Lat’s That

In a magic square of numbers, all rows, columns and diagonals have the same sum, or magic total. Here is an example:

1*5*9
8*3*4
6*7*2

(mt=15)

Here’s another:

06*07*11*10
15*02*14*03
04*13*01*16
09*12*08*05

(mt=34)

And another:

04*25*20*10*06
01*13*11*21*19
23*09*07*08*18
15*16*03*14*17
22*02*24*12*05

(mt=65)

And another:

35*15*10*18*11*22
05*25*33*12*07*29
34*30*04*14*21*08
02*16*27*17*23*26
03*24*09*19*36*20
32*01*28*31*13*06

(mt=111)

In all those magic squares, the magic total is fixed: the sum of all numbers from 1 to 36 is 666, so any individual line in a 6×6 magic square has to equal 666 / 6 or 111. In other kinds of magic figure, this rule doesn’t apply:

2*7*3
4***8
6*5*1

(mt=12)

6*3*4
2***8
5*7*1

(mt=13)

8*5*1
2***6
4*3*7

(mt=14)

8*1*6
4***2
3*5*7

(mt=15)


I call figures like that magic lattices. But the lattice is clearer in larger squares:

14*09*10*16*01
21****15****07
06*12*02*19*11
05****20****13
04*08*03*17*18

(mt=50)

08*06*13*16*15
18****01****11
10*12*20*07*09
05****03****19
17*02*21*14*04

(mt=58)

You can also use prime numbers, or numbers that are divisible only by themselves and 1. The prime 3×3 magic lattice can only have one magic total:

11*17*03
13****23
07*19*05

(mt=31)

But bigger squares can vary:

73*07*05*61*37
23****11****47
43*53*17*41*29
13****79****67
31*19*71*59*03

(mt=183)

37*29*47*17*59
67*61****05*03
19****41****73
53*71****07*11
13*23*31*79*43

(mt=189)

127*097*013*163*047*019*101
003*****109*****007*****037
139*149*067*041*071*011*089
083*****031*****179*****029
053*103*023*059*079*113*137
157*****151*****167*****043
005*061*173*073*017*107*131

(7x7, mt=567)

013*163*003*089*179*023*067
131*****107*****109*****173
053*103*037*061*047*157*079
059*****071*****007*****043
101*137*127*011*017*113*031
097*****151*****029*****139
083*073*041*019*149*167*005

(mt=537)

097*107*061*031*079*059*011
083*019*************101*053
041*****037*****007*****127
131*********137*********113
017*****013*****089*****071
003*103*************043*047
073*109*029*139*067*005*023

(mt=445)

31*16*09*56*25*54*19*48*23
65****38****36****42****27
34*62*63*11*53*15*08*02*33
21****50****04****35****61
01*59*24*07*26*52*22*32*58
57****17****44****05****43
06*47*49*39*13*18*46*51*12
29****28****60****64****14
37*55*03*45*20*30*40*41*10

(9x9, all integers, mt=281)

21*50*36*18*37*14*55*28*33
17****64****23****20****06
32*47*11*34*08*35*46*30*49
38****05****02****58****45
22*63*26*61*31*01*12*52*24
51****65****57****09****40
42*19*04*25*60*41*10*48*43
07****27****59****53****39
62*03*54*16*15*56*29*44*13

(all integers, mt=292)

127*173*023*137*073*197*007*103*061
131*************181*************101
047*************109*************019
031*************029*************191
199*043*017*041*179*193*059*013*157
167*************005*************037
003*************011*************107
113*************151*************089
083*149*079*053*163*067*071*097*139

(9x9, primes, mt=901)

043*139*109*137*013*053*193*023*029
181*157*********************163*079
101*****047*************149*****131
197*********019*****083*********071
007*************037*************127
113*********017*****191*********061
005*****151*************167*****031
089*107*********************067*199
003*173*059*041*103*073*179*097*011

(primes, mt=739)

089*193*127*007*079*003*181*101*109
173*149*********************107*139
059*****019*************103*****037
197*********017*****083*********131
043*************113*************167
071*********047*****157*********061
041*****013*************029*****011
053*151*********************179*097
163*199*005*191*031*067*073*023*137

(primes, mt=889)

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