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# 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.

Magic Cube, Magic Square, Tesseract

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## References

Adler, A. "Magic N-Cubes Form a Free Monoid." Electronic J. Combinatorics 4, No. 1, R15, 1-3, 1997. http://www.combinatorics.org/Volume_4/Abstracts/v4i1r15.html.Andrews, W. S. Magic Squares and Cubes, 2nd rev. ed. New York: Dover, 1960.Berlekamp, E. R.; Conway, J. H; and Guy, R. K. Winning Ways for Your Mathematical Plays, Vol. 2: Games in Particular. London: Academic Press, 1982.Boyer, C. "Multimagic Hypercubes." http://www.multimagie.com/English/Hypercubes.htm.Heinz, H. "The Tesseract." http://members.shaw.ca/tesseracts/.Heinz, H. "History of the Magic Tesseract (Indeed Magic Hypercubes, as Well)." http://members.shaw.ca/johnhendricksmath/tesseracts.htm.Heinz, H. "John Hendricks." http://www.magic-squares.net/hendricks.htm.Hendricks, J. R. "The Five and Six Dimensional Magic Hypercubes of Order 3." Canad. Math. Bull. 5, 171-189, 1952.Hendricks, J. R. "A Pan-4-agonal Magic Tesseract." Amer. Math. Monthly 75, 384, 1968.Hendricks, J. R. "Magic Tesseracts and -Dimensional Magic Hypercubes." J. Recr. Math. 6, 193-201, 1973.Hendricks, J. R. Erratum to 'Magic Tesseracts and -Dimensional Magic Hypercubes." J. Recr. Math. 7, 80, 1974.Hendricks, J. R. "Ten Magic Tesseracts of Order Three." J. Recr. Math. 18, 125-134, 1985-1986.Hendricks, J. R. Magic Squares to Tesseracts by Computer. Published by the author, 1999a.Hendricks, J. R. All Third Order Magic Tesseracts. Published by the author, 1999b.Hendricks, J. R. Perfect -Dimensional Hypercubes of Order . Published by the author, 1999c.Peterson, I. "Ivar Peterson's MathTrek: Magic Tesseracts." http://www.maa.org/mathland/mathtrek_10_18_99.html.Schroeppel, R. Item 51 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 18, Feb. 1972. http://www.inwap.com/pdp10/hbaker/hakmem/number.html#item51.Sloane, N. J. A. Sequence A021003 in "The On-Line Encyclopedia of Integer Sequences."Trenkler, M. "Magic -Dimensional Cubes of Order (mod 4)." Acta Arith. 92, 189-204, 2000.Trenkler, M. "A Construction of Magic Cubes." Math. Gaz. 84, 36-41, 2000.Trenkler, M. "Magic -Dimensional Cubes." Submitted to Acta Arith., 2000.

Magic Tesseract

## Cite this as:

Weisstein, Eric W. "Magic Tesseract." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MagicTesseract.html