Algorithms to make magic cubes and magic tesseracts

0. Table of contents

1. Magic cubes
   1. pantriagonal magic cubes (pan-3-agonal)    Updated!
   2. diagonal magic cubes (2,3-agonal)
   3. pantriagonal diagonal magic cubes (pan-3-agonal and 2,3-agonal)
   4. pandiagonal magic cubes (pan-2-agonal and 2,3-agonal)
   5. Nasik magic cubes (pan-2,3-agonal)
2. Magic tesseracts
   1. panmagic tesseracts (pan-4-agonal)
   2. pantriagonal magic tesseracts (pan-3-agonal and 3,4-agonal)
   3. pan-3,4-agonal magic tesseracts
   4. strictly magic tesseracts (2,3,4-agonal)
   5. pan and strictly magic tesseracts (pan-4-agonal and 2,3,4-agonal)    New!
   6. Nasik magic tesseracts (pan-2,3,4-agonal)

1. Algorithms to make magic cubes

   This section explains general algorithms to construct magic cubes of classes pantriagonal (pan-3-agonal), diagonal (2,3-agonal), pantriagonal diagonal (pan-3-agonal and 2,3-agonal), pandiagonal (pan-2-agonal and 2,3-agonal), and Nasik (pan-2,3-agonal). The algorithms for the two classes pantriagonal and diagonal support magic cubes of singly-even order (that is, order (4x+2)), which used to be said to be difficult to construct.
   Let m be the order of the cube and (a_ijk), where i,j,k = 0,...,m-1, be the cube (note that the indexes start from zero). The function gcd(x, y) means the greatest common divisor of x and y.
   Definitions of terms are here. Explanation of classes of magic cubes is here.

1.1 Algorithms for pantriagonal magic cubes (pan-3-agonal)

   A pantriagonal magic cube (or, panmagic cube) always exists if the order is higher than 3. In this case, an associated pantriagonal magic cube exists, too. See also algorithms for higher classes, pantriagonal diagonal and Nasik.

• Case m = 4x
  • Non-associated (complete, 2-compact)
  • Associated
• Case m = 2x+1 and m >= 5
  • Case gcd(m, 3) = 1 (associated)
  • Case gcd(m, 3) = 3 (associated)
• Case m = 4x+2 and m >= 6
  • Non-associated (complete)
  • Associated    Updated!

1.2 Algorithms for diagonal magic cubes (2,3-agonal)

   A diagonal magic cube (or, strictly magic cube) can exist only if the order is higher than 4. The following algorithms work for orders higher than 6. See also algorithms for higher classes, pantriagonal diagonal, pandiagonal, and Nasik.

• Case m = 4x and m >= 8 (associated)
• Case m = 2x+1 and m >= 7 (associated)
• Case m = 4x+2 and m >= 10 (non-associated)

1.3 Algorithm for pantriagonal diagonal magic cubes (pan-3-agonal and 2,3-agonal)

   A pantriagonal diagonal magic cube (or, a PantriagDiag magic cube) of even order can exist only if the order is higher than 7 and divisible by 4. The following algorithm works for orders higher than 11. See also algorithms for a higher class, Nasik.

• Case m = 4x and m >= 12 (non-associated, complete)

1.4 Algorithms for pandiagonal magic cubes (pan-2-agonal and 2,3-agonal)

   A pandiagonal magic cube can exist only if the order is divisible by 8, or odd and higher than 6. In this case, if the order is not equal to 8, an associated pandiagonal magic cube can exist. See also algorithms for a higher class, Nasik.

• Case m = 8x
  • Non-associated
  • Associated (m >= 16)
    • Case m = 16x
    • Case m = 16x+8 and m >= 24
• Case m = 2x+1 and m >= 7
  • Case gcd(m, 3x5) = 1 (associated)
  • Case 1 < gcd(m, 3x5) < m (associated)
  • Case gcd(m, 3x5) = m (associated)    (m = 15)

1.5 Algorithms for Nasik magic cubes (pan-2,3-agonal)

   A Nasik magic cube (or, a pan-2,3-agonal magic cube) can exist only if the order is divisible by 8, or odd and higher than 8. In this case, if the order is higher than 8, an associated Nasik magic cube exists. Furthermore, if the order is higher than 8 and divisible by 8, a Nasik magic cube exists which is both associated and 3-compact. There cannot exist a 2-compact Nasik magic cube.

