By Victor Bryant
Development from fundamentals and demonstrating the relationships one of the quite a few branches of combinatorics, Victor Bryant provides the consequences in an easy method. a number of examples and routines together with tricks and recommendations are integrated all through and serve to guide the reader to a few of the deeper result of the topic, lots of that are often excluded from introductory texts.
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Sample text
We can then start all over again and keep repeating the process until an n x n Latin square is D reached. The theorem showed that we can always extend p x n Latin rectangles to n x n Latin squares but we now turn attention to p x q rectangles: as we saw earlier, extension is not always possible. Example Can the Latin rectangle 6 1 2 3\ 5 6 3 1 1 3 6 2 3 2 4 6 be extended to a 6 x 6 Latin square? Solution No. One way to see this is to note that in any extension to a 6 x 6 Latin square as shown we would need three 5s in the box, but they will not fit because there are only two columns for them.
If n is odd the covering would again require an odd half of a domino. In the case of n even imagine the board chequered, like the one illustrated. Then the two removed corners are the same colour, so the remaining board has a different number of white squares from black. Since each domino covers precisely one white and one black square the covering is again impossible. Note that in these arguments we have used both an odd/even and a black/white parity check. Next consider covering a 6 x 6 board with 18 dominoes.
I} invites {... 'j'} B' . How do we know that this process continues until some new unengaged boy is invited? If no new unengaged boys have been invited then at a typical stage in the process, when we have a set of newly invited girls {k + 1,.... ,1}, whom will they invite next? So far the girls m + I and 1, 2, . , I and the boys 1', 2'.... 1' have been invited (and that includes all the boys known to girls 1, 2, ... and k), but overall that's fewer boys than girls. So, by the given condition of girls knowing at least as many boys, there still remains to be invited a boy known to one of the girls there (and hence to one of k + 1, ..