Discrete Math

(a) Consider a 3*3 matrix in which each entry is either 0; 1 or -1. Add up the sum of the three entries in each row, column and diagonal and form a list of the sums.

i. Give an explicit example of such a matrix and the corresponding list.

ii. Prove that the list must contain at least one repeated entry.

(b) Choose 16 distinct integers between 1 and 30. Prove that at least two of these differ by 3. Is this also true if we choose 15 distinct integers? Justify your answer.