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3. Find out if the following matrix can be arranged in "dictionary order" according to Benzer.
0 1 0 1 0 1 0 0 1 1 0 1 0 1 1 1 1 1 0 1 0 0 1 1 1 1 1 1 0 0 0 1 1 1 0 1 0 1 1 1 0 1 0 0 1 1 1 1 1 1 0 1 0 0 0 1 1 1 0 1 0 1 0 0 1 1 0 0 0 1 0 1 1 1 1 1 1 0 0 1 0
Answer:
If we select mutants 1, 3, 4, and 8 only, each of these mutants intersects with two of other mutants but not with the third one. This is exactly the case shown in Figure 2 of Benzer's 1959 paper, where it is shown that no solution is possible. Indeed, without loss of generality, we can choose any one of them as the first row. Let's pick mutant 1 as the first one. After this choice, three possible arrangements for this 4x4 recombination submatrix are:
Neither is a valid dictionary order. The other three possible arrangements are obtained by fixing the first row and reverting the order of the other rows, yield matrices identical to these three. It follows that this 4x4 matrix cannot be arranged in dictionary order. Adding more mutants does not change the relationship between these four mutants, and the extended matrix remains unable to be arranged. So, the matrix in Exercise 3 cannot be arranged in dictionary order.
© 2015 Joao Meidanis