Problem-Solving Strategies

by Arthur Engel

1 The Invariance Principle 2 Coloring Proofs 3 The Extremal Principle 4 The Box Principle 5 Enumerative Combinatorics 6 Number Theory 7 Inequalities 8 The Induction Principle 9 Sequences 10 Polynomials 11 Functional Equations 12 Geometry 13 Games 14 Further Strategies

Genres: Mathematics, Nonfiction, Textbooks, Puzzles, Technology, Science, Personal Development

416 pages, Paperback

First published December 12, 1997

Reviews

[Murilo Andrade · Rating 5 out of 5]

Another gem every problem solver or math olympic should have on the shelf. It encapsulates problem solving techniques in a handful of concepts, providing several examples and a whole bunch of problems (around 1300 I guess), most of them with solutions or advanced hints.

The book is divided into chapters consisting of a "strategy" or a subject. The range of problems is really wide, and sometimes are quite difficult. In my opinion there is a missing chapter on symmetry, for its applicability on problem solving.

"An experienced problem solver can often infer the road to the solution from the result." - grab all the information the problem gives you.

The chapters are:

Chapter 1 - The invariance principle

Here Engel describes invariants and monovariants. The idea is to construct a function ( often integer, positive) that does not change or is bounded and monotonic (in the latter case the algorithm must terminate)

- If you have (or can introduce a transformation), look for an invariant.
- Distance from the origin is frequently used on these problems
- parity is also useful
- If there is repetition, look for what does not change.

Chapter 2 - Coloring proofs

Weird chapter on solving board problems. Typical colorings are: chess-like, column (or row) based, etc. A few parity problems on this chapter.

Chapter 3 - The extremal principle

Very wide-applicable strategy. Many examples on Number Theory, Geometry, Combinatorics, etc are given.

- Pick an object that maximizes of minimizes some function. It frequently has nice properties, allows to simplify or even to arrive in a contradiction.
- Every finite nonempty set of nonnegative numbers has a min and a max
- Every nonempty set of positive integers has a min ( well ordering principle)

Chapter 4 - The box principle ( pigeonhole )

- If n+1 pearls are put into n boxes, then at least one box has more than one pearl.

The difficulty on this chapter is to find the box and the pearls for each problem.

Chapter 5 - Enumerative Combinatorics

- Count by bijection
- Recursion
- Divide and conquer : split a problem into smaller parts, solve the small problem and combine solutions.
- Sum rule, product rule, product-sum rule, sieving
- Construct of a graph which accepts the objects to be counted.
- Count in two ways
- If you cannot find the number of good objects, find the number of bad objects.

Chapter 6 - Number Theory

Basic introduction to NT. Lots of problems.

Chapter 7 - Inequalities

Basic Inequalities ( CS , x2> 0, AM-GM-HM, rearrangement, chebyshev, etc). A few hints for solving such problems:

-does the expression remind you AM-GM-HM? Cauchy-Schwarz?
-Can you apply rearrangement?
-An inequality homogeneous on its variables can be normalized.
-Is there any symmetry on the variables? In that case you can make additional hypothesis (e.g. a>= b >=c >=...)
-Trigonometrical substitution?
-Convexity and concavity ( Jensen)
-Can you transform it into a simpler form?

Chapter 8 - The induction Principle

Small Chapter on induction.

Chapter 9 - Sequences

Nice chapter on sequences. Includes linear recurrence, josephus problem, etc.

Chapter 10 - Polynomials

- Vieta's theorem
- Fundamental theorem of algebra
- Roots of unity
- reciprocal polynomials
- Symmetric polynomials

Chapter 11 - Functional Equations

Cauchy's functional equation and others. Small chapter

Chapter 12 - Geometry

Very long chapter, including:
- geometry and complex numbers
- transformation geometry
- classical euclidean geometry

Chapter 13 - Games

Describes nim, Bachet, Wythoff, etc

Chapter 14 - Further Strategies

- Graph theory
- Infinite descent
- Working backwards
- Conjugate numbers
- equations, functions and iterations
- integer funcitons (floor, ceiling, etc)

[Fri]

Number_theory

[Jessada Karnjana · Rating 5 out of 5]

เป็นเล่มที่ใช้งานคุ้มมาก Arthur Engel บอกว่าตั้งใจใช้เป็นหนังสือสำหรับเทรนเข้มทีมนักเรียน IMO ของเยอรมัน

[Manik Dhama · Rating 5 out of 5]

Summer vacation material
Very fun!

[Saatvik Jha]

Very nice book

[Sabyasachi Mukherjee · Rating 3 out of 5]

I really liked this book when I read it in high school.It has numerous problems ranging from hard to routine and can be used effectively for training for mathematical contests and for problem solving in general.

[paniz · Rating 5 out of 5]

I didn't read it yet!

[Raymond · Rating 4 out of 5]

Nice for training for IMO.

[Hessam]

Yeah! best book!!

[Maurizio Codogno · Rating 4 out of 5]

La mia recensione: http://xmau.com/notiziole/archives/00...

[Ainun Najib · Rating 5 out of 5]

A must read for mathletes :-)

[Akhil · Rating 5 out of 5]

Classic book for anyone interested in elementary mathematics and problem solving

[Anuwaya · Rating 5 out of 5]

it's very pure and nice book thanks engel for this marvelous book

[Suraj Kumar · Rating 5 out of 5]

Too tough for me. :(

[Soumyadip Banik]

happy