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 2014
7 June
Pfeffermann 1891
the first bimagic square
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(Mono)magic linesum 260
Bimagic linesum 11180
sv={260,
2 [1153+ 805121],
10 82 ,64,
2 [ 1153 - 805121],
8 3 ,8 2 }
Rank 7
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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.
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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
2
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 "infinitelymultimagic 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
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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
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4
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4
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Linesum 10
Weakly diagonal
Diagonals have 2 symbols
Lat4d
Rank 4
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Strongly diagonal
Diagonals have 4 symbols
Sud4a
Rank 3
8
kv5
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5
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5
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5
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5
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5
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Rank 5, R=750
SVs: 15 (linesum), and
[(5/2)(5± 5)] (twice)
9
kv5
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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)
1
4
9
16
25
16
25
1
4
9
4
9
16
25
1
25
1
4
9
16
9
16
25
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11
But also “Square Rooted”
less spread out than kv5
so lower entropy than kv5.
Compression C=31.82%, R= 6.57781
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5
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5
3
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12
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 (n1)},
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..n
S(1)=(n/2)(n+1), e.g. n=5: 1+2+3+4+5=15
S(2)= (n/6)(n+1)(2n+1),
e.g. n=5: S(2)=1+4+9+16+25=55
End