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Further for simplicity of a statement shift is understood as shift without repetitions of numbers 1, 2..., n designated (a1, a2..., an). The following basic concepts which are often going beyond a school course of mathematics lead to interesting algorithms.

Interestingly, with use of intersubject communications, it is possible to construct also lessons. Knowledge of bases of logic not only to development of cognitive interest of pupils, but also bases of successful mastering all course of an, promotes development of algorithmic thinking, in particular it is rational to ability to build the branching and algorithms, to the fastest acquisition of algorithmic language, helps with mastering any knowledge.

Use of intersubject communications. Work has to on the knowledge and abilities received by school students at other lessons both physical and mathematical, and natural, and, and a humanitarian cycle.

Divisibility of numbers. Let's give an example of intersubject communications when mathematical formulas and theorems are used for an algorithm. We will sort the task connected with Lagrange's theorem. The algorithm of its decision is simple, but gives the chance to school students with problems of the analysis of algorithms. These along with testing unfairly manage not only in school, but also in high school courses of programming.

Consistently applying this algorithm, it is possible to find a polynomial on not given multipliers. This task communication of representation of a polynomial as algebraic structure and functional dependence, and also the practical application of this communication.

To avoid these difficulties, it is expedient to suggest pupils to investigate real physical, chemical and other similar situations, independently to think over the mathematical model of the phenomenon leading to the equation or system of the equations. These equations are solved further by application of a method with use of the standard subprogramme, at the corresponding lesson of calculus mathematics. It is desirable that the equations describing the considered phenomena were not solved analytically or their decision was too difficult — it clearly demonstrates efficiency of a of the approached methods.

The second group is made by problems of calculus mathematics. In courses of mathematics and programming pupils get acquainted with the main methods of the approached solution of the equations, a of systems of the linear equations, with methods of interpolation and extrapolations, with methods of numerical integration. It to offer school students a big set of tasks. However thus there are difficulties of the methodical plan.

Lagrange's theorem claims that each natural number can be presented in the form of the sum of four squares of integers. It is proved structurally, i.e. the algorithm of a of such splitting for any number is given.

Applied orientation. The subject of work has to reflect the real situation arising in scientific and technical practice of use of the COMPUTER. Certainly, complexity level thus has to correspond to the school student's opportunities.