• Case m = 8x
  • Non-associated (complete, 3-compact)
  • Associated (m >= 16) (not 3-compact)
    • Case m = 16x
    • Case m = 16x+8 and m >= 24
  • Associated and 3-compact (m >= 16)
    • Case m = 16x
    • Case m = 16x+8 and m >= 24
• Case m = 2x+1 and m >= 9
  • Case gcd(m, 3x5x7) = 1 (associated)
  • Case 1 < gcd(m, 3x5x7) < m (associated)
  • Case gcd(m, 3x5x7) = m (associated)    (m = 15, 21, 35, or 105)

2. Algorithms to make magic tesseracts

   This section explains general algorithms to construct magic tesseracts of classes panmagic (pan-4-agonal), pantriagonal (pan-3-agonal and 3,4-agonal), pan-3,4-agonal, strictly magic (2,3,4-agonal), pan and strictly magic (pan-4-agonal and 2,3,4-agonal), and Nasik (pan-2,3,4-agonal). The algorithm for the class pantriagonal supports magic tesseracts of singly-even order (that is, order (4x+2)), which used to be said to be difficult to construct.
   Let m be the order of the tesseract and (a_ijkh), where i,j,k,h = 0,...,m-1, be the tesseract (note that the indexes start from zero). The function gcd(x, y) means the greatest common divisor of x and y.
   Definitions of terms are here. Explanation of classes of magic tesseracts is here.

2.1 Algorithms for panmagic tesseracts (pan-4-agonal)

   A panmagic tesseract (namely, a pan-4-agonal or panquadragonal magic tesseract) can exist only if the order is divisible by 4, or odd and higher than 6. In this case, if the order is higher than 6, an associated panmagic tesseract can exist. See also algorithms for higher classes, pan-3,4-agonal and Nasik.

• Case m = 4x
• Case m = 2x+1 and m >= 7
  • Case gcd(m, 3x5) = 1 (associated)
  • Case 1 < gcd(m, 3x5) < m (associated)
  • Case gcd(m, 3x5) = m (associated)    (m = 15)

2.2 Algorithms for pantriagonal magic tesseracts (pan-3-agonal and 3,4-agonal)

   A pantriagonal magic tesseract exists for any order except 3 and 6. (An order-3 pantriagonal magic tesseract cannot exist, and it is unknown whether an order-6 pantriagonal magic tesseract can exist or not.) If the order is odd or divisible by 4, an associated pantriagonal magic tesseract exists. See also algorithms for higher classes, pan-3,4-agonal and Nasik.

• Case m = 4x
• Case m = 2x+1 and m >= 5
  • Case gcd(m, 3) = 1 (associated)
  • Case gcd(m, 3) = 3 (associated)
• Case m = 4x+2 and m >= 10

2.3 Algorithms for pan-3,4-agonal magic tesseracts

   A pan-3,4-agonal magic tesseract can exist only if the order is divisible by 4, or odd and higher than 8. In this case, if the order is higher than 7, an associated pan-3,4-agonal tesseract can exist. See also algorithms for a higher class, Nasik.

• Case m = 4x
  • Non-associated (complete, 2-compact)
  • Associated (m >= 8) (2-compact)
• Case m = 2x+1 and m >= 9
  • Case gcd(m, 3x5x7) = 1 (associated)
  • Case 1 < gcd(m, 3x5x7) < m (associated)
  • Case gcd(m, 3x5x7) = m (associated)

2.4 Algorithm for strictly magic tesseracts (2,3,4-agonal)

   It is unknown what condition is required and sufficient for the existence of a strictly magic tesseract, namely, a 2,3,4-agonal magic tesseract. A strictly magic tesseract exists for any order divisible by 4 and higher than 7, and for any odd order higher than 14, and cannot exist for order 3, 4, or 5. The following algorithm works for orders divisible by 8. See also algorithms for higher classes, pan and strictly magic and Nasik.

• Case m = 8x (associated)
• Case m = 2x+1 and m >= 15
  • Case gcd(m, L₁₃) = 1 (associated, pan-2,3-agonal)
  • Case 1 < gcd(m, L₁₃) < m (associated, pan-2,3-agonal)
  • Case gcd(m, L₁₃) = m (associated, pan-2,3-agonal)

2.5 Algorithm for pan and strictly magic tesseracts (pan-4-agonal and 2,3,4-agonal)

   A pan and strictly magic tesseract, namely, a pan-4-agonal and 2,3,4-agonal magic tesseract, of even order can exist only if the order is divisible by 4 and higher than 7.

• Case m = 4x and m >= 8 (non-associated, complete)    New!

2.6 Algorithms for Nasik magic tesseracts (pan-2,3,4-agonal)

   A Nasik magic tesseract, namely, a pan-2,3,4-agonal magic tesseract, can exist only if the order is divisible by 16, or odd and higher than 16. In this case, if the order is higher than 16, an associated Nasik magic tesseract can exist. Furthermore, if the order is higher than 16 and divisible by 16, a Nasik magic tesseract exists which is both associated and 4-compact.
   There cannot exist a Nasik magic tesseract which is 2-compact or 3-compact.

• Case m = 16x
  • Non-associated (complete, 4-compact)
  • Associated (m >= 32) (non-4-compact)
    • Case m = 32x
    • Case m = 32x+16 and m >= 48
  • Associated and 4-compact (m >= 32)
    • Case m = 32x
    • Case m = 32x+16 and m >= 48
• Case m = 2x+1 and m >= 17
  • Case gcd(m, L₁₅) = 1 (associated)
  • Case 1 < gcd(m, L₁₅) < m (associated)
  • Case gcd(m, L₁₅) = m (associated)

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