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Some Comments on Philatelic Latin Squares from Pakistan

Some Comments on Philatelic Latin Squares from Pakistan

460

460 ong>Someong> ong>Commentsong> on Philatelic Latin Squares from Pakistan We find that the rank of the matrices in Figure 6.4.3 is equal to 6 and that a basis for the (column) null space of the matrix in the left-panel of Figure 6.4.3 is the set of columns of the matrix ⎛ ⎞ 1 ⎜ −1 ⎜ 0 ⎜ ⎜−1 ⎜ 1 ⎜ 0 ⎜ 0 ⎜ 0 ⎜ 0 ⎜ 0 ⎜ ⎝ 0 1 0 −1 −1 0 1 0 0 0 0 0 1 −1 0 0 0 0 −1 1 0 0 0 1 0 −1 0 0 0 −1 0 1 0 0 1 −1 0 0 0 0 0 0 0 −1 1 1 ⎟ 0 ⎟ −1 ⎟ 0 ⎟ ⎛ 0 ⎟ 1 0 ⎟ ⎜ ⎟ ⎜ ⎟ = ⎜−1 0 ⎟ ⎝ ⎟ 0 0 ⎟ 0 0 ⎟ −1 ⎟ 0 ⎠ 1 0 −1 0 ⎞ 1 ⎟ 0 ⎟ 0 ⎠ −1 0 0 0 0 0 1 ⊗ ⎛ ⎜ 1 ⎝−1 0 ⎞ 1 ⎟ 0 ⎠. −1 (6.4.7) 6.5 Philatelic Sudoku puzzle S5: stamps from the USA featuring “Beautiful Blooms” FIGURE 6.5.1: USA 2007, Scott 4185v. Our fifth philatelic Sudoku puzzle S5 is based on a booklet of stamps (Figure 6.5.1) from the USA featuring “Beautiful Blooms”, coded (from left-to-right in the top row) as follows: (1) coneflower, (2) tulip, (3) water lily, (4) poppy, (5) chrysanthemum, (6) orange Gerbera 30 daisy, (7) iris, (8) dahlia, (9) magnolia, (10) red Gerbera daisy. To solve puzzle S5 we block the stamps in pairs as follows: a = (1,2), b = (3,4), c = (5,6), d = (7,8), e = (9,10). (6.5.1) 30 Gerbera is a genus of ornamental plants from the sunflower family (Asteraceae) named in honour of the German naturalist Traugott Gerber (1710–1743), a friend of Carl Linnaeus (1707–1778).

y, (7) iris, (8) dahlia, (9) magnolia, (10) red Gerbera daisy. o solve puzzle S5 we block the stamps in pairs as follows: a = (1,2), b = (3,4), c = (5,6), d = (7,8), e = (9,10). (6.22) 5 6 n the stamps in Figure 6.5.1 may be represented by the matrix 9 10 7 8 3 4 1 2 d c a e ⎛ ⎞ 6 5 10 9 8 7 4 3 2 1 Ka Lok Chu et al. 461 ⎝a b c d e 7 8 5 6 1 2 9 10 3 4 ⎠, (6.23) � � a b c d e Then the ē ¯ stamps d b in aFigure c 6.5.1 may 8 be7 represented 6 5 by2the matrix 1 10 9 4 , where 3 the e d b a 9 10 7 8 4 3 2 ē d¯ b a c 1 5 6 super-bar reverses the numbers in the argument, e.g., ē = (10,9). Our solution to S5 is re the super-bar reverses theeasily numbers foundin to be theasargument, shown in Figure e.g., ē 6.5.2: = (10,9). Our solution to 10 9 8 7 3 4 1 2 6 5 easily found to be represented by the matrix ⎛ ⎞ a ⎜ ē ⎜ ⎜ā ⎜ ⎜e ⎜ ⎜b ⎜ ⎜c ⎜ ⎜d ⎜ ⎜¯b ⎜ ⎝ ¯c b d¯ ¯b d a e c ā ē c b ¯c ¯b e d a ē d¯ d a d¯ ā c b e ¯c ¯b e ⎟ c ⎟ ē ⎟ ¯c ⎟ d ⎟ a ⎟, ⎟ b ⎟ d¯ ⎟ ā⎠ 1 10 2 9 3 5 7 4 6 2 9 1 10 4 6 8 3 5 3 8 4 7 1 9 5 2 10 4 7 3 8 2 10 6 1 9 5 3 6 4 9 7 1 10 8 6 7 4 1 5 8 3 2 10 5 (6.24) 8 3 2 9 9 6 7 4 8 2 7 1 6 4 10 5 3 9 6 10 5 7 1 3 8 2 10 5 9 6 8 2 4 7 1 d¯ ¯c ā ē ¯b 8 7 6 5 2 1 10 9 4 3 FIGURE erbera is a genus of ornamental plants from the sunflower 6.5.2: family Solution (Asteraceae) to philatelic named in Sudoku honour puzzle of the S5. an naturalist Traugott Gerber (1710–1743), a friend of Carl Linnaeus Beautiful (1707–1778). blooms 32 Beautiful blooms 1 2 3 4 5 6 7 8 9 10 2 1 4 3 6 5 8 7 10 9 a b c d 3 4 1 2 9 10 5 6 7 8 b a e c 4 3 2 1 10 9 6 5 8 7 c e d b 1 2 3 4 5 6 7 8 9 10 2 1 4 3 6 5 8 7 10 9 3 4 1 2 9 10 5 6 7 8 1 2 3 4 5 6 7 8 9 10 4 3 2 1 10 9 6 5 8 7 2 1 4 3 6 5 8 7 10 9 a b c d e 5 6 9 10 7 8 3 4 1 2 3 4 1 2 9 10 5 6 7 8 b a e c d 6 5 10 9 8 7 4 3 2 1 4 3 2 1 10 9 6 5 8 7 c e d b a 7 8 5 6 1 2 9 10 3 4 5 6 9 10 7 8 3 4 1 2 d c a e b 8 7 6 5 2 1 10 9 4 3 6 5 10 9 8 7 4 3 2 1 e d b a c 7 8 5 6 1 2 9 10 3 4 8 7 6 5 2 1 10 9 4 3 9 10 7 8 4 3 2 1 5 6 10 9 8 7 3 4 1 2 6 5 FIGURE 6.5.3: Reduced-form solution to philatelic Sudoku puzzle S5. 1 2 3 4 5 6 7 8 9 10 10 9 8 7 3 4 1 2 6 5 2 1 4 3 6 5 8 7 10 9 9 10 7 8 4 3 2 1 5 6 3 4 1 2 9 10 5 6 7 8 5 6 9 10 7 8 3 4 1 2 7 8 5 6 1 2 9 10 3 4 4 3 2 1 10 9 6 5 8 7 6 5 10 9 8 7 4 3 2 1 8 7 6 5 2 1 10 9 4 3

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