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### The structure and geometry of 4×4 pandiagonal matrices

Barbara Turner, Ken Warner
1981 Discrete Mathematics
The sum of these columns would be 2 + l l l + 16 = 72, ; ' the equivalence class [ 1, 1, 0, O]invariant gives the fine permutation structure of these pandiagonal matrices.  ...  Furthermore, the elements of the 4~ 4 pandiagonal matrices in this equivalence class are obtained by the action of the group gen:erated by A and B on the single 4-tuple, (j, k, r, s) = (8, invariance property  ...

### Page 779 of Mathematical Reviews Vol. , Issue 97B [page]

1997 Mathematical Reviews
It is useful for the solution of some problems of distribution (modulo some prime number p) of combinatorial coefficients and numbers in some arithmetic structures. V. A.  ...  Normalized p-Latin squares and algorithms for their application to problems of the elements modulo p in arithmetic structures. (Russian. Russian summary) Voprosy Vychisl.i Prikl. Mat.  ...

### Page 2447 of Mathematical Reviews Vol. , Issue 2000d [page]

2000 Mathematical Reviews
The following are necessary conditions for the existence of a simple DTS(v, 4): Av(v — 1) =0 mod 3, 4 < 3(v —2). Let (X,A) be a DTS(v, 2) and (¥ B) be a DTS(u, 4).  ...  (English summary) Geometry, combinatorial designs and related structures (Spetses, 1996), 149-159, London Math. Soc. Lecture Note Ser., 245, Cambridge Univ. Press, Cambridge, 1997.  ...

### Magic Squares and Sudoku

John Lorch
2012 The American mathematical monthly
We introduce a family of magic squares, called linear magic squares, and show that any parallel linear sudoku solution of sufficiently large order can be relabeled so that all of its subsquares are linear  ...  I thank the referees and the Monthly editor for leaving this article far better than they found it.  ...  We show that if q > 3 then we lose nothing by replacing the word "sudoku" with the words "magic sudoku" in the previous sentence. The strategy and structure of the paper are as follows.  ...

### Page 2395 of Mathematical Reviews Vol. , Issue 82f [page]

1982 Mathematical Reviews
Praeger (Perth) Turner, Barbara; Warner, Ken 82f:20012 The structure and geometry of 4 <4 pandiagonal matrices. Discrete Math. 34 (1981), no. 3, 301-310.  ...  In this paper the authors find some generators and relations for G and investigate the structure of some of its subgroups. C. A. Rowley (Milton Keynes) VySenskii, V. A. [ViSins’kii, V. A.}; Klin, M.  ...

### Magic Mathematics Based on New Matrix Transformations (2D and 3D) for Interdisciplinary Physics, Mathematics, Engineering and Energy Management [chapter]

Wolfram Stanek, Maralo Sinag
2010 Products and Services; from R&D to Final Solutions
, ...) by Laplace Rule directly from Leibniz, too % Both rules of Sarrus and Laplace are visualised structures of the % Leibniz Formula detA_Leibniz (above shown for 4x4-Matrices) % This Leibniz Formula  ...  this point, W.Stanek provides an algorithm to obtain any 3D magic structure, especially 64 cells cube, primarily the visualised structure of the algorithm.  ...  step 2. 11 541 4 12 6 9 7 4 13 3 16 2 81 051 1 NOTES: • The matrices M1 until M4 are magic-matrices with the cells content of number 1 to 16  ...

### Superstochastic matrices and magic Markov chains

Karl Gustafson, George P.H. Styan
2009 Linear Algebra and its Applications
A brief account of the conceptual formulation of the two entities in this paper's title, plus an initial preliminary investigation of some of their mathematical properties, is given.  ...  Acknowledgments We mentioned in our proof of Theorem 2.1 that we need only doubly stochastic matrices there.  ...  In that connection, one of the referees pointed out three important books [19, 20, 21] where we may find  ...

### Polyhedral Cones of Magic Cubes and Squares [chapter]

Maya Ahmed, Jesús De Loera, Raymond Hemmecke
2003 Algorithms and Combinatorics
Using computational algebraic geometry techniques and Hilbert bases of polyhedral cones we derive explicit formulas and generating functions for the number of magic squares and magic cubes.  ...  The work of Ehrhart and Stanley [14, 15, 25, 26] when applied to the study of semi-magic squares showed that many enumerative and structural properties of magic arrays can actually be formulated in terms  ...  From now on, when referring to any of these structures, we will use the terminology magic arrays.  ...

