Solving Mathematical Problems, Tao
Man knows a lot of maths and solves a lot of maths problems.
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Notes
- Strategies in Problem Solving * 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?? * 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?? * What is Heron’s formula for the area of a triangle?? * What does $s$ represent in Heron’s formula?? * If you’re stuck on a problem, how could you modify the problem slightly?? * 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