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=0∑^m (H[[_ji,_q0]<_ql>])^k =
_l=0∑^m (H[[_j0,_qm-1]<_jm-1-l,_ql>])^k = (_l=0∑^{m^n} i^k) / m^n-1 ; k = 1..p} = Fh_k(mⁿ) / m^n-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
Fh_p(mⁿ) / m^n-1 Faulhaber's Formula (1631)
Fh_p(n) = _j=0∑ⁿ i^p = (1/(p+1))_k=1∑^p+1(-1)^δ(k,p) (^p+1_k) B_p+1-k n^k

  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
B_p (Bernouilli numbers) given by
B₀ = 1; B_p+1 = -(1/(p+2))_k=0∑^p (^p+2_k) B_k

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=0∑^m-1(x_i+c)^p = _i=0∑^m-1_j=0∑^p (^p_j)x_i^jc^p-j =
_j=0∑^p(^p_j) (_i=0∑^m-1x_i^j)c^p-j = _j=0∑^p(^p_j) [Fh_j(mⁿ) / m^n-1] c^p-j
where the fact is used that we started with a p-MultiMagic hypercube
Q.E.D.
coincidentially proven:
Fh_p(mⁿ) = _j=0∑^p(^p_j) Fh_j(mⁿ) [-(mⁿ+1)/2]^p-j <=>
_j=0∑^p-1(^p_j) Fh_j(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=0∑^m-1 (-s[i]])^p = (-1)^p _k=0∑^m-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)/2}i^p= ((-1)^p+1)_i=-0∑^{(m^2-1)/2}i^p,
which is 0 for odd p and even for even p, this sum divided by m^n-1 is the magic sum)
Q.E.D.
coincidentally proven:
selfcomplementairy sequence sums to hf_p(mⁿ)/m^n-1 sum when raised to an odd power p

p-MultiMagic
Multiplication
theorem The basic multiplication of p-MultiMagic hypercubes is p-Multimagic
Proof:
(A_m * B_q)_ji = A_ji%m (B_q,ji/m - 1)mⁿ
_i=0∑^mq-1(A_ji%m (B_q,ji/m - 1)mⁿ)^p =
_i=0∑^mq-1_k=0∑^p_l=0∑^p-k (^p_k)(^p-k_l) (A_ji%m^k (B_q,ji/m^l(-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

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=0∑^m (H[[_ji,_q0]<_ql>])^k = _l=0∑^m (H[[_j0,_qm-1]<_jm-1-l,_ql>])^k = (_l=0∑^{m^n} i^k) / m^n-1 ; k = 1..p} = Fh_k(mⁿ) / m^n-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 Fh_p(mⁿ) / m^n-1	Faulhaber's Formula (1631) Fh_p(n) = _j=0∑ⁿ i^p = (1/(p+1))_k=1∑^p+1(-1)^δ(k,p) (^p+1_k) B_p+1-k n^k
	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 B_p (Bernouilli numbers) given by B₀ = 1; B_p+1 = -(1/(p+2))_k=0∑^p (^p+2_k) B_k

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=0∑^m-1(x_i+c)^p = _i=0∑^m-1_j=0∑^p (^p_j)x_i^jc^p-j = _j=0∑^p(^p_j) (_i=0∑^m-1x_i^j)c^p-j = _j=0∑^p(^p_j) [Fh_j(mⁿ) / m^n-1] c^p-j where the fact is used that we started with a p-MultiMagic hypercube Q.E.D.
	coincidentially proven: Fh_p(mⁿ) = _j=0∑^p(^p_j) Fh_j(mⁿ) [-(mⁿ+1)/2]^p-j <=> _j=0∑^p-1(^p_j) Fh_j(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=0∑^m-1 (-s[i]])^p = (-1)^p _k=0∑^m-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)/2}i^p= ((-1)^p+1)_i=-0∑^{(m^2-1)/2}i^p, which is 0 for odd p and even for even p, this sum divided by m^n-1 is the magic sum) Q.E.D.
	coincidentally proven: selfcomplementairy sequence sums to hf_p(mⁿ)/m^n-1 sum when raised to an odd power p
p-MultiMagic Multiplication theorem	The basic multiplication of p-MultiMagic hypercubes is p-Multimagic
p-MultiMagic Multiplication theorem	Proof: (A_m * B_q)_ji = A_ji%m (B_q,ji/m - 1)mⁿ _i=0∑^mq-1(A_ji%m (B_q,ji/m - 1)mⁿ)^p = _i=0∑^mq-1_k=0∑^p_l=0∑^p-k (^p_k)(^p-k_l) (A_ji%m^k (B_q,ji/m^l(-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.