Magic Tesseract

A magic tesseract is a four-dimensional generalization of the two-dimensional magic square and the three-dimensional magic cube. A magic tesseract has magic constant

so for , 2, ..., the magic tesseract constants are 1, 17, 123, 514, 1565, 3891, ... (OEIS A021003).

Berlekamp et al. (1982, p. 783) give a magic tesseract. J. Hendricks has constructed magic tesseracts of orders three, four, five (Hendricks 1999a, pp. 128-129), and six (Heinz). M. Houlton has used Hendricks' techniques to construct magic tesseracts of orders 5, 7, and 9.

There are 58 distinct magic tesseracts of order three, modulo rotations and reflections (Heinz, Hendricks 1999), one of which is illustrated above. Each of the 27 rows (e.g., 1-72-50), columns (e.g., 1-80-42), pillars (e.g., 1-54-68), and files (e.g., 1-78-44) sum to the magic constant 123.

Hendricks (1968) has constructed a pan-4-agonal magic tesseract of order 4. No pan-4-agonal magic tesseract of order five is known, and Andrews (1960) and Schroeppel (1972) state that no such tesseract can exist.

The smallest perfect magic tesseract is of order 16, having magic constant , and has been constructed by Hendricks (Peterson 1999).

-dimensional magic hypercubes of order 3 are known for , 6, 7, and 8 (Hendricks). Hendricks has also constructed a perfect 16th order magic tesseract (where perfect means that all hyperplanes are perfect).

In 2003, Christian Boyer constructed the first bimagic and trimagic tesseracts.