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For example, a normal 8 × 8 square will always equate to 260 for each row, column, or diagonal. The normal magic constant of order n is n 3 + n / 2 . The largest magic constant of normal magic square which is also a: triangular number is 15 (solve the Diophantine equation x 2 = y 3 + 16y + 16, where y is divisible by 4); square number ...
The number zero for n = 6 is an example of a more general phenomenon: associative magic squares do not exist for values of n that are singly even (equal to 2 modulo 4). [3] Every associative magic square of even order forms a singular matrix, but associative magic squares of odd order can be singular or nonsingular. [4]
* In 4 × 4 most-perfect magic squares, any 2 cells that are 2 cells diagonally apart (including wraparound) sum to half the magic constant, hence any 2 such pairs also sum to the magic constant. A semimagic square is an n × n square with the numbers 1 to n 2 in its cells, in which the sum of each row and column is the same.
In the example below, the left square is the original square, while the right square is the new square obtained by this transformation. In the middle square, rows 1 and 2 and rows 3 and 4 have been swapped. The final square on the right is obtained by interchanging columns 1 and 2 and columns 3 and 4 of the middle square.
A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. [1]
Since each 2 × 2 subsquare sums to the magic constant, 4 × 4 pandiagonal magic squares are most-perfect magic squares. In addition, the two numbers at the opposite corners of any 3 × 3 square add up to half the magic constant. Consequently, all 4 × 4 pandiagonal magic squares that are associative must have duplicate cells.
So, in an n × n magic square using the numbers from 1 to n 2, a magic series is a set of n distinct numbers adding up to n(n 2 + 1)/2. For n = 2, there are just two magic series, 1+4 and 2+3. The eight magic series when n = 3 all appear in the rows, columns and diagonals of a 3 × 3 magic square. Maurice Kraitchik gave the number of magic ...
For each square, cells with the same colour (excluding grey) sum to the magic constant. Note *: The second requirement of most-perfect magic squares imply that any 2 cells that are 2 cells diagonally apart (including wraparound) sum to half the magic constant, hence any 2 such pairs also sum to the magic constant. Width: 100%: Height: 100%