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Squares of odd numbers are odd, and are congruent to 1 modulo 8, since (2n + 1) 2 = 4n(n + 1) + 1, and n(n + 1) is always even. In other words, all odd square numbers have a remainder of 1 when divided by 8. Every odd perfect square is a centered octagonal number. The difference between any two odd perfect squares is a multiple of 8.
Perfect numbers are natural numbers that equal the sum of their positive proper divisors, which are divisors excluding the number itself. So, 6 is a perfect number because the proper divisors of 6 are 1, 2, and 3, and 1 + 2 + 3 = 6. [2] [4]
In 1939, B. Rosser and R. J. Walker published a series of papers on diabolic (perfect) magic squares and cubes. They specifically mentioned that these cubes contained 13m 2 correctly summing lines. They also had 3m pandiagonal magic squares parallel to the faces of the cube, and 6m pandiagonal magic squares parallel to the space-diagonal planes ...
In their book, Kathleen Ollerenshaw and David S. Brée give a method of construction and enumeration of all most-perfect magic squares. They also show that there is a one-to-one correspondence between reversible squares and most-perfect magic squares. For n = 36, there are about 2.7 × 10 44 essentially different most-perfect magic squares.
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.
In mathematics, a P-multimagic square (also known as a satanic square) is a magic square that remains magic even if all its numbers are replaced by their kth powers for 1 ≤ k ≤ P. 2-multimagic squares are called bimagic , 3-multimagic squares are called trimagic , 4-multimagic squares tetramagic , and 5-multimagic squares pentamagic .
The folks at Minyanville did a comprehensive calculation and came up with the following chart. The numbers are based on a $50 a square game, with a $625 payout for the 1st and 3rd quarters, a ...
The square of an integer may also be called a square number or a perfect square. In algebra, the operation of squaring is often generalized to polynomials, other expressions, or values in systems of mathematical values other than the numbers. For instance, the square of the linear polynomial x + 1 is the quadratic polynomial (x + 1) 2 = x 2 ...