# Category Archives: MAT-STEP

# “A picture is worth a thousand equations”

# Geometry and “crap”

# Using simple special cases in problem-solving

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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

# The shape of the answer

# STEP and MAT

*Terry Tao’s book “Solving Mathematical Problems” will help*

# 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.

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:

1^{2} and 2^{1}

2^{3} and 3^{2}

3^{4} and 4^{3}

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.