Perfect Magic Cube


A perfect magic cube is a magic cube for which the rows, columns, pillars, space diagonals, and diagonals of each n×n orthogonal slice sum to the same number (i.e., the magic constant M_3(n)). While this terminology is standard in the published literature (Gardner 1976, Benson and Jacoby 1981, Gardner 1988, Pickover 2002), it has been suggested at various times that such cubes instead be termed Myers cubes, Myers diagonal cubes, or diagonal magic cube (Heinz).


There is a trivial perfect magic cube of order one, but no perfect cubes exist for orders 2-4 (Schroeppel 1972; Benson and Jacoby 1981, pp. 23-25; Gardner 1988). While normal perfect magic cubes of orders 7 and 9 have been known since the late 1800s, it was long not known if perfect magic cubes of orders 5 or 6 could exist (Wells 1986, p. 72), although Schroeppel (1972) and Gardner (1988) note that any such cube must have a central value of 63. (Confusingly, Benson and Jacoby (1981, p. 5) give a table containing the entry "5×5×5--pandiagonal only," incorrectly suggesting that a perfect magic cube of order 5 is not possible.) Then, on Nov. 14 2003, C. Boyer and W. Trump discovered the order five perfect magic cube illustrated above in three-dimensional form and in cross sections (Schroeppel 2003, Augereau 2003, Weisstein 2003). As expected, this cube has central value 63.


Boyer and Trump's discovery followed closely Trump's discovery of the first known 6th order perfect magic on September 1, 2003 (Boyer), illustrated above.


The first published perfect magic cube was the order 7 cube found by the Reverend A. H. Frost of St. John's College, Cambridge (Frost 1866). Because Frost had been a missionary in Nasik, India, he termed the special type of magic cube he constructed a Nasik cube. Langman (1962) subsequently constructed another perfect magic cube of order seven, and still others were found by R. Schroeppel and Ernst Straus (Wells 1986, p. 72).


The first published perfect cube of order 8 was constructed by Gustavus Frankenstein and published in the March 11, 1875 edition of the The Cincinnati Commercial newspaper (Barnard 1888, Gardner 1976, Benson and Jacoby 1981, Gardner 1988; Boyer). Waxing poetic on his discovery, Frankenstein went on to note "This discovery gives me greater satisfaction than if I had found a gold mine under my door-sill; and it is delight like this that makes poverty sweeter than the wealth of Croesus." The construction of an order 8 perfect magic cube is discussed in Ball and Coxeter (1987). Rosser and Walker rediscovered the order-eight cube in the late 1930s but did not publish it, and Myers independently discovered the cube shown above in 1970 (Wells 1986, p. 72; Gardner 1988).

Frost (1878) found a perfect magic cube of order 9, but it did not use consecutive numbers. The first published normal perfect cube of order 9 was found by Planck (1905). The first perfect cube of order 10 was constructed it in 1988 by Li Wen and communicated to C. Boyer in Dec. 2003. Perfect magic cubes of orders 11 and 12 are also known (Barnard 1888, Benson 1981, Boyer). The following table summarizes known perfect cubes and their discoverers (Boyer).

4impossible (Schroeppel 1972)
5Trump and Boyer (Boyer, Weisstein 2003)
6Trump (Boyer, Weisstein 2003)
7Frost (1866)
8Frankenstein (1875)
9Planck (1905)
10Li Wen (1988, Boyer)
11Barnard (1888)
12Benson (1981)

See also

Bimagic Cube, Magic Cube, Magic Square, Magic Tesseract, Nasik Cube, Pandiagonal Perfect Magic Cube, Semiperfect Magic Cube

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Perfect Magic Cube

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Weisstein, Eric W. "Perfect Magic Cube." From MathWorld--A Wolfram Web Resource.

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