Views
5 years ago

Some Comments on Philatelic Latin Squares from Pakistan

Some Comments on Philatelic Latin Squares from Pakistan

458

458 ong>Someong> ong>Commentsong> on Philatelic Latin Squares from Pakistan FIGURE 6.4.1: Abkhazia 2006, BiStamp 29 252. 29 Stamps from Abkhazia are catalogued by BiStamp [14] but not by �Scott [47].

Ka Lok Chu et al. 459 1 2 3 4 5 6 7 8 9 10 11 12 3 1 2 1 2 3 4 5 6 7 8 9 10 11 12 3 1 2 6 4 5 9 7 8 12 10 11 Abkhazia sea animals: 8x3 on 12 stamps 1 5 9 2 6 10 3 7 11 4 8 12 Abkhazia sea animals: 8x3 on 12 stamps 9 1 5 1 5 9 1 9 10 2 6 2 10 6 2 10 311 7 311 7 3 11 4 8 12 12 4 8 4 12 9 1 5 5 1 6 4 5 FIGURE 6.4.2: Row-wise (left panel) and column-wise (right panel) coding and blocking 9 7 8 of philatelic Sudoku puzzle S4 based on Abkhazia “sea animals” stamps (Figure 6.4.2). 12 10 11 11 3 7 7 3 1 2 3 4 15 26 37 48 59 10 6 11 7 12 8 9 10 11 12 78 12 4 8 8 4 2 3 1 5 26 34 48 19 67 11 7 12 8 10 5 10 11 12 9 78 1 2 3 14 25 36 7 8 9 10 11 12 3 1 2 6 1 34 2 45 3 419 5 27 6 7 8 8 12 8 9 10 5 11 11 6 12 78 11 12 9 10 78 2 3 1 45 56 64 8 9 7 11 12 10 2 3 4 1 6 7 8 5 10 11 12 9 78 4 5 6 7 48 19 10 2 11 3 12 8 51 62 73 12 9 10 11 78 3 1 2 76 84 95 9 7 8 12 10 11 3 4 1 2 7 8 5 6 11 12 9 10 78 4 5 6 10 7 11 8 12 9 10 11 512 16 2 4 3 8 4 59 1 67 2 311 7 8 12 85 10 69 7 10 2 12 11 39 1012 1 11 178 2 3 4 78 5 6 4 38 19 27 11 12 10 6 2 4 3 1 5 9 5 6 67 7 8 8 9 12 8 10 10 5 11 10 11 12 1 11 3 2 12 1 3 92 4 78 2 3 4 1 78 6 4 5 69 47 58 12 10 11 3 1 2 6 7 8 5 10 11 12 9 2 3 4 1 78 7 8 9 10 9 11 7 12 8 1 2 73 48 5 9 6 10 7 11 7 8 12 85 651 11 62 12 11 93 1012 4 3 95 4 110 6 2 378 4 1 2 78 8 9 7 12 11 10 12 11 10 2 3 81 59 6 7 4 8 5 6 7 12 9 10 11 4 1 2 3 78 11 12 8 10 5 62 73 12 1 95 10 6 11 4 4 1 2 3 78 9 7 8 12 10 11 3 1 2 6 4 5 9 10 11 12 1 2 3 4 5 6 7 8 78 9 7 8 12 10 9 10 11 11 3 12 1 12 26 34 45 5 6 7 8 78 10 11 12 1 2 3 4 5 6 7 8 9 10 10 11 11 12 12 9 2 3 4 1 6 7 8 5 78 11 12 10 2 3 1 5 6 104 4 11 8 9 127 7 11 1 11 12 10 2 12 10 11 39 10 12 4 3 95 4 12 6 2 37 7 4 8 5 19 6 78 6 7 8 5 78 12 10 11 3 1 2 6 4 5 11 9 12 7 8 10 12 2 12 10 911 10 11 3 12 1 11 95 4 1 10 6 2 34 3 8 48 5 19 6 27 7 78 7 8 5 6 78 78 78 78 78 78 78 78 78 78 78 78 4 5 6 7 8 9 10 11 12 1 2 3 FIGURE 6.4.3: Reduced-form block-Latin solutions to philatelic Sudoku puzzle S4 12 10 11 3 12 1 92 10 6 11 4 45 19 27 38 5 6 4 8 9 7 11 12 10 2 3 1 78 8 5 6 7 78 blocked row-wise B (4,3) (left panel) and column-wise B (3,4) (right panel). 4 6 4 5 9 7 8 12 10 11 3 178 2 78 78 78 78 78 78 78 78 78 78 78 7 8 9 10 11 12 1 2 3 4 5 6 It is interesting 8 9 to 7 compare 11 12 10 the 2generalized 3 1 5 Kronecker-sum 6 4 solutions (6.4.1) and (6.4.2) of size 12×12 9 to7 this 8 philatelic 12 10 11Sudoku 3 1 puzzle 2 6 S4 4(r = 5 8,c = 3,s = 12) with the generalized 10 11 12 1 2 3 4 5 6 7 8 9 Kronecker-sum solutions (6.3.3) and (6.3.5) of size 6 × 6 to the philatelic Sudoku puzzle S3 (r = 6,c = 3,s = 6), 11 12 10 2 3 1 5 6 4 8 9 7 12 10 11 3 1 2 6 4 5 9 7 8 10 2 6 6 2 B (2,3) 3 = E2×2 ⊗ K3×3 + 3K2×2 ⊗ E3×3 − 3E2×2 ⊗ E3×3, (6.4.4) B (3,2) 3 = E3×3 ⊗ K2×2 + 2K3×3 ⊗ E2×2 − 2E3×3 ⊗ E2×2, (6.4.5) where the 2 × 2 one-step (backwards) circulant ⎛ ⎞ 1 K2×2 = ⎝ 2 ⎠. (6.4.6) 2 1 4

Comments on 5×5 philatelic Latin squares - Chance
some comments on philatelic latin squares from pakistan1
April 09 - Pakistan Philatelic Club
Magic, Latin, and Sudoku Squares - Math Teachers' Circles
some properties of latin squares - Department of Computer Science
Biembeddings of Latin squares obtained from a voltage construction
A class of latin squares derived from nite abelian groups
Some comments about the Finite Square Well
Anything but square: from magic squares to Sudoku - Plus Magazine
some uredinales from northern areas of pakistan - Pakistan Journal ...
to See Some Comments for This Page from Previous ... - Marmaris
Some Orb Weaver Spiders From Punjab, Pakistan - Zsp.com.pk
Some Examples from Latin America. - Consultative Group on Early ...
effects of some common weeds from pakistan on plant-parasitic
Land Reform Bill- Some general Comments from CUTS The law ...
Page 1 of 11 Comments from some of the participants in the High'n ...