Estimate of the number of magic squares of order 6 |

Their result was

They published it in May, 1998: Number of Magic Squares From Parallel Tempering Monte Carlo (external link)

I could confirm this result in August, 2001 using another method that may be called 'Monte Carlo Backtracking'. One single estimate takes about one minute on a standard PC (2 GHz). In March, 2002 the data of more then 100,000 runs gave the following mean value:

Where

One part of my estimate considers all magic squares (A) with the smallest Integer in any main-diagonal and the other part all other squares (B).

The mean values of all single estimates are in detail:

Source code of the program (html) The program was written in gb32, a German programming language. |

The backtracking strategy (word document) See in which order the cells were filled. |

Transformations of order-6 magic squares (word document) This file is in German Language (sorry), it shows how transformations can be used in order to reduce the number of squares that have to be calculated. Only the first parts were used in the program. |

All data of the 100,000 runs (zipped txt) The first column contains the estimates for A, the second for B. Parts of this file may be imported to excel. Each number must be multiplied by the transformation factor 24 and by the statistical factor 1,057,496,832,000. |

Some remarks about the program (html) I try to explain some details of the program. |

Walter Trump, Nürnberg, Germany, (c) 2003-06-03 (last modified: 2003-06-03)