Strategy and Tactics in Problem Solving

Solving problems is an activity not unique to mathematics. There are strategies of problem solving that are applicable to solving problems in any human endeavor. But every field and every situation call for specific knowledge and specific habits of mind to apply to solving problems. I therefore distinguish the universal strategies from field specific tactics.

As a young assistant professor I was looking for a desk. I found one new plain table of suitable dimensions in a carpenter shop for a price that in my estimate would not even cover the cost of materials. To my pointing that out the owner replied, "That's OK. This is because I know how." 35 years later I still have that table, although it now plays a different role in our household.

There is nothing specific to mathematics in George Polya's four stages of problem solving:

1. Understanding the problem
2. Devising a plan
3. Carrying out the plan
4. Looking back

To these I can add considering special cases as a stepping stone in understanding the problem; seeking an analogous or related problem; splitting the problem into smaller ones (which may be also regarded as moving to Polya's second stage.) Generalization is the opposite of specialization and is often throws revealing light on the problem's true nature. These all are strategic tools of problem solving. However, details of implementation are, of necessity, field specific.

At the time of writing, this site contained about 5500 pages, at least one third of which is devoted to solving problems, many of olympiad caliber. It may take time to classify them as to the problem solving tactics employed, but I plan to do that in a systematic manner.

I found a detailed classification in an article by Murray Klamkin, cited below. I added a few, and for some have not yet come across suitable examples. I'll be looking to fill the gaps.

I would very much appreciate comments and additional examples.

There is always satisfaction in solving a problem. The most common way to approach a problem is by identifying a general class to which the problem belongs and using a method (if such exists) that is applicable to the problems of that class. Enjoyment of having a problem solved is more a result of detecting and exploiting the problem's peculiarities. As a rule, those peculiarities once observed define a refined class of problems which succumb to the same method of solution. This is actually a problem solver learning process: looking back at the just solved problem, make a note as to what features made the problem solvable by the method you employed.

One of the best problem solving strategies is do something; do not get (and stay) flustered if you do not see a solution to a problem right away. Try one of the tactics, force yourself to speak aloud - something will come out. Ray Bradbury, when short on ideas, taught himself to work with a dictionary picking random words and trying to put them together into something meaningful and relevant. So try.

Also, note that some of the tactics was previously mentioned - with additional advice - in a short assay of what constitutes a proof.

Further reading

I highly recommend the 1981 article The heuristic of George Polya and its relation to artificial intelligence by Alan Newell that offers penetrating analysis of Polya's problem solving heuristics prior to the discussion on its applicability to AI.

1. R. Gelca, T. Andreescu, Putnam and Beyond, Springer, 2007
2. A. Engel, Problem-Solving Strategies, Springer Verlag, 1998
3. M. S. Klamkin, Mathematical Creativity in Problem Solving II, in In Eves' Circles, J. M. Anthony (ed.), MAA, 1994
4. G. Polya, How to Solve It, Princeton University Press, 1973
5. G. Polya, Mathematical Discovery, John Wiley & Sons, 1981
6. G. Polya, Mathematics and Plausible Reasoning, v 1, Princeton University Press, 1954
7. P. Zeitz, The Art and Craft of Problem Solving, John Wiley & Sons, 1999

Specializaton, i.e., considering special cases, or introducing additional restrictions, while trying to solve a problem may highlight aspects of the original problem that have perhaps been missed at first glance. Success in solving special cases may be reassuring of the validity of the original claim, while failure to solve a special case may prompt the solver to look for a counterexample which will also serve to disprove the larger problem.

Specialization is opposite to generalization; it is wonderful that both are valuable tools in the arsenal of a problem solver.

• Arithmetic and geometric means
• Probability on Discrete Infinite Sets
• Isoperimetric Theorem For Quadrilaterals
• Rectified, Halved, Sheared, Eyeballs Still Surprise

There is a whole page at the site devoted to G. Polya's thesis that quite often more general or more strict problems are easier to solve than the more special ones. The page explains the concept and features several lists of solved sample problems.

• Infinitely many proofs that there are infinitely many primes
• 1089 and a Property of 3-digit Numbers

The Pythagorean theorem has two distinct analogues in the $3D$ space. For the parallelipiped with sides $a,b,c$ and diagonal $d,$ we have $a^{2}+b^{2}+c^{2}=d^{2}.$ For a tetrahedron with three faces having right angles at the shared vertex and areas $A,B,C,$ $A^{2}+B^{2}+C^{2}=D^{2},$ where $D$ is the area of the remaining face.

Being analogous is very much close to being similar but the latter is usually more specific.

