Pandiagonal perfect multiplicative magic cubes



What are pandiagonal perfect multiplicative magic cubes?

As explained in the multiplicative magic cubes page, a pandiagonal perfect magic cube has all its lines (rows, columns, pillars, triagonals, plane diagonals, broken triagonals and broken diagonals) magic. It is a perfect magic cube that remains perfect magic when any face of the cube is moved parallel to itself, from its side to the opposite side of the cube. They are the BEST possible cubes: any line, entire or broken, in any direction gives always the same product!

Because a cell in a cube has 3^3 - 1 = 26 adjacent cells, a pandiagonal perfect multiplicative magic cube of order n has all its 13n lines which are magic.

My 4 best pandiagonal perfect multiplicative magic cubes, and Shirakawa's cube, extracted from the general table, are:

Thanks again to Edwin Clark, Mathematics Department of the University of South Florida, USA, for checking in 2006 all my pandiagonal perfect multiplicative magic cubes, confirming that they have all the announced properties.


What is the smallest possible magic product P?
(of pandiagonal perfect multiplicative magic cubes)

This problem is also a factorization problem in 3 dimensions:

What is the smallest composite integer P from which we can pick and organize distinct factors in a multiplicative cube? The result of the multiplication of these numbers in any line of any direction (including any broken diagonal and broken triagonal) has always to be equal to this integer P. And what is the size (=order) of this cube?

>> In 2 dimensions (squares), the answer is known. The smallest possible integer is 14400, organizing distinct factors in a 4th-order square that way:

When we multiply the integers of any row, column, or diagonal (including all broken diagonals), we get always 14400. There are 32 different ways to get 14400.

>> In 3 dimensions (cubes), the question is an open problem. I do not have the answer, but my smallest composite integer is:

        89 518 183 823 250 314 294 722 560 000

organizing 512 of its factors in a 8th-order magic cube. Because it is a 8th-order pandiagonal perfect cube, there are 138 = 832 different ways to get the same product P.


What is the smallest possible maximum number?
(of pandiagonal perfect multiplicative magic cubes)

My best result is 24 992, used in a 11th-order cube.

It means that the numbers used in this cube are smaller than the numbers used in smaller cubes! The smallest known Max nbs for 8th-order and 9th-order are bigger: 17 297 280 and 591 192.

Because it is a 11th-order pandiagonal perfect cube, there are 1311 = 1573 different ways to get the same product P.


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