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 M_4(n) = 1/2 n(n^4 + 1), so for n = 1, 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.
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