Ben A. answered • 05/05/20

PhD Student in Computer Science with 10+ years of tutoring experience

Hint #1: This problem has a symmetry to it. If you rotate the triangle the movements of the counters should look the same. They all follow a similar path.

Hint #2: When you see, "prove this is true for all integers n > 0" that's a hint you need induction.

To do induction properly you need a base case, an inductive hypothesis, and an inductive step.

Here's how I would approach solving it:

For the base case start with a side length n = 2. You can work out the paths for this case by hand (they each move in a path that looks like 3 sides of a trapezoid and end up in a different corner).

Now you need to show that if you can prove it for n = 2, that it must be true for n = 3.

Draw out the triangle for n = 3 and solve it by hand. This is a little harder. Once you do, you should see a pattern. Use this to create your recursive relation.

Hint #3: When n = 3, the drawing will have 18 one-unit segments that need to be painted. So each counter will paint 6 of the segments.

Hint #4: For any size n, look at the small triangles with side = 1 that are in the corners of the big triangle. If the counter follows one of the edges out of the corner, how do can a counter color the other edge leading into the corner? Every counter must end up in one of these corners at the end. So the counter must start in a corner, somehow paints some of the edges in the middle, and heads back to a corner.

If you need more help with this problem or other problems, please feel free to message me!

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