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The cube of a number n is denoted n 3, using a superscript 3, [a] for example 2 3 = 8. The cube operation can also be defined for any other mathematical expression, for example (x + 1) 3. The cube is also the number multiplied by its square: n 3 = n × n 2 = n × n × n. The cube function is the function x ↦ x 3 (often denoted y = x 3) that
Perfect numbers are natural numbers that equal the sum of their positive proper divisors, which are divisors excluding the number itself. So, 6 is a perfect number because the proper divisors of 6 are 1, 2, and 3, and 1 + 2 + 3 = 6. [2] [4]
A perfect parallelepiped is a parallelepiped with integer-length edges, face diagonals, and body diagonals, but not necessarily with all right angles; a perfect cuboid is a special case of a perfect parallelepiped. In 2009, dozens of perfect parallelepipeds were shown to exist, [19] answering an open question of Richard Guy. Some of these ...
In mathematics, a perfect magic cube is a magic cube in which not only the columns, rows, pillars, and main space diagonals, but also the cross section diagonals sum up to the cube's magic constant. [ 1 ] [ 2 ] [ 3 ]
In 1939, B. Rosser and R. J. Walker published a series of papers on diabolic (perfect) magic squares and cubes. They specifically mentioned that these cubes contained 13 m 2 correctly summing lines. They also had 3 m pandiagonal magic squares parallel to the faces of the cube, and 6 m pandiagonal magic squares parallel to the space-diagonal planes.
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Perfect: All 3m planar arrays must be pandiagonal magic squares. In addition, all pantriagonals must sum correctly. These two conditions combine to provide a total of 9m pandiagonal magic squares. The smallest normal perfect magic cube is order 8; see Perfect magic cube. Nasik; A. H. Frost (1866) referred to all but the simple magic cube as Nasik!