[Math Lair] Heuristics

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Heuristics can be defined as the study or practice of the methods and rules of learning, discovery, and problem solving. The word heuristics can also be used to describe these methods and rules. A good reference on heuristics is George Polya's classic book How to Solve It.

There are several differences between heuristics and algorithms that are worth mentioning:

For well-defined, straightforward problems such as those commonly found in mathematics textbooks, the method of solving the problem is obvious, and all that is required is to directly apply the theorems, formulae, and algorithms that you already know to solve the problem. However, other problems, both in everyday life and in advanced mathematical inquiry, are not always well-defined, and having a knowledge of what methods may be useful in solving them can be helpful.

Common Heuristics

Here is a list of several common heuristics used in solving mathematical problems. This list is by no means complete.

Assume the Opposite
If you're trying to prove something, you could assume the opposite of what you're trying to prove and show that it has to be false. If the opposite assumption to what you want to prove is false, then, by the law of excluded middle, the assumption that you want to prove must be true. This approach is used in three related methods of proof:
In backtracking you make an assumption about the solution and see what follows from that assumption. If it seems to work out, make another assumption and so on until you've solved the problem. If it doesn't work (e.g. it results in a contradiction), backtrack to a previous choice point and try another assumption. If you've ever worked on a Sudoku puzzle, you've likely used backtracking.
Break the Problem Up
Find a way to solve part of the problem. Then, look at the rest of the problem as a sub-problem and solve it, breaking it up in turn as required.
Consider Another Problem
This method involves considering a problem into a problem that is either logically equivalent to (but usually simpler than) the original problem, a (usually simpler) variant of the problem for which the answer can lead to an answer for the original problem, or some other related problem that may be helpful.
Create a Diagram, Graph, Table, Model, or Equation
Often, a problem is stated in an unhelpful way. How to proceed may become clear if you represent the data in the form of a diagram, graph, table of values, model, or equation.
Guess and Check (also known as Trial and Error, Generate and Test, and other names)
Generate possible solutions and then determine whether they meet the solution criteria. While it's often discouraged by teachers (it can, after all, be excessively time-consuming), it's as valid a method of reaching a solution as any other.
Inductive Reasoning
If you're given a bunch of data that may be related, it may help to reason inductively to seek a pattern that ties the data together. Once you've found a pattern, you can use regular mathematical methods of proof to confirm it.
Mathematical Induction
The method of mathematical induction can be useful in proofs.
Reasoning by Analogy
Given a previously solved problem, use that problem to determine an analogous solution to the current problem. Related to "Consider Another Problem", above.
Restate the Problem
It can be helpful to restate the problem in another way. You could turn an algebraic problem into a geometrical problem or vice versa, or you could turn a problem into a logically equivalent problem.
Work Backwards
If you know what the goal or solution is, you can analyze the last step needed to achieve the goal, then the next-to-last step and so on until you get to the initial state. When you were young, you may have used this approach to solve mazes. As well, I find myself using this method for a real-life problem as I'm writing this. Tomorrow morning I have an appointment for 9:00 am. For what time should I set my alarm clock? Well, if I know that I need to be at the appointment at 9, that means that I have to be in the car at 8:50, which means that I need to take the dog for a walk at 8:20, which means that... and so on until I've worked backwards to what time I need to get up.
Sometimes a problem can be easier to understand when it is generalized. Abstracting can provide a broader perspective and suggest other techniques.

See also: Pigeonhole Principle

Sources used (see bibliography page for titles corresponding to numbers): 8, 41.