Counting in two ways Reid Barton Summary: Counting in two ways Reid Barton June 28, 2005 1. (a) Find the number of triangles and diagonals in a triangulation of a regular n-gon. (b) Same question, but for "quadrangulations"; which regular n-gons can be quadrangulated? 2. In each cell of a 5 × 5 square grid is written either +1 or -1. The product of the values in each row and each column is computed. Is it possible that the sum of these ten values is zero? Same problem for a 4 × 4 square grid. 3. Is it possible to place 10 numbers from {1, 2, . . ., 15} into the following circles such that the absolute values of the differences of pairs of adjacent circles are 1, 2, . . . , 14 in some order? 4. (Iran '96) The top and bottom edges of a chessboard are identified together, as are the left and right edges, yielding a torus. Find the maximum number of knights which can be placed so that no two attack each other. 5. There are n pieces of candy in a pile. One is allowed to separate a pile into two piles, and add the product of the sizes of the resulting piles to a running total. The process terminates when each piece of candy is in its own pile. Show that the final sum is independent of the sequence of operations performed. 6. Twenty-five people form several committees. Each committee has five members, and any two com- mittees have at most one common member. Determine, with justification, the maximum number of committees. Collections: Mathematics