Infinite Numerical Magic Square

Introduction to Magic Squares

Magic squares are ancient mathematical puzzles full of charm. Below is a detailed introduction:

1. What is a Magic Square?

A Magic Square is an n×n square matrix composed of distinct numbers that satisfies:

• The sum of numbers in each row and column is equal
• The sum of numbers in both main diagonals is also equal
• Typically uses consecutive natural numbers from 1 to n² (some variations may allow repeated numbers)

2. Key Concepts

Magic Constant

The common sum of each row/column/diagonal. Calculation formula:
Magic Constant = n(n² + 1)/2

• 3×3 Magic Square: Magic Constant = 15
• 4×4 Magic Square: Magic Constant = 34

Classification by Order

1. Odd-order (e.g., 3×3, 5×5)
2. Doubly even-order (4×4, 8×8, i.e., multiples of 4)
3. Singly even-order (6×6, 10×10, i.e., even numbers not multiples of 4)

Odd-Order Magic Squares

Note: After downloading and running the program, view the detailed step-by-step example diagram on the program webpage.
Note: Create a magic square.xlsx file by inputting numerical values, including detailed step-by-step reference tables.
Program download address：https://github.com/jiqi136/Magic-square/releases/download/untagged-8e3467f4e2a4ee5e386f/Infinite.Numerical.Magic.Square.V1.1.zip

Step-by-step explanation for creating a 5x5 odd-order magic square:

Table: A-Single

Fill number 1 in the first cell of the top row, then fill increasing odd numbers spaced apart until reaching the square number 25.
picture

Table: B-Dual

Fill number 2 in the last cell of the bottom row, then fill increasing even numbers spaced apart until reaching 24 (square number minus one).
picture

Table: C-Spaced

Insert blank rows and columns between each existing row and column.
picture

Table: D-Shift

1. Connect the midpoints of the four edges to form a diamond-shaped frame.
2. Numbers outside the diamond frame are folded into the frame along the diagonal axis like paper folding.
   pictureA
   pictureB

Table: E-Merged

Rotate the diamond frame 45 degrees to align, then copy the numbers within the frame to a new table.
picture

Complete: Batch verify sum values using 'Auto Sum' in the 'Home' tab at the top right of the xls sheet.

Single-Even Magic Square

Singly even: Even numbers not divisible by 4 (e.g., 6,10,18...)
Note: After downloading and running the program, view the detailed step-by-step example diagram on the program webpage.
Note: Create a magic square.xlsx file by inputting numerical values, including detailed step-by-step reference tables.
Program download address：https://github.com/jiqi136/Magic-square/releases/download/untagged-8e3467f4e2a4ee5e386f/Infinite.Numerical.Magic.Square.V1.1.zip

Step-by-step explanation for creating a 6x6 singly even magic square:

Table:A-Half 3

Introduce the creation of a single even 6 divided by 2 to form a 3 magic square.
picture

Table:B-4x Minus 3

Multiply each value of the 3 magic square by 4 and subtract 3.
picture

Table:C-Split Table

Split the 3 magic square into Table A and Table B with alternating values.
picture

Table:D-Forward Order

Expand each value in Table A with 4 adjacent numbers to form sub-squares.
Increment by 1 in forward order: sub-square top row [n+0, n+1], bottom row [n+2, n+3].
picture

Table:E-Reverse Order

Decrement by 1 in reverse order: sub-square top row [n+3, n+2], bottom row [n+1, n+0].
pictureA
pictureB

Table:F-Adjust

1. Swap left-right in first row's sub-squares, ignoring last corner.
2. Swap top-bottom in last column's sub-squares, ignoring last corner.
3. Swap left-right in bottom-right corner of last row.
4. Swap top-bottom in bottom-left corner of first column.
   pictureA
   pictureB
   pictureC

Complete: Batch verify sum values using 'Auto Sum' in the 'Home' tab at the top right of the xls sheet.
picture

Double-Even Magic Square

Doubly even: Even numbers divisible by 4 (e.g., 8,12,16...)
Note: After downloading and running the program, view the detailed step-by-step example diagram on the program webpage.
Note: Create a magic square.xlsx file by inputting numerical values, including detailed step-by-step reference tables.

Program download address：https://github.com/jiqi136/Magic-square/releases/download/untagged-8e3467f4e2a4ee5e386f/Infinite.Numerical.Magic.Square.V1.1.zip

Step-by-step explanation for creating an 8x8 doubly even magic square:

Table:A-Single

Introducing the creation of a doubly even 8 magic square divided by 2 into 4 magic squares.
picture

Table:B-4x Minus 3

Each value of the 4 magic square is multiplied by 4 and then subtracted by 3.
picture

Table:C-Upper Section

Expand each value into 4 adjacent numbers:
Top row [n+0, n+3], bottom row [n+2, n+1].
Example with n=1: Small square's top row [1, 4], bottom row [3, 2].
picture

Table:D-Lower Section

Expansion method for the lower section:
Top row [n+3, n+0], bottom row [n+1, n+2].
picture

Complete: Batch verify sum values using 'Auto Sum' in the 'Home' tab at the top right of the xls sheet.
picture