You’ve seen how mathematical problems can be solved with less, or even no, detailed calculation by looking at what the problem tells you about the shape of the solution.
Another useful method, at least for checking answers, is to look at simple special cases of the problem, or simplified versions of the problem. We saw some of that when we talked about how to check your answers in FP1 and M2. Continue reading
- Click here for MAT papers and solutions
- Click here for STEP papers and solutions; or here
- My own suggestions on problem-solving in mathematics
- Click here for pdf of Terry Tao’s excellent book “Solving Mathematical Problems”
Hints for the problems at “Using simple special cases in problem-solving”.
1. Suppose it’s a 1×1 square. Which of the inequalities is valid? And suppose it is a very tiny square (say side 0.0001). Which of the inequalities is valid?
You can also do this by symmetry-type arguments. For any given shape of rectangle, A increases with the square of P. So the “shape” of the answer must be an equation connecting what power of P with what power of A?
Or another symmetry-type argument. The greatest rectangle area for any given perimeter P is got by a square. (Generally, the greatest polygon area for any given perimeter is got by a regular polygon, and the greatest area overall for any given perimeter is got by a circle). So A≤(the area of a square with perimeter P.
2. Think about the maximum area when θ=π/2, and when θ is very small.
3. Think about n=1.
4. Can y be negative? What value are all its maxima? Are its maxima equally spaced?
5. Never mind about 100 for the moment. Find n for when the sum≥2, then n for when the sum≥3, then n for when the sum ≥4. See the pattern…
6. Never mind about 99 and 100 for the present. Compare:
12 and 21
23 and 32
34 and 43
See the trend. If you want to be sure the trend is real, work out:
7. Never mind about 1,000,000 for the moment. Work out the 2-and-5 problem for 0≤n<10, and the 3-and-7 problem for 0≤,n<21. See if you can build on that.