MATC44 2008 - s1 - University of Toronto at Scarborough Department of Computer and Mathematical Sciences MAT C44 Winter 2008 Solutions to Assignment#1

MATC44 2008 - s1 - University of Toronto at Scarborough...

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University of Toronto at Scarborough Department of Computer and Mathematical Sciences MAT C44, Winter 2008 Solutions to Assignment #1 Problem 38. on page 24: The following is a symmetric, idempotent Latin square of order 3: 1 3 2 3 2 1 2 1 3 Now we shall prove that there does not exist a symmetric, idempotent Latin square L of even order n . Assume there is such an L . In L there are n copies of each number 1 , 2 , ..., n . One copy of these numbers in positioned along the main diagonal, since L is idempotent. Since L is symmetric, if number a k is in the position ( i, j ) above diagona, there is also k in the position ( j, i ) below diagonal. Therefore, there is an even number of copies of each number 1 , 2 , ..., n not on the main diagonal Hence, L contains an odd number of copies of each number 1 , 2 , ..., n . However, n is even, giving a contradiction. Problem 16. on page 41: If there exists a person in the group with no acquaintances, then there cannot exist a person in the group who knows everyone else, and vice versa. Let pigion hole i contain people who have i acquaintances, i = 0 , ..., n - 1. Then holes 0 and n - 1 cannot both contain an element. Assuming that either one of them is empty, there are
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