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I am searching for some books that describe useful, interesting, not-so-common, (possibly) intuitive and non-standard methods (see note *) for approaching problems and interpreting theorems and results in number theory, analysis, algebra, linear algebra, and other branches of mathematics.

(*) Such methods can be (but not limited to) from the areas of

  • probability;
  • combinatorics;
  • graph theory;
  • physics;
  • geometry.

Examples of such books can be Uspenskii's Some Applications of Mechanics to Mathematics or Apostol's and Mnatsakanian's New Horizons in geometry.

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  • $\begingroup$ A bounty on a question with a big-list tag? I'm not sure that makes any sense. $\endgroup$ – Gerry Myerson Oct 2 '14 at 11:14
  • $\begingroup$ This question is very broad. Almost every large branch of mathematics is useful in some way for each other large branch. Can you be more specific about the types of problems you are interested in? $\endgroup$ – Jair Taylor Oct 4 '14 at 22:10
  • $\begingroup$ @JairTaylor I am mostly interested in non-standard approaches. For example, New Horizons in Geometry uses methods from geometry to solve problems which normally require calculus; or The Mathematical Mechanic uses ideas from physics to deal with problems from geometry, calculus, or even number theory. $\endgroup$ – Dal Oct 4 '14 at 22:15
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The Probabilistic Method by Alon and Spencer is a classic.

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The classroom resource materials of MAA(mathematical association of america) can be very useful as it presents unusual approaches to mathematical ideas:

Some of its book I have used are as follows:

1.)Combinatorics A Problem Oriented Approach by Daniel A. Marcus.

2.)Visual Group Theory by Nathan Carter.

3.)Exploratory Examples in Real Analysis by J.E.Snow and K.E.Weller.

You can find books suitable for you as you described in question from the link below:

http://digital.ipcprintservices.com/publication/?i=140026&p=24

you'll enjoy maths through these books in above link!

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    $\begingroup$ nice catalog link !!!! $\endgroup$ – spectraa Oct 2 '14 at 11:23
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    $\begingroup$ Good. Thank you very much for your suggestions. $\endgroup$ – Dal Oct 2 '14 at 13:44
  • $\begingroup$ Also the Group Explorer is the accompanying software for the "Visual Group Theory" book. $\endgroup$ – Alexander Konovalov Oct 2 '14 at 22:16
  • $\begingroup$ @AlexanderKonovalov yes you are right!!! $\endgroup$ – coool Oct 3 '14 at 2:45
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Mark Levi, The Mathematical Mechanic: Using Physical Reasoning to Solve Problems.

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  • $\begingroup$ This book has been mentioned in an answer to another question of mine; since you suggest it too, I will definitely give it a look. Thank you very much. $\endgroup$ – Dal Oct 2 '14 at 13:44
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We're using A Combinatorial Introduction to Topology by Henle as a supplemental resource in my general topology class right now. Some of the alternate proofs are interesting, like using Sperner's lemma to give a combinatorial proof of the Brouwer Fixed point theorem in $\mathbb R ^2$

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