Magic Tesseracts

The traditional magic square consists of a set of integers arranged in the form of a square so that the numbers in each row, column, and diagonal add up to the same total. If the integers are consecutive numbers from 1 to n², the square is said to be of nth order (see More than Magic Squares, Oct. 14, 1996).

There are no magic squares of order 2, and only one (not counting its rotations and reflections) of order 3.

4 9 2

3 5 7

8 1 6

There are 880 magic squares of order 4, again excluding reflections and rotations of each pattern. Here's one remarkable example.

7 12 1 14

2 13 8 11

16 3 10 5

9 6 15 4

Notice that not only the rows, columns, and diagonals add up to 34 but also the corner 2 x 2 subsquares. Other subsquares give the same result.

In 1973, mathematician and computer scientist Richard Schroeppel, now at the University of Arizona in Tucson, used a computer to count the number of magic squares of order 5. The total comes to 275,305,224!

1 15 24 8 17

23 7 16 5 14

20 4 13 22 6

12 21 10 19 3

9 18 2 11 25

It's natural to extend the concept of magic squares to three and four dimensions.

In an n x n x n magic cube, each of the n² rows, columns, and pillars and the four space diagonals sum to a number known as the magic constant. If the cross-section diagonals also add up to the same constant, you have a so-called perfect magic cube.

It has been proved that there are no perfect magic cubes of order 2, 3, or 4. It is possible to construct one of order 7, 8, 9, 11, and higher. No one has yet found one of order 5, 6, or 10, or proved that such cubes cannot exist.

By relaxing the requirements, you can form a wide variety of different types of magic cubes. Here's one of order 3.

Bottom Layer Middle Layer Top Layer

4 12 26 20 7 15 18 23 1

11 25 6 9 14 19 22 3 17

27 5 10 13 21 8 2 16 24

In four dimensions, the equivalent of a cube is a hypercube, or tesseract. A tesseract has 16 vertices, 32 edges, 24 squares, and 8 cubes (see http://mathworld.wolfram.com/Tesseract.html for an illustration or http://www.geom.umn.edu/docs/outreach/4-cube/for a guided tour).

The smallest perfect magic tesseract is of order 16 and is made up of 16 x 16 x 16 x 16 numbers. The sum of the numbers in each of the rows, columns, pillars, and files (rows in a fourth spatial direction) and along the eight major quadragonals, which pass through the center and join opposite corners, is 524,296.

John R. Hendricks, a retired meteorologist in British Columbia, has been studying magic squares, cubes, and tesseracts for years. With the help of Cliff Pickover at the IBM Thomas J. Watson Research Center in Yorktown Heights, N.Y., he has now computed the placement of all 65,536 values in a perfect magic tesseract of order 16.

It's the world's first known 16th-order perfect magic tesseract, Pickover says.

"In four-dimensional space, we checked different directions along the various kinds of diagonals, row, columns, and so on," he notes. "We used a systematic method to check the sums of 16 numbers along each of these routes through each of the 65,536 points."

The computation required about 10 hours on an IBM Intellistation running the Windows NT operating system.

References:

Gardner, M. 1988. Magic squares and cubes. In Time Travel and Other Mathematical Bewilderments. New York: W.H. Freeman.

______. 1961. Magic squares. In The 2nd Scientific American Book of Mathematical Puzzles & Diversions. New York: Simon & Schuster.

Information about magic cubes and tesseracts is available at http://mathworld.wolfram.com/MagicCube.html. A starting point for exploring magic squares is http://www.geocities.com/~harveyh/magicsquare.htm.

Cliff Pickover has a Web site at http://www.pickover.com.

Comments are welcome. Please send messages to Ivars Peterson at ipeterson@maa.org.