Don’t Always Trust Hints
I was recently tackling a math exam question for very able high school students. The question came with a hint on how to tackle it.
After some thought, it struck me that I had an easier way to tackle the problem if I ignored the hint. For a moment, I put myself in the shoes of a keen 18-year old who wants to do well in the exam. How many such students would be brave enough to ignore the hint and try their own approach? Probably not many.
It’s a good reminder that hints usually reflect one person’s way of attacking a problem, but that’s no guarantee that there aren’t alternative or even better ways to tackle it. Let’s take a look at the problem.
Ignoring the hint
The hint tells us to focus on the area that the guard cannot see. But hold on. Let’s pretend we are the guard. As we walk back and forth from A to E, what can we see? Here is an alternative sketch of the diagram above:
I want to make a couple of simple observations:
- The guard can always see the blue shaded area, no matter where he is along AE. This area is 12⨉4+8⨉4 = 80 units.
- In addition, triangles A and B will be visible to the guard. These will vary according to the position x, and at one of the the extreme points — x = 4 — triangle B will be degenerate (its area will be zero).
Therefore, our question reduces to simply studying the areas of triangles A and B and finding when their sum is at a maximum and minimum. I think this is a significant simplification that is unlikely to be arrived at if the student follows the hint given in the question.
Finding the sum of the areas
For this we can apply some basic trigonometry. Here’s another version of the diagram: