What is a fifth-order phantom cube? What are the rules?

A fifth-order phantom square is a fifth-order planar and phantom square, which is a 5X5 square grid filled with 25 different numbers so that the sums of each row, column, and two diagonals are equal.

Magic Square is a method of arranging numbers in a square grid so that the sum of the numbers in each row, column, and diagonal is equal. Magic Square is also a traditional Chinese game. In the olden days, it was common in government offices and schools. It is a way of arranging natural numbers from one to a number of numbers into squares with a number of numbers in each direction, so that the sums of several numbers in the same row, column and diagonal are equal.

People have studied and come up with the formula for calculating the sum of all the numbers in the rows, columns, and diagonals of a phantom square of any order as

S=n(n^2+1) /2 where n is the order of the phantom square, and the number sought is S.

Expanded

Laws of the third-order phantom squares: phantom sum vs. center number, phantom sum = 3 x center number

Proof:

There are 4 lines through the center number. Adding all these 4 lines gives:

Phantom sum x 4 = sum of all numbers + center number x 3

And we know that in a third order phantom square, the sum of all numbers = 3 x phantom sum (in three rows or columns) so we have:

Phantom sum x 4 = phantom sum x 3 + center number x 3

Simplifying gives:

Phantom sum = 3 x center number

Source: Baidu Encyclopedia - Phantom Square