# Solving Mathematical Problems: A Personal Perspective

> Source: https://ollybritton.com/notes/books/solving-mathematical-problems/ · Updated: 2021-07-01 · Tags: notes, books

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> Man knows a lot of maths and solves a lot of maths problems.

* [Notes](#notes)
	* [Strategies in Problem Solving](#strategies-in-problem-solving)
			* [What are the three main types of problems??](#what-are-the-three-main-types-of-problems)
			* [If you use $a$, $b$ and $c$ for the sides of a triangle, what notation could you use for the angles??](#if-you-use-a-b-and-c-for-the-sides-of-a-triangle-what-notation-could-you-use-for-the-angles)
			* [What two things should you do after selecting good notation for a problem??](#what-two-things-should-you-do-after-selecting-good-notation-for-a-problem)
			* [Instead of sides $a$, $a + b$ and $a + 2b$ for lengths in an arithmetic progression, how could you exploit symmetry to simplify??](#instead-of-sides-a-a--b-and-a--2b-for-lengths-in-an-arithmetic-progression-how-could-you-exploit-symmetry-to-simplify)
			* [What is Heron's formula for the area of a triangle??](#what-is-herons-formula-for-the-area-of-a-triangle)
			* [What does $s$ represent in Heron's formula??](#what-does-s-represent-in-herons-formula)
			* [If you're stuck on a problem, how could you modify the problem slightly??](#if-youre-stuck-on-a-problem-how-could-you-modify-the-problem-slightly)
			* [What are some of the ways you can reformulate a problem??](#what-are-some-of-the-ways-you-can-reformulate-a-problem)

## Notes
### Strategies in Problem Solving
##### What are the three main types of problems??
* Show that...
* Find...
* Is there...

##### If you use $a$, $b$ and $c$ for the sides of a triangle, what notation could you use for the angles??
$$
\alpha, \beta, \gamma
$$

##### What two things should you do after selecting good notation for a problem??
* Write down basic facts you in the notation
* Draw a diagram if possible

##### Instead of sides $a$, $a + b$ and $a + 2b$ for lengths in an arithmetic progression, how could you exploit symmetry to simplify??
$$
b - d, b, b + d
$$

##### What is Heron's formula for the area of a triangle??
$$
\sqrt{s(s-a)(s-b)(s-c)}
$$

##### What does $s$ represent in Heron's formula??
Half the perimeter of the triangle.

##### If you're stuck on a problem, how could you modify the problem slightly??
* Consider a special or simple case of the problem
* Solve a simpler version of the problem
* ...
* Formulate a conjecture which would imply the problem, and try to solve that first
* Derive some consequence of the problem, and try to prove that first
* Reformulate the problem
* Examine solutions of similar problems
* Generalize the problem

##### What are some of the ways you can reformulate a problem??
* Prove by contradiction
* Try a substitution
* Take the contrapositive

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