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In mathematics, a magic tesseract is the 4-dimensional generalization of a magic square and magic cube, that is, a number of integers arranged in an n × n × n × n pattern such that the sum of the numbers on each pillar (along any axis) as well as the main space diagonals is equal to a single number, the so-called magic constant of the tesseract, denoted M4(n). It can be shown that if a magic tesseract consists of the numbers 1, 2, ..., n4, then it has magic constant (sequence A021003 in OEIS)
- <math>M_4(n) = \frac{1}{2}n(n^4+1)<math>
If, in addition, the numbers on every cross section diagonal also sum up to the tesseract's magic constant, the tesseract is called a perfect magic tesseract; otherwise, it is called a semiperfect magic tesseract. The number n is called the order of the magic tesseract.
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