# Hints for “MAT problems using simple special cases”

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.

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:

nn+1/(n+1)n(n−1)n/n(n−1)

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.

# Dirichlet’s principle

This morning, 13 April 2015, the Financial Times has a reference to mathematical method in one of its articles on economic affairs.

# How many marks does Edexcel take off for bad writing in S2?

How many marks do Edexcel take off for bad writing in S2? I’ve looked again at the recent mark schemes, and I’m not sure.