Understand the problem

Read the problem and make sure you understand it. Ask the following:

• What is the unknown?
• What are the data?
• What are is the condition?

Often it will be useful to draw a diagram and identify the required quantities/data depicted. Usually it will be necessary to introduce suitable notation.

For resources on techincal/mathematics writing I have found the following to be useful

• Technical writing for computer scientists - Mihir Bellare
• Mathematical Writing - Franco Vivaldi

Specifically the following from Mihir Bellare is crucial to keep in mind > The main thing to grasp is that it is your responsiblity to communicate clearly. It is not the other person’s fault if they don’t understand you. It is your job to make yourself clear. Once you see it that way, you start putting in the effort, and things get better.

Devise a plan

Find a contection between the given information and the unkown that will enable you to calculate the unknown. When a connection is not readily apparent, the following ideas can be helpful when devising the plan.

Try to recognize something familiar

Relate to previous knowledge. See if you can’t recall a familar problem that had a similar unknown.

Try to recognize patterns

Some problems are solved by recognizing that some pattern is occuring. If the regularity or repetition in a problem can be seen, you might can guess the pattern and prove it.

Use analogy

Think of something similar, a problem that is related but easier. Through solving the related but simpler problem, it can key you into what is needed to solve the more difficult problem. E.g. if working on an N-Dimensional problem, first try to think of it in just the first or second dimensions.

Introduce something extra

Auxiliary aids can be helpful in making connections. E.g. on a diagram a new line drawn can be useful, or in algebra a new but related unknown.

Take cases

Try splitting the problem in to several cases, with a different argument for each of the cases. E.g. with an absolute value it is benefical to consider different cases.

Work backward

Assume your problem is solved. Step by step work from the imagined solution back to the original data. Then you might be able to reverse the steps and establish a solution to the original problem. E.g. solving equations commonly employs this.

Establish subgoals

Complex problems will often require different building blocks which come together to solve the problem. If we can establish these seperate blocks, we can work towards them, partially fulfilling our established criteria with each one completed, before combining them all together.

Indirect reasoning

At times, attacking the problem indirectly can be appropriate. E.g. using proof by contradiction to prove that \(P\) implies \(Q\), we assume that \(P\) is true and \(Q\) is false and try to see why this can’t happen. Somehow this information has to be used to arrive at a contradiction that we know to be true absolutely.

Mathematical induction

Let \(S_n\) be a statment about the positive integer \(n\). Suppose

1. \(S_1\) is true
2. \(S_{k+1}\) is true whenever \(S_k\) is true.

Then \(S_n\) is true for all positive integers \(n\).

Carry out the plan

Ensure the in carrying out the plan, that at each stage you check the plan and write the details to prove that each stage is correct.

Look back

Once the solution is completed, look back over it. Do this to A) see if you have made errors in the solution, and B) see if you can think of an easier way to solve the problem. It is also important that by looking back you will familarize yourself with the method of the solution in the hope that it may useful for a future problem.