Ways of finding the sums of some interesting sequences of numbers are discussed. Finding the patterns creates a challenge, but the patterns are not too difficult for average pupils to discover. Mathematical induction can then be used to prove the formulas. (LS)

A device is suggested for constructing triangles with the same area and the same perimeter. A proof is given as is a proof of necessary conditions for a special case. The question of congruence of the triangles is left to the reader. (LS)

Details of an investigation into polygonal numbers are given. From a study of the tabled series, patterns are discovered both within and between the various polygonal series. From a study of these results and the corresponding figures, formulas are developed. (LS)

A method for generating Pythagorean triples is presented and demonstrated with several examples. Its relationship to another method is shown and several interesting related facts are proven. (LS)

By changing the Euclidean metric definition, a new geometry is developed. Properties of figures are analyzed and theorems are proven in this new system. (DT)

Methods of finding the total number of gifts given in The Twelve Days of Christmas'' are described. (DT)

Fundamental concepts of symmetry and rigid motion are developed through activities using polygonal shapes made up of six congruent equilateral triangles. (DT)

The usual Euclidean metric definition is modified; new figures and measures are defined and analyzed using this different distance function. (DT)

Three worksheets are provided; the aim is to develop skill in visualization of the cube from two-dimensional patterns. (DT)