Works on magic tesseracts and hypercubes

magic tesseracts

• order 4
  • an order-4 2,4-agonal magic tesseract (associated) (CSV file) [Nakamura, December 2004]
       This tesseract is associated and 2,4-agonal. An order-4 strictly magic tesseract (that is, a 2,3,4-agonal magic tesseract) cannot exist.
  • an order-4 pan-3-agonal magic tesseract (associated) (CSV file) [Nakamura, September 2007]
       This tesseract is associated and every pantriagonal is magic. Of course, every quadragonal is also magic.
• order 8
  • an order-8 pan-3,4-agonal magic tesseract (associated) (CSV file) [Nakamura, August 2004]
       This tesseract is associated and pan-3,4-agonal.
  • an order-8 pan-3,4-agonal magic tesseract (associated and 2-compact) (CSV file) [Nakamura, September 2007]
       This tesseract is associated, 2-compact, and pan-3,4-agonal.
  • an order-8 2,3,4-agonal magic tesseract (associated) (CSV file) [Nakamura, December 2004]
       This tesseract is associated and strictly magic (that is, 2,3,4-agonal).
  • an order-8 pan-4-agonal & 2,3,4-agonal magic tesseract (non-associated) (CSV file) [Nakamura, September 2006]
       This tesseract is both panmagic (that is, pan-4-agonal) and strictly magic (that is, 2,3,4-agonal).
  • an order-8 pan-3,4-agonal & 2,3,4-agonal magic tesseract (non-associated) (CSV file) [Nakamura, September 2006]
       This tesseract is pan-3,4-agonal and 2,3,4-agonal. In other words, it is panmagic, strictly magic, and pantriagonal.
       An order-8 Nasik (that is, pan-2,3,4-agonal) magic tesseract cannot exist.
• order 10
  • an order-10 pan-3-agonal magic tesseract (non-associated) (CSV file) [Nakamura, August 2007]
       Every pantriagonal of this tesseract is magic. Of course, every quadragonal is also magic.
• order 12
  • an order-12 pan-4-agonal magic tesseract (associated) (CSV file, 116KB) [Nakamura, August 2004]
       This tesseract is associated and panmagic (that is, pan-4-agonal).
  • an order-12 pan-3,4-agonal magic tesseract (associated and 2-compact) (CSV file, 116KB) [Nakamura, September 2007]
       This tesseract is associated, 2-compact, and pan-3,4-agonal.
  • an order-12 2,3,4-agonal magic tesseract (associated) (CSV file, 116KB) [Nakamura,September 2006]
       This tesseract is associated and strictly magic (that is, 2,3,4-agonal).
  • an order-12 pan-4-agonal & 2,3,4-agonal magic tesseract (non-associated) (CSV file, 116KB) [Nakamura, September 2006]
       This tesseract is both panmagic (that is, pan-4-agonal) and strictly magic (that is, 2,3,4-agonal).
       An order-12 Nasik (that is, pan-2,3,4-agonal) magic tesseract cannot exist.
• order 14
  • an order-14 pan-3-agonal magic tesseract (non-associated) (CSV file, 222KB) [Nakamura, August 2007]
       Every pantriagonal of this tesseract is magic. Of course, every quadragonal is also magic.
       No magic tesseract of singly even order can be pandiagonal or panquadragonal.
  • an order-14 3,4-agonal magic tesseract (non-associated) (CSV file, 222KB) [Nakamura, June 2005]
       This tesseract is 3,4-agonal and 1176 of the 2352 diagonals are also magic.
       It is unknown whether an order-14 strictly magic tesseract (that is, 2,3,4-agonal magic tesseracts) exists or not.
    

