Discrete Mathematics: Elementary and Beyond

14 Finite Geometries, Codes, Latin Squares, and Other Pretty Creatures 14.1 Small Exotic Worlds The Fano plane is a really small ...

212 14. Finite Geometries, Codes,Latin Squares,and Other Pretty Creatures If you check this out and admit it’s true, but reply t ...

14.1 Small Exotic Worlds 213 (e)all our points have the same number of lines through them. “No doubt you can prove this yourself ...

214 14. Finite Geometries, Codes,Latin Squares,and Other Pretty Creatures FIGURE 14.4. The Tictactoe plane. they showed that thi ...

14.1 Small Exotic Worlds 215 14.5). Then we would have properties (a) through (g) ourselves.^2 But we prefer to distinguish betw ...

216 14. Finite Geometries, Codes,Latin Squares,and Other Pretty Creatures (one can be moved onto any other by appropriately rota ...

14.2 Finite Affine and Projective Planes 217 the circle, if it contains exactly one point of the circle. Show that at every poin ...

218 14. Finite Geometries, Codes,Latin Squares,and Other Pretty Creatures if a finite projective plane has ordern, thenn+ 1 line ...

14.2 Finite Affine and Projective Planes 219 of the affine plane, rather than on a line of the projective plane, to be called th ...

220 14. Finite Geometries, Codes,Latin Squares,and Other Pretty Creatures (0, 0) (0, 4) (4, 4) (4, 0) (0, 2) (1, 3) (2, 4) (3, 0 ...

14.3 Block Designs 221 Therefore, they don’t allow larger and smaller clubs (because they are afraid that larger clubs might sup ...

222 14. Finite Geometries, Codes,Latin Squares,and Other Pretty Creatures (k=v,b=r=λ= 1). This is so uninteresting that we exclu ...

14.3 Block Designs 223 Here is our proof: If this were possible, then for the number of clubsb we would get from (14.1) that b= ...

224 14. Finite Geometries, Codes,Latin Squares,and Other Pretty Creatures Clubs Citizens member of... FIGURE 14.9. Representing ...

14.4 Steiner Systems 225 How many inhabitants must a town have to allow a system of clubs that is a Steiner system? In other wor ...

226 14. Finite Geometries, Codes,Latin Squares,and Other Pretty Creatures This problem sounds similar to the badge problem discu ...

14.4 Steiner Systems 227 Carl). This implies that the committee has at least (v−3)/2+2 = (v+1)/ 2 members, a contradiction. So a ...

228 14. Finite Geometries, Codes,Latin Squares,and Other Pretty Creatures points form a representative committee, and as we have ...

14.5 Latin Squares 229 walk together form a block design again, but now this is what we called above “trivial” (all triples out ...

230 14. Finite Geometries, Codes,Latin Squares,and Other Pretty Creatures second: 0, 0 1, 1 2, 2 3, 3 1, 3 0, 2 3, 1 2, 0 2, 1 3 ...