Multimagic series for cubes
See also the multimagic series for squares

As we have seen for the smallest bimagic cube, it may be interesting, in order to try to construct a p-multimagic cube of order-n, to find all the p-multimagic series of order n, that is to say the series of n different integers, from 1 to n3, having the correct magic, bimagic,... to p-multimagic sums (= S1, S2,... Sp):

The order 3 allows one to get bimagic series. Here are the 4 possible bimagic series:

So that means that:

  1. 3 + 19 + 20 = 4 + 15 + 23 = 5 + 13 + 24 = 8 + 9 + 25 = 42 = S1
  2. 3 + 19 + 20 = 4 + 15 + 23 = 5 + 13 + 24 = 8 + 9 + 25 = 770 = S2

For the order 4, there are 8 bimagic series for which the list is given on the Smallest bimagic cube page.

Here is a summary of the number of multimagic series, some of these lists being downloadable as Excel files, from 44Kb to 540Kb each. The numbers of bimagic series of orders 9 and 10 were computed by Walter Trump, Germany, in October 2005. The number of trimagic series of order 9 was computed by Gildas Guillemot, France, in December 2006 and later confirmed by Michael Quist, USA, in May 2008. The numbers of bimagic series of orders 11 and 12 were computed in May and June 2013 by Walter Trump, using a program written by Lee Morgenstern, USA. In August 2015, Dirk Kinnaes, Belgium, confirmed the previously known numbers of bimagic series, and computed the number of bimagic series of order 13: see his algorithm (PDF file).

In June 2013, Michael Quist wrote a paper estimating the numbers of magic and multimagic series, including bimagic series for cubes of order N. See Here is his formula, and the obtained numeric values for 10 ≤ N ≤ 18. The error decreases with higher orders. In August 2015, Dirk Kinnaes improved Quist's formula, adding a new term (see also his PDF file given above).

In March 2017, Dirk Kinnaes wrote a paper "Estimating the number of K-multimagic hypercube series" (PDF file) including a formula estimating the numbers of trimagic series for cubes of order N. Here is his formula, and the obtained numeric values for 7 ≤ N ≤ 15. The error will decrease with higher orders, we should expect much smaller errors for orders > 10.

The bimagic and trimagic series are referenced respectively under the numbers A090653 and A092312 in the On-Line Encyclopedia of Integer Sequences, OEIS Foundation.

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