On this site I present a collection of investigational projects in mathematics. The purpose is to provide open-ended problems (and some guidance) that high school and college-level students can approach creatively and in their own way. I believe that these kinds of investigations are important in one's education (mathematical and otherwise), and they can be a lot of fun. Most of the projects are investigations that I worked through as "research" problems. Students are encouraged to ask their own questions and formulate their own ideas and hypotheses about the structures they explore.

I hope that these projects are able to inspire, enlighten, and delight other students of mathematics, as they have certainly done for me.

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Projects in Mathematics
More about how these projects work.

• Pascal's Simplices Project (combinatorics, number theory): Pascal's triangle gives the coefficients of binomial expansions (a + b)n. What is the generalization of Pascal's triangle? What are the coefficients of general multinomial expansions ?

• Pythagorean Triples Project (number theory): What are all the integer solutions to the equation x2 + y2 = z2? To how many such solutions does a given number belong?

• Regular Polygons Project (geometry, trigonometry, graph theory): What is the area of a regular polygon with n sides? What is the angle between each side? What is the radius of the circumscribed circle? How many intersection points do the diagonals of a regular polygon form?

• Regular Polyhedra Project (geometry): There are only five convex regular solids. What are their volumes and surface areas?

• Regular Polytopes Project (geometry, combinatorics): Both the cube and the regular tetrahedron have generalizations in n dimensions. What are the volumes of these n-dimensional objects? How many vertices/sides/faces/etc. does each have?

• Sums of Consecutive Powers Project (number theory): Is there a way to expediently calculate the sum of consecutive natural numbers raised to a constant power p? Bernoulli gave the answer, and along the way we meet an interesting family of polynomials.

 Supplementary Topics Mathematical Induction: Induction is a method of proof by which a statement can be proven for infinitely many cases. Modular Arithmetic: Modular arithmetic is a powerful tool in the theory of numbers. Solutions to Polynomial Equations: Here are derivations of the quadratic and cubic formulas. Expository Solutions to the Projects Pascal's Simplices Pythagorean Triples Regular Polygons Regular Polyhedra Regular Polytopes Sums of Consecutive Powers
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