Lattice points : applications in problem solving

The purpose of this paper is to investigate specific problems in mathematics which have solutions on a set of lattice points. In each problem, the introduction of a set of points simplifies the solution and gives a means of discovering a general solution which can be applied to similar problems. The paper begins with a study of the lattice points as integers. Chapter I also gives a method of approximating irrational numbers. Later a finite lattice is applied to the game of billiards. Chapter II is a discussion of the area of simple polygons. It includes a proof of Pick's Theorem, Pick's Theorem adapted to polygons with "holes", as well as two methods of approximating pi using lattice points. Chapters III, IV and V consider the solution of two puzzles from recreational mathematics. The "difficult crossing" problem uses a square lattice as one method of solution. The "pouring" problem uses a triangular lattice. In the discussion of these puzzles questions such as the following are answered. 1. Is there a systematic way of finding possible solutions? 2. Is there a general method for finding the solution which can be used when the conditions of the problem are changed? 3. Under what circumstances will there be no solution?