Toolbox

Some general suggestions for problem solving:

• Exploit Symmetry
• Work backward
• While counting, look at things from different angles and always try to exchange order of summation
• Try smaller problems, lower dimensions, extreme cases

General principles:

• Pigeonhole principle (Dirichlet)
• Parity checks (odd vs. even)
• Apply arithmetic modulo N, remember your divisibility criteria
• Induction, strong induction
• Use of geometric transformations

Links:

Terry Tao on problem solving