Question

A square array of numbers is said to be a magic square, if the sum of the numbers in
each of its row, column and main diagonals is equal. In the following 4 × 4 magic square,
find (a + b + c + d) if the sum of the remaining 12 entries is 150.

Answer 1

n recreational mathematics and combinatorial design, a magic square[1] is a {\displaystyle n\times n}n\times n square grid (where n is the number of cells on each side) filled with distinct positive integers in the range {\displaystyle 1,2,...,n^{2}}{\displaystyle 1,2,...,n^{2}} such that each cell contains a different integer and the sum of the integers in each row, column and diagonal is equal.[2] The sum is called the magic constant or magic sum of the magic square. A square grid with n cells on each side is said to have order n.

The mathematical study of magic squares typically deals with its construction, classification, and enumeration. Although completely general methods for producing all the magic squares of all orders do not exist, historically three general techniques have been discovered: by bordering method, by making composite magic squares, and by adding two preliminary squares. There are also more specific strategies like the continuous enumeration method that reproduces specific patterns. The magic squares are generally classified according to their order n as: odd if n is odd, evenly even (also referred to as "doubly even") if n = 4k (e.g. 4, 8, 12, and so on), oddly even (also known as "singly even") if n = 4k + 2 (e.g. 6, 10, 14, and so on). This classification is based on different techniques required to construct odd, evenly even, and oddly even squares. Beside this, depending on further properties, magic squares are also classified as associative magic squares, pandiagonal magic squares, most-perfect magic squares, and so on. More challengingly, attempts have also been made to classify all the magic squares of a given order as transformations of a smaller set of squares. Except for n ≤ 5, the enumeration of higher order magic squares is still an open challenge.

Answer 2

Step-by-step explanation:

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