Pandiagonal perfect magic cubes

As defined in the book Magic Cubes New Recreations of William H. Benson and Oswald Jacoby, a pandiagonal perfect magic cube 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. It means that a pandiagonal perfect magic cube must have all its broken diagonals magic and all its broken triagonals magic.

Gabriel Arnoux (1831-1913)

It seems that the first pandiagonal perfect magic cube has been created by Gabriel Arnoux, former French navy officer.

After I discovered this small sentence printed on page 61 in his book Arithmétique Graphique - Les espaces arithmétiques hypermagiques I went in 2003 to the Archives of the Académie des Sciences in Paris. I had the joy of retrieving this cube in the folder of the session held on May 9th 1887: Arnoux's letter is perfectly kept, with the complete cube, so 173 = 4 913 handwritten numbers! But it remained to check if this cube had really the announced properties. After I completely photographed all the handwritten pages, I sent all the .JPG files on a CD-ROM to my friend Harvey Heinz, Canada. Then Harvey had the courage to type on his computer all the 4 913 numbers, and confirmed to me on July 6th, 2003, that this Arnoux's cube, with only 8 badly written numbers, is truly a pandiagonal perfect magic cube. Thanks Harvey!

Frederick A.P. Barnard (1809-1889)

After this Arnoux's cube, other pandiagonal perfect magic cubes were created. The American Frederick A.P. Barnard in his remarkable Theory of Magic Squares and of Magic Cubes published in the Memoirs of the National Academy of Sciences of 1888, the following year but completely independently of Arnoux, gives an 8th-order cube and two 11th-order cubes. Barnard was President of Columbia College New York, now Columbia University, during 25 years, from 1864 to 1889.

It seems impossible to create a pandiagonal perfect magic cube of an order smaller than 8.

Return to the home page