Multimagic hypercubes

Why limit ourselves to 2 dimensions of multimagic squares and to 3 dimensions of multimagic cubes? While nature limits (or appears to limit) itself to three dimensions, there is no limit to mathematical objects.

So it is possible to construct multimagic hypercubes of dimension 4, 5, and so on... using similar methods to those already used for dimensions 2 and 3. Here are the first multimagic hypercubes of dimension 4, also called multimagic tesseracts:

 Hypercube of dimension 4,or tesseract(*) Order File to be downloaded Magic degree of rows, columns, pillars, files Magic degree of quadragonals (or dim. 4 diag.) Magic degree of triagonals (or dim. 3 diag.) Magic degree of diagonals (or dim. 2 diag.) Bimagic 32 Zipped Excel file of 4.2Mb 2 2 1 0 32 Zipped Excel file of 4.0Mb 0 1 64 Too big to be downloaded!Contact C. Boyer. 1 1 Trimagic 243 3 3 1 1 Trimagic (and perfect bimagic) 256 2 2

(*) All these hypercubes were created from February to April 2003 by Christian Boyer.

 I dedicate all these multimagic hypercubes to John-R. Hendricks. John R. Hendricks has done the most impressive work on magic tesseracts in the world, including enumerating all 58 different tesseracts of order 3.            Christian Boyer

At first glance, something seems to be strange in this table: bimagic hypercubes of order 32 should be small, but the associated file is enormous. In fact, these hypercubes are surprisingly big because they use as many numbers as the pentamagic square of order 1024. Just because 324 = 10242.

Here are some characteristics of the trimagic hypercube of order 256. It uses numbers from 0 to 4,294,967,295. Its 16,777,216 rows, 16,777,216 columns, 16,777,216 pillars and 16,777,216 files are trimagic. Its 8 quadragonals are trimagic. Its 4,096 triagonals and its 786,432 diagonals are bimagic. This hypercube is also perfect bimagic, since the x-agonals of all the dimensions are bimagic (and its quadragonals are even trimagic). Its magic sums are:

• S1  = 549755813760
• S2  = 1574122160406792590720
• S3  = 5070602398551734364826868121600

These 5 hypercubes have been checked separately by Yves Gallot (France), then by Renaud Lifchitz (France) from March to June 2003. Yves Gallot is the author of the famous Proth program used by searchers of big prime numbers.
See http://www.utm.edu/research/primes/programs/gallot/.