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What is Magic Cube?

the Magic cube are the three-dimensional mathematical phenomenon originating in the two-dimensional option - a magic square. Having dealt with a magic square, to present a magic cube will not make such work any more.

So, a magic square call the square consisting of a certain quantity of cages of n*n (in case of n=3 it is 3*3), in each of which there is a number from 1 to n2 (respectively, from 1 to 9=32) at what the sums of numbers will be equal in any vertical, horizontal or direct diagonal row, it is a constant of a concrete magic square, it is called a constant. Its formula looks so: To =n/2 (1+n2), and if to substitute figure 3 instead of n, we will receive the constant equal 3/2(1+32)=15. Thus, the example of such square can look as follows:



the Number n also determines by 4 9 2

3 5 7

8 1 6

a square order, for example if in one party of a square of 5 cages, then, respectively, it will be a square of the fifth order.

Such magic square in which are equal to a constant of the sum not only cages in ranks and the main diagonals, but also in " is possible; broken lines diagonals, it is accepted to call it pan-diagonal (vsediagonalny). Broken lines diagonal (them still call torichesky parallel translations) can be presented, having arranged a row two identical pan-diagonal squares where the constant will be found on all visible diagonals, on two from each of the squares entered in this rectangle (rice). Such square represents a rapport which regardless of number of repetitions can be closed in a ring (Torahs), and the number of diagonals will be equal to the number n increased by the number of repetitions.

The magic square can also have property of symmetry or associativity when each couple of numbers in it symmetrized concerning the center of a square at addition yields result of n2+1.

We sorted that is a magic square. Now we will consider how it is possible to construct a magic cube by the same principles, as a magic square.

Now, armed with terms, we undertake studying of a magic cube. The simple magic cube of n - an order is difficult from n of cubes - bricks, the number from 1 to n3 is appropriated to each of which. And cubes - numbers are located in such order that the sums of numbers in any direct row parallel to cube edges, and also in four spatial diagonals (which are connecting corners of a cube and passing through its center), will be equal among themselves and are defined by a formula: To =n/2 (1+n3)

Perfect a magic cube is called when the sums not only four diagonals connecting cube corners, but also the diagonals connecting the numbers finding on cube edges are equal to a constant (in other words, diagonals of any square which is a part of this cube). It is interesting that formally the simplest perfect magic cube is a cube of the first order though in practice it is not anything remarkable.

The magic cube can also be pan-diagonal if all main and broken diagonals on each of sections of a cube parallel to any of cube sides are equal to a constant (each square of which the cube as if consists), and superpan-diagonal if at it all main and broken diagonals are equal to a magic constant in all six diagonal sections passing through cube diagonals.

At the same time the superpan-diagonal cube will be magic only at observance of a condition that its any sides can be transferred in parallel, like transferring of lines and columns in a magic pan-diagonal square, and will correspond to a constant in each turned-out diagonal.

There are those who consider that these mathematical phenomena are useless and inapplicable in life, however not all achievements of mathematics process of understanding of a magic square and magic cube, and also studying of the principles of their construction, and construction in practice have to be applied to specific objectives, give fine training of a brain and development of spatial thinking by means of which the person finds ampler opportunities in various spheres today`s, all of the accelerating life.