### Polyhedral Cones of Magic Cubes and Squares [article]

M. Ahmed, J. De Loera, R. Hemmecke
2002 arXiv   pre-print
Using computational algebraic geometry techniques and Hilbert bases of polyhedral cones we derive explicit formulas and generating functions for the number of magic squares and magic cubes.  ...  The work of Ehrhart and Stanley [14, 15, 25, 26] when applied to the study of semi-magic squares showed that many enumerative and structural properties of magic arrays can actually be formulated in terms  ...  For example, the stochastic semi-magic squares are the well-known bistochastic matrices (n × n matrices whose row and column sums are one) and P is the famous Birkhoff-von Neumann polytope [9, 26] .  ...

### Page 749 of Mathematical Reviews Vol. 17, Issue Index [page]

Mathematical Reviews
(with Proskurowski, Andrzej) Construction of pandiagonal magic squares from circulant pandiagonal matrices. Ars Combin. 15 (1983), 153-162. (J.  ...  Construction of magic squares from circulant pandiagonal matrices. Ars Combin. 15 (1983), 153-162. (J. Dénes) 85b:05041 05B15 (with Systo, Maciej M.) Edge-coloring of trees and unicyclic graphs.  ...

### Page 141 of Mathematical Reviews Vol. , Issue Index [page]

Mathematical Reviews
(English summary) 98¢:05031 Pott, Alexander * Finite geometry and character theory. 98j:05032 Qiu, Weisheng On the multiplier conjecture.  ...  The use of combinatorial structures in communication signal design 98m:94005 Gysin, Mare New D-optimal designs via cyclotomy and generalised cyclotomy. (English summary) 98a:05039 Hamilton, G. M.  ...

### Page 195 of Mathematical Reviews Vol. 14, Issue Index [page]

Mathematical Reviews
Fundamental regions in S, for the simple quaternary Gj... 82b:20007 Turner, Barbara (with Warner, Ken) The structure and geometry of 44 pandiagonal matrices. 82f:20012 Visins’kii, V. A.  ...  (Russian) (82g:05049) Wefelscheid, Heinrich Verallgemeinerte Minkowski-Geometrie. (82b:51014) Chelwood Gate * Finite geometries and designs. (82g:05003) Conference: Isle of Thorns, Finite Geometries and  ...

### Page 535 of Mathematical Reviews Vol. , Issue Index [page]

Mathematical Reviews
Foregger) 87e:15012 15A15 Minimum permanent on faces of staircase type of the polytope of doubly stochastic matrices. Linear and Multilinear Algebra 18 (1985), no. 4, 271-306. (Thomas H.  ...  Banchoff) 87¢:53011 53A10 (51M20, 52A25) (with Andersson, Sten) Differential geometry of crystal structure descriptions, relationships and phase transformation. Z.  ...

### Subject Index Volumes 1–200

2001 Discrete Mathematics
Erdös and Sós, 4925 Erdös and Szekeres, 4879 Erdös et al., 5529, 5888 Erdös Rubin Taylor, 3815 Erdös Sós Turán, 506 Everett, 2901 Faigle and Sands, 2122 Faudree et al., 4399 Faudree Gould Jacobson  ...  Slater, 5299 Hertz and de Werra, 3748 Hibi, 4839 Hoàng, 3748, 4811 Holton, 1660 Holyer and Cockayne, 1118 Howorka, 2636  ...  5284, see also three-way regular 3-way regular tripartite graph, 5284 of degree four, 5284 3-wise, see also three-wise balanced design, 3912 4 × 4 pandiagonal matrix, 1160 4 array, 2059 4 colour conjecture  ...

### Page 456 of Mathematical Reviews Vol. , Issue Index [page]

Mathematical Reviews
On pandiagonal symmetrical magic squares of order 6. J. Recreational Math. 16 (1983/84), no. 4, 263-267. (Robert Girse) 86a:05024 05B15 Johnson, Bruce McK.  ...  Linear and Muitilinear Algebra 16 (1984), no. 1-4, 29-38. (Robert Beauwens) 86b:15022. 15A48 (15A15) (with Rodman, L.) Inertia possibilities for completions of partial Hermitian matrices.  ...
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