Knut Vik Designs are “Multimagic”

Peter Loly (speaker) with Adam Rogers and Ian Cameron

Department of Physics and Astronomy, University of Manitoba, Winnipeg

CMS Summer 20147 June

2

Pfeffermann 1891the first bimagic square

• (Mono)magic linesum 260• Bimagic linesum 11180• sv={260,• 2 [1153+ 805121],• 10 82 ,64,• 2 [ 1153 - 805121],• 8 3 ,8 2 }• Rank 7

• Main diagonal and two parallel pandiagonals (PDs) shown at right in colour:

• The green PD has non-magic linesum of 286, so this bimagic square is not pandiagonal magic.

56 34 8 57 18 47 9 31

33 20 54 48 7 29 59 10

26 43 13 23 64 38 4 49

19 5 35 30 53 12 46 60

15 25 63 2 41 24 50 40

6 55 17 11 36 58 32 45

61 16 42 52 27 1 39 22

44 62 28 37 14 51 21 3

Background

• Boyer has popularized multimagic squares where the rows, columns and main diagonals (RCD) of full cover magic squares maintain RCD lineums for integer powers p of each element.

• C. Boyer, http://www.multimagie.com/English/• Bimagic squares correspond to p=2 begin at orders 8

and 9, trimagic for p=3 begin at order 12, etc.• Recently Walter Trump and Francis Gaspalou have

used a form of backtracking and bimagic series to count exactly the order 8 bimagics (2014).

This Talk• Rogers observed before 2007 that Latin squares (rows and columns of

m symbols) are similarly "multi-semi-magic" for rows and columns. • Some Latin squares are also diagonal. • Christian Eggermont called diagonal Latin squares "infinitely-

multimagic squares" in a 2004 talk: www.win.tue.nl/(tilde)ceggermo/math/Multimagic(ul)Cube(ul)Day(ul)v3.ppt;

• Now after Nordgren (2013) drew our attention to Knut Vik Latin designs (orders 5,7,11,...) which have all pandiagonals parallel to the main diagonals containing one of each of the n symbols, we realized that Knut Viks are "multimagic" to all integer powers.

• R. Nordgren, Pandiagonal and Knut Vik Sudoku Squares, Mathematics Today, 49 (2013) 86-87 and Appendix.

Continued• Moreover these can be compounded to construct Knut Vik

designs of multiplicative orders 5×5,5×7,... [Rogers and Loly 2013].• Some properties which can be compounded and iterated are

discussed. See also:• A. Hedayat and W. T. Federer, Ann. Statist., 3(2)(1975), 445-447,

On the Nonexistence of Knut Vik Designs for all Even Orders; • We use entropic measures based on singular value analysis to

identify related squares (“clans”). • Ian Cameron, Adam Rogers and Peter D. Loly, “Signatura of magic

and Latin integer squares: isentropic clans and indexing”, Discussiones Mathematicae Probability and Statistics, 33(1-2) (2013) 121-149, or http://www.discuss.wmie.uz.zgora.pl/ps;

Shannon Entropy and Magical Squares

• Newton, P.K. & DeSalvo, S.A. (2010) [“NDS”]“The Shannon entropy of Sudoku matrices”,

Proc. R. Soc. A 466:1957-1975 [Online Feb. 2010.]

• Immediately clear to us that we could extend NDS using our studies of the singular values of magic squares at IWMS16 in 2007, published in Lin. Alg. Appl. 430 (10) 2659-2680, 2009.

Shannon Entropy

• NDS formula: h= -∑ni=1 σi ln(σi); < σi are normalized σ>

• hmax = ln(n)• NDS compression measure: • C=1-h/ln(n)*100%• [100% - uniform; 0% - random]• Effective rank: erank=exp[h]• N.B. h,C and erank are related.• An index, R, of the sum of the 4th powers of the SVs

(omitting the first linesum SV) is also a useful metric.

Diagonality of Latin squares

1 2 3 4

2 4 1 33 1 4 24 3 2 1

• Linesum 10• Weakly diagonal• Diagonals have 2 symbols• Lat4d• Rank 4

1 2 3 4 3 4 1 2 4 3 2 1 2 1 4 3

• Strongly diagonal• Diagonals have 4 symbols • Sud4a • Rank 3

8

kv5

9

1 2 3 4 5

4 5 1 2 3

2 3 4 5 1

5 1 2 3 4

3 4 5 1 2

Rank 5, R=750SVs: 15 (linesum), and [(5/2)(5± 5)] (twice)

kv5

• Rank 5 – non-singular (full rank)• SVs: 15 (linesum), and • [(5/2)(5± 5)] (twice)• One of 7 Singular Value Clans at 5th order• R-index 750 • R is the sum of the fourth powers of the last

(n-1) SVs.• Compression C=16.61%

“infinitely”-MULTImagic, e.g. kv5 elements squared

(more spread out – higher entropy,Compression C=7.39%, R=864,190)

11

1 4 9 16 25

16 25 1 4 9

4 9 16 25 1

25 1 4 9 16

9 16 25 1 4

But also “Square Rooted”less spread out than kv5

so lower entropy than kv5. Compression C=31.82%, R= 6.57781

12

1 2 3 4 5

4 5 1 2 3

2 3 4 5 1

5 1 2 3 4

3 4 5 1 2

Compounding Knut Viks• Rogers (2004-), Rogers and Loly (2005-2013), used a weighted

sum of Kronecker products to compound Latin squares of orders m and n to their multiplicative order mn.

• All are highly singular: • Orders:• 5*5=25, C=47.24%, rank 5+5-1=9;• 5*7=35, C=47.32% (aggregated) AND 49.98% (distributed), • Both rank 5+7-1=11;• 7*7=49, C=50.41%, rank 7+7-1=13.• Our order 49 is in the same SV clan as Nordgren’s order 49 Super

Sudoku, suggesting that they are related by row and column permutations.

Linesums for Knut Vik’s“Multimagic” Latins <MML> using 0..(n-1): s(p)=Sigma[i^p]{p=0 to (n-1)}, e.g. s(1)=0+1+2+3+4=10 from Mathematica: 1/2 (-1+n) n [Check n=5: (1/2)(-1+5)5=10]Mathematica : Sum[i^2,{i,0,n-1}]: s(2)= 1/6 (-1+n) n (-1+2 n)

Alternatively:S(p)=Sigma[i^p]{p=1 to n} using 1..nS(1)=(n/2)(n+1), e.g. n=5: 1+2+3+4+5=15S(2)= (n/6)(n+1)(2n+1), e.g. n=5: S(2)=1+4+9+16+25=55

End