Problem Solver's Toolbox

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The goal of this page is to collect simple problem solving strategies and tools. We hope that students interested in the Wisconsin Math Talent Search would find the described ideas useful. Our hope is that this page and the discussed topics can be used as a starting point for future exploration.

General ideas

There is no universal recipe for math problems that would work every time, that's what makes math fun! There are however a number of general strategies that could be useful in most cases, here is a short list of them. Many of these ideas were popularized by the Hungarian born Mathematician George Pólya in his book How to Solve It).

Should we add some general problem solving ideas here? (E.g. like this: How to Solve It).

Modular arithmetic

When we have divide two integers, they don't always divide evenly, and there is a quotient and a remainder. For example when we divide 10 by 3 we get a remainder of 1. It turns out that these remainders behave very well under addition, subtraction, and multiplication. We say two numbers are the same "modulo m" if they have the same remainder when divided by m. If a and a' are the same modulo m, and b and b' are the same modulo m, then a+b and a'+b' are the same modulo m, and similarly for subtraction and multiplication. This often makes calculation much simpler. For example, see 2016-17 Set #2 problem 3.

See Art of Problem Solving's introduction to modular arithmetic for more information.

Mathematical induction

Angles in the circle