• Tesseract

• An Inequality for Grade 8
• Triangle of Minimum Perimeter
• Concurrence in Arbelos

• Probability of Matching Socks

• Pythagorean Theorem
• Square root of 2 is irrational
• Infinitude of Primes
• Butterfly Theorem
• Concyclic Circumcenters: A Dynamic View
• Napoleon's Theorem
• Morley's Theorem
• Bottema's Theorem
• Another simple integral
• Middle School Problem from a Moscow Olympiad
• Breaking Chocolate Bars
• Four Travelers Problem
• Chameleons of Three Colors
• Fractions on a Binary Tree
• Countability of Rational Numbers
• All about altitudes
• Angles in 4-5-6 Triangle
• Probability of Two Integers Being Coprime
• Circle through the Incenter

• Square Inscribed in Triangle
• PLL: Apollonius' Problem with Two Lines and a Point
• Equilateral Triangle in Equilateral Triangle

• What Is Symmetry?
• A Not Symmetric Inequality
• Symmetry Principle in Probability
• A simple integral
• Another simple integral
• Pure Angle Chasing III

An argument by continuity assumes the presence of a continuous function whose properties could be used to solve a given problem. For example, the area can be assumed (and eventually proved) to be a continuous function of a geometric shape (Note, though, that not all planar shapes may be assigned a numeric area in any sensible way. The common ones - polygons, circles, ellipses - do have areas which depend continuously on their shape.)

• Equilateral Triangle in Equilateral Triangle
• Cutting a Hole in Half
• Equilateral Triangle on a Closed Curve

In mathematics, the word "similarity" may be encountered in several different contexts. There is a special page with explanations and a variety of examples: What Is Similarity?

Without loss of generality - WLOG, for short - is a frequent stratagem in problem solving. It is incredibly powerful, although, at first sight, often appears innocuous. The essence of WLOG is in making a random selection among a number of available ones for the reason of all possible variants being equipotent, i.e., leading to exactly same procedure and result. For further explanation, examples and link, check a separate file.

One of the methods of proof, is "reductio ad absurdum" - assuming what is to be proved false and getting a contradiction as a result.

Explanations and examples

Many many examples have been collected on a separate page.

Many many examples have been collected on a separate page.

Many many examples have been collected on a separate page.

• Distance to line
• Optical Property of Ellipses
• Cassini's Ovals and Geometric Optimization
• Reflections in Ellipse
• Triangle of Minimum Perimeter
• Angle Bisectors in Ellipse II
• Maximum Perimeter Property of the Incircle

• Determinants, Area, and Barycentric Coordinates
• 100 Prisoners and a Light Bulb
• Superposition Principle

What Is the Extremal Principle?

• An Inequality for Grade 8
• An Extension of the AM-GM Inequality
  • An Extension of the AM-GM Inequality: A second look
• Schur's Inequality
• Newton's and Maclaurin's Inequalities
• Rearrangement Inequality
• Chebyshev Inequality
• A Mathematical Rabbit out of an Algebraic Hat
• An Inequality With an Infinite Series
• An Inequality: 1/2 * 3/4 * 5/6 * ... * 99/100 less than 1/10
• A Low Bound for 1/2 * 3/4 * 5/6 * ... * (2n-1)/2n
• An Inequality: Easier to prove a subtler inequality
• Inequality with Logarithms
• An inequality: 1 + 1/4 + 1/9 + ... less than 2
• Inequality with Harmonic Differences
• An Inequality by Uncommon Induction
• From Triangle Inequality to Inequality in Triangle
• Area Inequality in Triangle II
• An Inequality in Triangle
• Hlawka's Inequality
• An Application of Hlawka's Inequality
• An Inequality in Determinants
• An Application of Schur's Inequality
• An Inequality from Tibet
• Application of Cauchy-Schwarz Inequality
• Area Inequalities in Triangle
• An Inequality from Tibet
• An Inequality with Constraint
• An Inequality with Constraints II

• Half an Egg Wonder
• Reverse Solitaire
• Morley's Theorem
• Wythoff's Nim
• Subtraction Games

• Two Cyclic Inequalities
• Another simple integral
• A Quartic Equation

• Modular Arithmetic

• Tribute to Invariance
• Jumping Isn't Everything
• Breaking Chocolate Bars

• Geometric Transformations of the Plane

The method of angle chasing is explained on a separate page.

Counterexample is an example with a negative connotation. Whereas an example may be used to support or illustrate a claim, a counterexample is used to refute an assertion. How an example is being used often depends on the purpose or the formulation. For example, the word "nth" is an example of an English word without a vowel. It is a counterexample to the claim that every English word contains a vowel.

For a few more words and many additional examples see a separate file.

• The Three Jugs Problem
• Cabart's Collinearity
• Complex Numbers and Geometry
• http://www.cut-the-knot.org/triangle/Morley/Morley.shtml
• Two Butterflies Theorem
• Carnot's Theorem
• Semicircle on Square
• Eyeball Theorem Rectified

• What Is Infinite Descent?

• Graphing Equations Is Useful
• Graphing Equations Is Useful, II
• Graphing Equations Is Useful, III
• Graphing Equations Is Useful, IV
• Point common to two similar rectangles
• Equilateral Triangle in Equilateral Triangle
• Construction of n-gon from the midpoints of its sides

• Gergonne in Ellipse
• Expressions Invariant under Projective Mappings
• Cevians and Conic through Six Points on Sidelines of Triangle
• A Matter of Collinearity: Metamorphosis of a Problem
• A Property of Perspective Triangles
• Two Circles, Tangents and Two Collinearities
• Projective Invariants For Pascal And Desargues