Magic Cubes
A magic cube is a three-dimensional (or higher dimensional) analogue of a magic square. Specifically, an order n magic cube is an n x n x n array of numbers chosen from the set {1,2,...,n3} so that each number appears exactly once and the sum of the numbers in every row, column, file, and diagonal is the same, magic constant. The magic constant must therefore be the sum of all the numbers in the cube divided by the number of rows (columns or files). So, it is (1 + 2 + ... + n3)/n2 = n(n3 +1)/2.
A perfect magic cube is such that each square parallel to a face of the cube is a magic square. Here is one:
If a magic cube is not perfect then it is called semi-perfect. A pandiagonal cube has the property that all of its broken space diagonals sum up to the magic constant. In other words, if copies of the cube were stacked on top of each other and to the right and left as well (without rotating), then any diagonal string of length n will have all of its numbers adding up to the magic constant. Here is a pandiagonal magic cube.
Methods are known for constructing magic cubes of every order other than 2. If A and B are magic cubes and A can be rotated or reflected to look just like B, then A and B are considered to be the same magic cube. Accordingly, there are 4 different order 3 magic cubes. The number of order 4 magic cubes is not known. There are 7680 pandiagonal order-4 magic cubes. Methods are known for constructing perfect magic of any order other than 2, 3, 4, 5, 6, or 10. However, it has been proved that there are no perfect magic cubes of order 2, 3, or 4 so only the order 5, 6, and 10 are in question.
Here is a proof (reproduced from Martin Gardner's Time Travel and other Mathematical Bewilderments) that there are no order 3 perfect cubes. Suppose that such a cube exists. Consider any square parallel to a face of the cube. Let A, B, C be the numbers on the first row and D, E, F be the numbers on the third row. Let X be the middle number. Since the magic constant of any order 3 magic cube is 42,
3X + A + B + C + D + E + F = 3 x 42, A + B + C = 42, and D + E + F = 32.
This implies that 3X = 42 and X = 14. In a pure magic cube, however, no number is repeated twice. So, 14 cannot be the middle number of every cross-section. Thus, a perfect order-3 magic cube is impossible.
A magic tesseract is a 4-dimensional magic cube. Here is one:
There are 58 magic tesseracts of order 3. Hendricks has constructed a pandiagonal order 4 magic tesseract and a perfect order 16 magic tesseract. He also proved there are no perfect magic tesseracts of order less than 16.
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