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# Works on magic tesseracts and hypercubes

The following are my works on magic tesseracts and hypercubes (CSV files). Every magic tesseract and hypercube in this page is normal.
Definitions of terms are here. Works on magic cubes are here.

## 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 2D-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 2D-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
(1) I proved in December 2004 that a strictly magic hypercube of order m and dimension n exists if the order m is divisible by 2e, where e is the least integer such that 2e > 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.

(2) I proved in January 2013 that a strictly magic tesseract (that is, 2,3,4-agonal magic tesseract) exists for any even order greater than 7, and for any odd order greater than 14. Examples for orders 8, 10, 12, 14, 15, and 18 are shown above. (For order 16 and 17, there exists a Nasik (stronger than strictly magic) tesseract.) It is an open problem whether a strictly magic tesseract of order 6, 7, 9, 11, or 13 exists or not (it is impossible for order 3, 4, or 5).

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: Home Site Map

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