square Sets

Problem Description

We propose a collection of magic square problems for System Competition under ASP-RFC. These problems can be easily encoded using aggregates, in a nontrivial way (meaning they are not easily reduced to ASP-core), and thus may serve as benchmarks for testing the effectiveness of the aggregate handling by a solver. The problem instances can also be generated by adding additional constraints.

Input Predicates: data/1, num/1

Output Predicates: sqr/3

Problem Description:

Normal magic squares: a magic square of order 'n' is an arrangement of n² numbers -- 1..n² -- in a square, such that the 'n' numbers in all rows, all columns, and both diagonals sum to the same constant n*(n^2+1)/2. It is known that the magic square of any order exists. But the exact complexity is unknown to our best knowledge.

Semimagic squares: it is a relaxed normal magic square in the sense that only the rows and columns but not necessarily the diagonals sum to the magic constant n*(n²+1)/2.

Panmagic squares: a panmagic square is a magic square with the additional property that the broken diagonals, i.e. the diagonals that wrap round at the edges of the square, also add up to the magic constant n*(n²+1)/2. The Number of distinct (modulo rotation and reflection) panmagic square of order n=1,2,3,4,5,6 are 1, 0, 0, 48, 3600, 0 respectively. (Cf. http://oeis.org/searchq=panmagic+square&sort=&language=english&go=Search)

Associative magic squares: An associative magic square is a magic square for which every pair of numbers symmetrically opposite to the center sum up to the same value n^2+1. The number of distinct (modulo rotation and reflection) associative magic squares of order n=1,2,3,4,5,6 are 1,0,1,48,48544,0 respectively. (cf. http://oeis.org/search?q=associative+magic+square&sort=&language=english&go=Search)

Bimagic squares: A bimagic square is a magic square that also remains magic if all of the numbers it contains are squared.

Trimagic squares: A trimagic square is a magic square that also remains magic if all of the numbers it contains are squared or cubed. In the benchmark, we require for the cubed cases only.

Author(s)

• Yisong Wang
  • Guizhou University, China
• Jia-Huai You
  • University of Alberta, Canada