 
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 ngon.
(b) Same question, but for "quadrangulations"; which regular ngons 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. Twentyfive 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.
