Tool to generate magic squares size N, kind of matrices composed of distinct integers set such as the sum of any line or column are equal.
Magic Square  dCode
Tag(s) : Number Games, Fun/Miscellaneous, Arithmetics
dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!
A suggestion ? a feedback ? a bug ? an idea ? Write to dCode!
Tool to generate magic squares size N, kind of matrices composed of distinct integers set such as the sum of any line or column are equal.
Formula: Set number 1 on the left of the median line, the other numbers are written following the rule: if the cell is empty, in the cell in the bottom right of the previous one, else directly to the left of the occupied cell. When the cell does not exist (after last column or last row), go to the other side of the square/matrix.
The constant values $ M $ of the sums of the magic squares have a minimum value (for nonzero integer positive values).
$$ M = n (n ^ 2 + 1) / 2 $$
For a size 3x3, the minimum constant is 15, for 4x4 it is 34, for 5x5 it is 65, 6x6 it is 111, then 175, 260, ...
Any lower sum will force the use of either negative numbers or fractions (not whole numbers) to solve the magic square.
Franklin's square is a panmagic square with a magic constant of 260.
Example:
52  61  4  13  20  29  36  45 
14  3  62  51  46  35  30  19 
53  60  5  12  21  28  37  44 
11  6  59  54  43  38  27  22 
55  58  7  10  23  26  39  42 
9  8  57  56  41  40  25  24 
50  63  2  15  18  31  34  47 
16  1  64  49  48  33  32  17 
This is a 3x3 magic square used in Feng Shui which is represented as well
4 Wealth  9 Fame  2 Relationship 
3 Family  5 Health  7 Children 
8 Wiseness  1 Career  6 Help/Friends 
Kaldor's magic square is a square used in economics, which has nothing to do with digits or numbers of mathematics but rather with concepts from economic policy.
Yes, there are magic cubes, their magic value is $$ M = n(n^3+1)/2 $$ (which may or may not have magic diagonals)
Example:



dCode retains ownership of the online 'Magic Square' tool source code. Except explicit open source licence (indicated CC / Creative Commons / free), any algorithm, applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any function (convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (PHP, Java, C#, Python, Javascript, Matlab, etc.) no data, script, copypaste, or API access will be for free, same for Magic Square download for offline use on PC, tablet, iPhone or Android !
Please, check our community Discord for help requests!