5-dimensional magic hypercubes

• order 4
  • an order-4 pan-3,5-agonal 5-D magic hypercube (associated) (CSV file) [Nakamura, August 2004]
       This hypercube is associated and panmagic (that is, pan-5-agonal). It is also pan-3-agonal.
  • an order-4 2,5-agonal 5-D magic hypercube (non-associated) (CSV file) [Nakamura, May 2008]
  • an order-4 4,5-agonal 5-D magic hypercube (associated) (CSV file) [Nakamura, May 2008]
  • an order-4 3,5-agonal 5-D magic hypercube (non-associated, knight tour) (CSV file) [Nakamura, April 2009]    New!
       The 1,024 consecutive integers of this hypercube trace out a magic knight tour. It is also 3-agonal.
• order 6
  • an order-6 pan-5-agonal 5-D magic hypercube (non-associated, complete) (CSV file) [Nakamura, November 2004]
       This hypercube is panmagic (that is, pan-5-agonal) and complete.
  • an order-6 pan-5-agonal 5-D magic hypercube (associated) (CSV file) [Nakamura, March 2008]
       This hypercube is associated and panmagic (that is, pan-5-agonal).
• order 8
  • an order-8 pan-3,5-agonal 5-D magic hypercube (associated) (CSV file, 197KB) [Nakamura, August 2004]
       This hypercube is associated and panmagic (that is, pan-5-agonal). It is also pan-3-agonal.
  • an order-8 pan-3,5-agonal 5-D magic hypercube (associated, 3-compact) (CSV file, 197KB) [Nakamura, June 2009]    New!
       This hypercube is associated, 3-compact (that is, 2x2x2-compact), and panmagic (that is, pan-5-agonal). It is also pan-3-agonal.
       There is no normal magic hypercube of odd dimension which is both associated and 2-compact (that is, 2x2-compact).
  • an order-8 2,3,4,5-agonal 5-D magic hypercube (associated) (CSV file, 197KB) [Nakamura, December 2004]
       This hypercube is associated and strictly magic (that is, 2,3,4,5-agonal).
  • an order-8 3,5-agonal 5-D magic hypercube (non-associated, knight tour) (CSV file) [Nakamura, May 2009]    New!
       The 32,768 consecutive integers of this hypercube trace out a magic knight tour. It is also 3-agonal.
• order 10
  • an order-10 pan-3,5-agonal 5-D magic hypercube (associated) (zipped CSV file, 245KB) [Nakamura, May 2008]
       This hypercube is associated and panmagic (that is, pan-5-agonal). It is also pan-3-agonal.

6-dimensional magic hypercubes

• order 4
  • an order-4 pan-3,5-agonal 6-D magic hypercube (associated) (CSV file) [Nakamura, September 2007]
       This hypercube is associated and pan-3,5-agonal.
  • an order-4 2,5,6-agonal 6-D magic hypercube (associated) (CSV file) [Nakamura, May 2008]
  • an order-4 4,6-agonal 6-D magic hypercube (associated) (CSV file) [Nakamura, May 2008]
• order 8
  • an order-8 pan-3,5,6-agonal 6-D magic hypercube (associated) (zipped CSV file, 717KB) [Nakamura, August 2004]
       This hypercube is associated and panmagic (that is, pan-6-agonal). It is also pan-3,5-agonal.
  • an order-8 pan-3,5,6-agonal 6-D magic hypercube (associated and 2-compact) (zipped CSV file, 659KB) [Nakamura, September 2007]
       This hypercube is associated, 2-compact, and panmagic (that is, pan-6-agonal). It is also pan-3,5-agonal.
  • an order-8 2,3,4,5,6-agonal 6-D magic hypercube (associated) (zipped CSV file, 773KB) [Nakamura, December 2004]
       This hypercube is associated and strictly magic (that is, 2,3,4,5,6-agonal).

7-dimensional magic hypercube

• order 4
  • an order-4 pan-3,5,7-agonal 7-D magic hypercube (associated) (CSV file, 112KB) [Nakamura, August 2004]
       This hypercube is associated and panmagic (that is, pan-7-agonal). It is also pan-3,5-agonal.
  • an order-4 3,5,7-agonal 7-D magic hypercube (non-associated, knight tour) (CSV file, 112KB) [Nakamura, April 2009]    New!
       The 16,384 consecutive integers of this hypercube trace out a magic knight tour. It is also 3,5-agonal.
• order 6
  • an order-6 pan-7-agonal 7-D magic hypercube (associated) (zipped CSV file, 863KB) [Nakamura, May 2008]
       This hypercube is associated and panmagic (that is, pan-7-agonal).

