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magic_square_mip

<table align="left"> <td> <a href="https://colab.research.google.com/github/google/or-tools/blob/main/examples/notebook/contrib/magic_square_mip.ipynb">Run in Google Colab</a> </td> <td> <a href="https://github.com/google/or-tools/blob/main/examples/contrib/magic_square_mip.py">View source on GitHub</a> </td> </table>

First, you must install ortools package in this colab.

python

%pip install ortools

Magic square (integer programming) in Google or-tools.

Translated from GLPK:s example magic.mod ''' MAGIC, Magic Square

Written in GNU MathProg by Andrew Makhorin [email protected]

In recreational mathematics, a magic square of order n is an arrangement of n^2 numbers, usually distinct integers, in a square, such that n numbers in all rows, all columns, and both diagonals sum to the same constant. A normal magic square contains the integers from 1 to n^2.

(From Wikipedia, the free encyclopedia.) '''

Compare to the CP version: http://www.hakank.org/google_or_tools/magic_square.py

Here we also experiment with how long it takes when using an output_matrix (much longer).

This model was created by Hakan Kjellerstrand ([email protected]) Also see my other Google CP Solver models: http://www.hakank.org/google_or_tools/

python

import sys
from ortools.linear_solver import pywraplp

#
# main(n, use_output_matrix)
#   n: size of matrix
#   use_output_matrix: use the output_matrix
#


def main(n=3, sol='CBC', use_output_matrix=0):
  # Create the solver.
  print('Solver: ', sol)
  solver = pywraplp.Solver.CreateSolver(sol)
  if not solver:
    return

  #
  # data
  #
  print('n = ', n)

  # range_n = range(1, n+1)
  range_n = list(range(0, n))

  N = n * n
  range_N = list(range(1, N + 1))

  #
  # variables
  #

  # x[i,j,k] = 1 means that cell (i,j) contains integer k
  x = {}
  for i in range_n:
    for j in range_n:
      for k in range_N:
        x[i, j, k] = solver.IntVar(0, 1, 'x[%i,%i,%i]' % (i, j, k))

  # For output. Much slower....
  if use_output_matrix == 1:
    print('Using an output matrix')
    square = {}
    for i in range_n:
      for j in range_n:
        square[i, j] = solver.IntVar(1, n * n, 'square[%i,%i]' % (i, j))

  # the magic sum
  s = solver.IntVar(1, n * n * n, 's')

  #
  # constraints
  #

  # each cell must be assigned exactly one integer
  for i in range_n:
    for j in range_n:
      solver.Add(solver.Sum([x[i, j, k] for k in range_N]) == 1)

  # each integer must be assigned exactly to one cell
  for k in range_N:
    solver.Add(solver.Sum([x[i, j, k] for i in range_n for j in range_n]) == 1)

  # # the sum in each row must be the magic sum
  for i in range_n:
    solver.Add(
        solver.Sum([k * x[i, j, k] for j in range_n for k in range_N]) == s)

  # # the sum in each column must be the magic sum
  for j in range_n:
    solver.Add(
        solver.Sum([k * x[i, j, k] for i in range_n for k in range_N]) == s)

  # # the sum in the diagonal must be the magic sum
  solver.Add(
      solver.Sum([k * x[i, i, k] for i in range_n for k in range_N]) == s)

  # # the sum in the co-diagonal must be the magic sum
  if range_n[0] == 1:
    # for range_n = 1..n
    solver.Add(
        solver.Sum([k * x[i, n - i + 1, k]
                    for i in range_n
                    for k in range_N]) == s)
  else:
    # for range_n = 0..n-1
    solver.Add(
        solver.Sum([k * x[i, n - i - 1, k]
                    for i in range_n
                    for k in range_N]) == s)

  # for output
  if use_output_matrix == 1:
    for i in range_n:
      for j in range_n:
        solver.Add(
            square[i, j] == solver.Sum([k * x[i, j, k] for k in range_N]))

  #
  # solution and search
  #
  solver.Solve()

  print()

  print('s: ', int(s.SolutionValue()))
  if use_output_matrix == 1:
    for i in range_n:
      for j in range_n:
        print(int(square[i, j].SolutionValue()), end=' ')
      print()
    print()
  else:
    for i in range_n:
      for j in range_n:
        print(
            sum([int(k * x[i, j, k].SolutionValue()) for k in range_N]),
            ' ',
            end=' ')
      print()

  print('\nx:')
  for i in range_n:
    for j in range_n:
      for k in range_N:
        print(int(x[i, j, k].SolutionValue()), end=' ')
      print()

  print()
  print('walltime  :', solver.WallTime(), 'ms')
  if sol == 'CBC':
    print('iterations:', solver.Iterations())


n = 3
sol = 'CBC'
use_output_matrix = 0
if len(sys.argv) > 1:
  n = int(sys.argv[1])

if len(sys.argv) > 2:
  sol = sys.argv[2]
  if sol != 'GLPK' and sol != 'CBC':
    print('Solver must be either GLPK or CBC')
    sys.exit(1)

if len(sys.argv) > 3:
  use_output_matrix = int(sys.argv[3])

main(n, sol, use_output_matrix)