History is replete with examples of Mathematical amateurs who have made a lasting contribution. One notable instance is Pierre de Fermat, whose 'Last Theorem' confounded Mathematicians for 350 years and led to the construction of whole new disciplines intended to lay a foundation for the solution.

More recently, San Diego homemaker Marjorie Rice took up the question of which types of shapes can tile the plane, developing a fruitful, unorthodox notation to solve previously unanswered questions. Mathematical icon Paul Erdos relished the story of a twelve year old boy who instantly solved a problem he, Erdos, had needed ten minutes to solve. (Given n + 1 positive integers less than or equal to 2n, prove there are always two of them that are relatively prime).

Modern problem solvers, however humble, are part of this long tradition of taking delight in tackling a puzzle. The enterprise can be deeply satisfying and often depends on little accumulated knowledge. Take this problem:

Problem. The faces of each of 15 pennies, packed as shown here, are colored either black or gray. Prove that there exist three pennies of the same color whose centers are the vertices of an equilateral triangle.

The solution is here, but try to solve the problem before peeking. The solution requires little more than understanding the 'reductio ad absurdum' method of proof and knowing what an 'equilateral triangle' is (a triangle whose three sides are equal) ... and applying an intricate logical chain of reasoning.

Here's another problem once found on a placemat at a restaurant to occupy children:

Problem. Make a 3 x 3 'checkerboard' square, placing each of the numbers one through nine in the nine cells, such that the sum of each row, each column, and each diagonal, is the same.

Try it yourself first! But here is the solution, using simple arithmetic and logical reasoning. Such number squares are called 'Magic Squares'. The problem calls for a Magic Square of order three. A related problem asks whether the solution is 'unique', meaning the only solution up to rotations and reflections. Durer's engraving Melancholia shows a Magic Square of order four. Is it unique?

Of course many problems require knowledge of Mathematical definitions, principles, and methods. An amazing amount can be done with little overhead, however. Consider this problem:

Problem. Find the smallest possible value of f(x) = x + 1/x, where x can be any positive real number.

Calculus can be used to solve such problems, but simpler approaches are possible, sometimes yielding more insight. One tack is to try different values of x, guess the answer, then prove it, as is done here.