Solve the problem of the odd order magic square and add a note Thank you

Odd Tier Magic Formula:

1 in the first column is **, in turn obliquely fill in the upper left, the left out of the grid when the most right, the top out of the grid when the most down, in case of repetition out of the grid nowhere to fill, retreat to the original number of the right line.

Let's take a 5th-order magic square as an example:

1 column, 2 columns, 3 columns, 4 columns, 5 columns.

1 line 15 16 22 3 9

2 lines 8 14 20 21 2

Line 3 1 7 13 19 25

Line 4 24 5 6 12 18

Line 5 17 23 4 10 11

[General fourth-order magic square].

Four groups of arbitrary numbers, as long as the difference between the four numbers in each group is the same, can form a fourth-order magic square. Such as the following four groups of numbers:

Using the above four sets of numbers, the 4th magic square composed of the Latin square:

The general solution method of the fourth-order magic square].

The Latin square a is completed with the row difference of 0, a, b, and c, and the orthogonal Latin square b is completed with the column difference of 0, x, y, and z, and then the Latin magic square c of the array is obtained by a+b+n. As shown below:

A simple example is intuitive:

Perfect fourth-order magic square].

If a+b=c, x+y=z, i.e., a=c-b, x=z-y, i.e., row difference row difference column difference column difference column difference, such an array can form a perfect magic square. As shown in the following figure, it is an example

A perfect magic square is one in which not only the sum of rows, columns, and two diagonals is equal to the magic sum, but also the sum of the pan-diagonals parallel to the diagonal. Imagine tiling the magic square like a tile, and then taking any 4 or 4 squares is a magic square.

1-16 is a special case of the above array, i.e. 16 numbers are equally different numbers. Such as consecutive numbers or 16 numbers of equal differences.

Arrays that can form a perfect magic square of the 4th order can be completed in the simplest way: [sequential numbering, symmetrical exchange of numbers at the center point] to complete the magic square. As shown below:

Consecutive numbers can be made very quickly with orthogonal Latin squares

The formula [c=4a+b+n], (n is the starting number, the number from 1 to 16, n is 1).

Steps: (for any odd magic square).

Fill in the 1 in the middle of the first row and 2 in the top right of the 1 (i.e. move one square to the left and one square up). Where:

If the number is on the first row (e.g. 1 is), the bottom row is assumed to be above the first row, and the next number is filled in on the hypothetical row; Once filled, put the hypothetical line back at the bottom.

Similarly, if the number is in the last column, assume that the first column is to the right of the last column, and fill in the next number in the hypothetical column. Once completed, put the hypothetical column back in the first column.

Let's use a fifth-order magic square as an example: (to click to see a larger image).