Mathematics

For every two couples of continues numbers, holds that: If the SECOND number from the first couple of numbers is added to the FIRST number from the second couple of numbers, if we then subtract from the result of the addition the FIRST number from the first couple of numbers then the result will be the SECOND number of the second couple of numbers.

When a curve, s, rolls without slipping on another curve, t, the path of a point, P, whose position is fixed with respect to s is called a Roulette. The point P can be on the curve s, or any point in the plane of s, subject to the conditions set. When one circle rolls on another, the Roulette is known as a Trochoid. External contact gives the Epi-trochoid, while internal contact gives the Hypo-trochoid. Absence of a prefix indicates that the fixed circle has become a straight line. When the tracing point is on the circumference of the variable circle, we obtain the Cycloid. The same terminology as in the case of the Trochoid holds for the case of external and internal contact, and for the case of the straight line. When the variable circle becomes a straight line, the Roulette is known as an Involute. In fact, this name is applied to all Roulettes when the variable curve is a straight line.

When one basic question, in many different forms, receives some attention and an effort is made for an answer to be given, we are usually faced with the same problem as when faced with the problem of our incomplete understanding of the mechanisms of nature. A number of questions inevitably arise: "What is this, if any, missing from the mathematical description of the basic laws of physical science? Can the dimension of Time suddenly disappear, or become so weak that can be neglected, from within the mathematical de- scription of these laws of proven ability to describe accurately the physical phenomena? Is there something we have neglected because its activity remains, in most of the cases, hidden under a non-transparent surface, which only provides us with a description of an effect? Behind a number of variables and constants, in a mathematical expression, is there another power which remains hidden, actually making possible the result and determining, that way, the direction things will evolve?"

If we draw a 3x3 grid (three squares per Row & three squares per Column), we will obtain a total of nine (9) squares overall in which we can arrange numbers in different combinations. If we consider at first (because of the total number of squares in the grid and because this is the series of numbers consisting the basis of the Decimal numerical system) arranging in the grid the nine first Natural Numbers 1…9, we can obtain a 3x3 "Magic Square".

“Because Mathematicians frequently make use of Time, they ought to have a distinct idea of the meaning of the Word, otherwise they are Quacks”, Issac Barrow (1630 - 1677).

This is a collection of Mathematical Problems along with their solutions, solved by strictly mathematical methods. Can be real-world problems, or problems of a more abstract nature. In some cases even problems referring to the very nature of mathematics itself will be questioned, fully addressed and, if possible, solved. In every case, all results of mathematical problems solved are demonstrated and examined formally.