The Magic Encyclopedia ™

the p-MultiMagic Feature
(by Aale de Winkel)

The magic hypercube is qualified as p-Multimagic when all 1-agonals and n-agonals sum to the same sum when all numbers are raised to the power k for all k from 1 to p. For the expressions for the p-multimagic sums see: Faulhaber's Formula

the p-MultiMagic Feature
Given a magic object in its regular sense, the figure remains summing to the same number
when all numbers are rasised to the same power. The figure is classified as p-Multimagic
when this is the case for all powers uptil the number p
mIN-Magic Hypercube {H[ji] ε [1..mn] | l=0m (H[[ji,q0]<ql>])k =
l=0m (H[[j0,qm-1]<jm-1-l,ql>])k = (l=0m^n ik) / mn-1 ; k = 1..p} = Fhk(mn) / mn-1
The exact sum terms merely depicts all 1-agonals and all n-agonals to sum over. (see hypercube article)
With help of tri-digital equations this author found 80 families of order 8 bimagic squares
224 order 9 bimagic square families where found with help of bi-digital equations by this author
and known bimagic squares by Collisson and Benson Jacobi identified by used methods
each "family" of order m squares holds (m-odd(m))!!/2 different squares by main n-agonal permutation
currently only order 25 bimagic cubes where found (by John R Hendricks)
(see: Authors Webpages for details, database section for full listing)
p-MultiMagic sum
Fhp(mn) / mn-1
Faulhaber's Formula (1631)
Fhp(n) = j=0n ip = (1/(p+1))k=1p+1(-1)δ(k,p) (p+1k) Bp+1-k nk

  Fh1(n) =   (1/2)    (n2   +     n)
  Fh2(n) =   (1/6)   (2n3  +   3n2   +   n)
  Fh3(n) =   (1/4)    (n4   +   2n3   +   n2)
  Fh4(n) = (1/30)   (6n5   + 15n4   + 10n3 -     n)
  Fh5(n) = (1/12)   (2n6   +  6n5   +   5n4 -     n2)
  Fh6(n) = (1/42)   (6n7   + 21n6   + 21n5 -   7n3 +     n)
  Fh7(n) = (1/24)   (3n8   + 12n7   + 14n6 -   7n4 +   2n2)
  Fh8(n) = (1/90) (10n9   + 45n8   + 60n7 - 42n5 + 20n3 -   3n)
  Fh9(n) = (1/20)   (2n10 + 10n9   + 15n8 - 14n6 + 10n4 -   3n3)
 Fh10(n) = (1/66)   (6n11 + 33n10 + 55n9 - 66n7 + 66n5 - 33n3 + 5n)

notes: δ(k,p) kroenecker delta (1 iff k = p; 0 otherwise) and
Bp (Bernouilli numbers) given by
B0 = 1; Bp+1 = -(1/(p+2))k=0p (p+2k) Bk




p-MultiMagic Theorems
A few general theorems regarding the p-Multimagic feature are given below
addition of constant
to each number
The addition of a constant to each number in a p-MultiMagic hypercube
the hypercube remains p-Multimagic
Proof:
i=0m-1(xi+c)p = i=0m-1j=0p (pj)xijcp-j =
j=0p(pj) (i=0m-1xij)cp-j = j=0p(pj) [Fhj(mn) / mn-1] cp-j
where the fact is used that we started with a p-MultiMagic hypercube
Q.E.D.
coincidentially proven:
Fhp(mn) = j=0p(pj) Fhj(mn) [-(mn+1)/2]p-j <=>
j=0p-1(pj) Fhj(mn) [-(mn+1)/2]p-j = 0
for odd p, since the magic sum for odd p = 0 for the centralized number range and
-(mn+1)/2 is the appropriate shifting constant
Q.E.D.
p-MultiMagic
Complementairy
invariance
theorem
The p-MultiMagic sum is invariant for the complementairy transformation
s[i] --> (mn+1) - s[i]
Proof:
using the addition of constant theorem move the range to [-(mn-1)/2 .. (mn-1)/2]
thus the complementairy number of s[i] = -s[i] thus transformed the sum of powers
of the complement yield:
k=0m-1 (-s[i]])p = (-1)p k=0m-1 s[i]p
for odd p these sums are 0 while for even p (-1)p = 1 and the equation holds
(The odd p sums are 0 since i=-(m^2-1)/2(m^2-1)/2ip= ((-1)p+1)i=-0(m^2-1)/2ip,
which is 0 for odd p and even for even p, this sum divided by mn-1 is the magic sum)
Q.E.D.
coincidentally proven:
selfcomplementairy sequence sums to hfp(mn)/mn-1 sum when raised to an odd power p
p-MultiMagic
Multiplication
theorem
The basic multiplication of p-MultiMagic hypercubes is p-Multimagic
Proof:
(Am * Bq)ji = Aji%m (Bq,ji/m - 1)mn
i=0mq-1(Aji%m (Bq,ji/m - 1)mn)p =
i=0mq-1k=0pl=0p-k (pk)(p-kl) (Aji%mk (Bq,ji/ml(-1)k-lmn)
notice that the k and l sums are summing over power < p since A and B are p-MultiMagic
these sums are constant on all the considered lines, hence the product is p-MultiMagic
Q.E.D.

With thanks to <Walter Trump> who suggested the move to the symmetric number range to proof the complementairy invariance theorem