I'm doing my final year of under graduation through distance education and would be appearing for entrance tests for various graduate schools in a few weeks. I am looking for a database of algorithms/flowcharts for solving several types of general problems often encountered in these tests, giving an exhaustive list of techniques and methods available and the order in which they should be applied. Here is a good example of a flowchart for solving the general problem of

"Testing a series for convergence" https://www.math.ucdavis.edu/~egoldwyn/math21c/series_flowchart.pdf.

Another example could be this for "Testing a function for uniform continuity" (just here as an example, may contain errors).

  1. Try to draw the graph of the function.
  2. Test for Lipschitz Condition.
  3. Use $|f(x)-f(y)|<e$ to find a $d(e)$ such that $|x-y|<d(e) \implies |f(x)-f(y)|<e$.
  4. Test if a continuous extension over a closed and bounded set (compact set) exists.
  5. Test for points of discontinuity.
  6. Fix an $e$ and try to find $x$ and $y$ such that $|f(x)-f(y)|\geq e$ while $|x-y|$ can be made as small as desired.

Currently when I try to solve problems under time constrains I often miss a few methods and get stuck. Since a lot of these tests are designed to test one's speed as well as understanding, such a resource would be quite helpful to improve speed after I've understood each technique thoroughly. The syllabus is roughly the topics covered in a $3$ year undergraduate course in the following areas:

Abstract Algebra
Linear Algebra
Real Analysis
Complex Analysis
General Topology
Differential Equations
Probability Theory

If it doesn't exist and if it would be useful to others, perhaps we can create such a collection here.



A general problem solving research paper, that also includes some practical ideas is the University of Georgia's page Mathematical problem solving, essentially a synthesis of the current research in problem solving theory and techniques.

Now to address some of your specific requests:

A very good algebra problem solving document is Algebra Concepts Equation Solving Flow Chart,providing steps and examples for several different types of equations. Another one that emphasises backtracking is Solving Equations Using Backtracking.

A flowchart, examples and some more information are provided in the document Differential Equations,in fact there are several flow charts for different methods in this document. Some more information and steps are provided in the document Differential equations: 2nd Order differential equations.

A combinatorics flow chart was developed and discussed on the gmat forum here.

The MCGraw-Hill textbook chapter Linear Equations and Inequalities have examples and steps to solve main problems involved with inequalities.

Hope this helps

On that point, as a Maths teacher, I have found that several textbooks have the steps outlined for solving problems included in the worked examples.

  • $\begingroup$ Thank you for your response. The articles on DE and heuristics were helpful while others were probably directed towards high school students. I was unable to access the gmat forum article even after registering. Perhaps my question is too broad and demanding; posting different questions for each general problem type might bring more responses. $\endgroup$ – Akash Gaur Dec 26 '13 at 15:26

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