Notes on magic hypercubes

1. Associated panmagic hypercubes
(1) Order m = 4
   I proved in August 2004 that an associated panmagic hypercube exists for order m = 4 and any odd dimension n >= 3.
(Examples for dimension 5 and dimension 7 are shown above, and an example for dimension 3 is here.)
   On the other hand, every order-4 panmagic hypercube of even dimension is complete, so an order-4 (normal) panmagic hypercube of even dimension cannot be associated.
In particular, no order-4 associated pandiagonal magic square exists, and no order-4 associated panquadragonal magic tesseract can exist.

(2) Order m = 4x (m >= 8)
   I proved in August 2004 that an associated panmagic hypercube exists for any order m = 4x >= 8 and any dimension n >= 2.
(Examples for order 8 and dimension 4, dimension 5, and dimension 6 are shown above, and an example for order 8 and dimension 3 is here.)
   In addition, if the dimension n is even, an associated 2-compact panmagic hypercube can exist for any order m = 4x >= 8.
(Examples for order 8 and dimension 4 and dimension 6 are shown above. A magic hypercube of odd dimension cannot be both associated and 2-compact.)

(3) Order 4x+2 (m >= 6)
   I proved in May 2008 that an associated panmagic hypercube exists for any order m = 4x+2 >= 6 and any odd dimension n >= 3.
(Examples for order 6 and dimension 5 and dimension 7 are shown above, and an example for order 6 and dimension 3 is here.)
   On the other hand, if the dimension is even and the order is singly-even, a magic hypercube can be neither panmagic nor associated.

2. Strictly magic hypercubes
   I proved in December 2004 that a strictly magic hypercube of order m and dimension n exists if the order m is divisible by 2^e, where e is the least integer such that 2^e > n+1.
   For example, a strictly magic hypercube of order 8 exists for any dimension lower than 7, and a strictly magic hypercube of order 16 exists for any dimension lower than 15.
(Examples for order 8 and dimension 4, dimension 5, and dimension 6 are shown above, and an example for order 8 and dimension 3 is here.)
   I proved in May 2008 that a strictly magic hypercube of order 8 and dimension 7 cannot exist.

3. Order-4 diagonal magic hypercubes
   I proved in May 2008 the following theorem on the existence of a diagonal (namely, 2-agonal) magic hypercube of order 4:
A diagonal magic hypercube of order 4 and dimension n >= 2 cannot exist if n = 3 (mod. 4), and exists if not.
   For example, such a magic hypercube exists for dimension 2, 4, 5, 6, or 8, and cannot exist for dimension 3, 7, or 11.
(Examples for dimension 4, dimension 5, and dimension 6 are shown above. It is well known that an order-4 diagonal magic hypercube cannot exist for dimension 3.)
   On the other hand, a triagonal (namely, 3-agonal) magic hypercube of order 4 and dimension n exists for any dimension n >= 3.

4. Magic hypercubes with knight tours
   I proved in May 2009 that a magic hypercube with a knight tour exists for any order m = 4x >= 4 and any odd dimension n >= 3.
(Examples for dimension 5 and order 4, dimension 5 and order 8, and dimension 7 and order 4 are shown above, and an example for dimension 3 and order 4 is here.)
   On the other hand, a magic hypercube with a knight tour is not found yet for even dimension n >= 4. Awani Kumar constructed an order-4 semimagic tesseract with a knight tour, whose quadragonals are not magic, in September 2008. He constructed various (semi)magic hypercubes with knight tours, for example, a magic square of order 12 with a knight tour, magic cubes of orders 8, 12, and 16 with knight tours, a 5-dimensional semimagic hypercube of order 4 with a knight tour (whose quintagonals are not magic), 2-dimensional magic tours of order 8 on the surfaces of a cube and a tesseract (whose diagonals are not magic), etc.

   See the following sites to study magic knight tours:

• Harvey D. Heinz's site : for dimensions 2 and 3, for dimensions higher than 3
• Guenter Stertenbrink and Jean Charles Meyrignac's site

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