Only SM with number 64 in the first row and first column were considered.

We divided the whole search into four different tasks:

A: SM with symmetric first row including number 1.

B: SM with non-symmetric first row including number 1.

C: SM with number 1 not in the first row and first column. And further (complicated) conditions for the row and the column with 1.

D: SM with number 1 not in the first row and first column. And the row with 1 consists of the complementary numbers of the first row.

In the cases B and C only one SM of each complementary pair was calculated.

The exhaustive search was done on different computers with GB32-programs created by Walter Trump.

Confirmation of the Results |

Francis Gaspalou made a C-program by only using rough descriptions of the methods.

He checked about one hundred of combinations first row - first column.

For each combination, he enumerated all the SM and all the SQ. No error could be found.

He also checked that the following known sets of squares were inside the whole set:

* the 841 associative SQ

* the 1 836 pandiagonal complete SQ

* the 1 344 Greco-Latin SQ (Tarry's SQ)

* the 10 317 Coccoz's SQ

* the 2 543 Rilly's SQ

Francis Gaspalou applied to some subsets all the possible permutations of rows and columns

for checking that the obtained squares were inside the initial subset.

Of course we checked whether the complements of all SQ were inside the whole set.

(You get the complement of a 8x8-square when replacing all entries j by 65-j.)

The used methods are principally proved correct.

On smaller problems like semi-magic 5x5-squares we get long known correct results.

To be true: a complete verification of the total result is still outstanding.

The exhaustive search should be done by other persons and other programs.

But that means a lot of work.

Therefore also verifications of partial results are welcome.

- Choose bimagic order-8 series with 64 inside for the first row and the first column.

- Calculate all SM for this row-col-couple with our methods or better your's.

- Derive all possible SQ from the found SM.

- Compare your results with our's using the following tables.

- ZIP : SQ sorted by the indices of the row and the column with 64 inside (9 MB)

- XLSX: Large spreadsheet with the numbers of SM per row-column-couple (60 MB)

Back to Bimagic 8x8-Squares | Walter Trump, Nürnberg, Germany, (c) 2014-06-05, last update: 2014-06-20 |