<p>Magic</p>

6f20e1b3-eef8-4afd-a34c-331b0bc11ada 83 4 5 false false 0 decimal

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Magic Squares

<p>Magic</p>

magic(n) with n≥3 returns an n·n matrix constructed from the 
integers 1 through n^2 with equal row and column sums. n magic I n n matrix : M I : V 1 n range : R V : C V : I R C el V R el : J I transpose : 17 mat n 4 mod 0 ≡ Q 1 n 2^range : P Q Q stack : M C R el P R C 1 - (n * + el : 13 mat k Q I k el 4 mod 2 / trunc J k el 4 mod 2 / trunc ≡ M k el n 2^1 + M k el - : 0 if for 21 line n 2 mod 0 ≡ p n 2 / : κ n 2 - 4 / : M p magic : 13 mat M M M 2 p 2^* + augment M 3 p 2^* + M p 2^+ augment stack : M I J IJ A M : I.1 I I p + stack : I.2 I p + I stack : 12 mat r 1 I.1 length range c 1 J length range A I.1 r el J c el el M I.2 r el J c el el : for for A eval 41 line : n 6 ≡ M M 1 p range 111 mat IJ : M M 1 p range 1 κ range n κ - 2 + (n range stack IJ : if M M κ 1 + 1 κ 1 + augment IJ : 51 line M n I J + n 3 - 2 / + n mod * I 2 J * + 2 - n mod + 1 + vectorize : if if M eval 31 line :

Reference: https://www.mathworks.com/content/dam/mathworks/mathworks-dot-com/moler/exm/chapters/magic.pdf

More elegant could be:

M 1 n 2^n range transpose 0 n 1 - (range + :

Using find function:

M find

F I 4 mod 2 / trunc J 4 mod 2 / trunc ≡ vectorize find : M F el n 2^1 + M F el - : 21 line

This vectorization don't works:

A I.1 J el M I.2 J el :

Choose the bigest magic squares matrix to do, and generate random numbers:

N 50 :

η n random 1 + 11 line :

n N 2 - random 3 + : 27

Program subdivides into three cases:

M n magic :

Doubly even n 4 mod ¬ Singly even n 2 mod ¬ Odd cases Odd

Magic constant:

M tr 9855

M transpose tr 9855

n 2 / n 2^1 + (* 9855

1 n / k k 1 n 2^sum * 9855

For any row or column:

c η : 26

r η : 6

M c col sum 9855

M η row sum 9855

The matrix is constructed from the integers 1 through n^2:

M sum 266085

1 n 2^range sum 266085

Note: this isn't the prove, just shows the property.

The magic squares matrix:

M 38040943846749652555458361264167069972813059408437466495524553582611640669698727272958874364654945235525816106396686977262628578611546449352255158060963866769672525545685114143492521550579608637666695724245355841131421715205495786076366656947232352818311214117019954857760663566469372222518082111140169198227576605634663692721215079108110139168197226255604633662691720204978107109138167196225254283632661690719194877106135137166195224253282311660689718184776105134136165194223252281310339688717174675104133162164193222251280309338367716164574103132161163192221250279308337366395154473102131160189191220249278307336365394423437210113015918819021924827730633536439342245171100129158187216218247276305334363392421450479991281571862152172462753043333623914204494785071271561852142432452743033323613904194484775065351551842132422442733023313603894184474765055345631832122412702723013303593884174464755045335625912016 mat

Alvaro