I was casually browsing YouTube for nice puzzles (like one does) and stumbled upon this amazing point-line geometry puzzle that I had never seen before, so I couldn’t resist writing it up.
The full video introduces a series of nice puzzles designed to challenge common intuition and this was one of them.
You can see the introduction to the problem I am about to discuss in the clip below:
Fortunately (or unfortunately depending on your preference) the presenter leaves out the full explanation, although he does give the answer and some good hints on where to begin for the proof.
I thought I’d fill in the gaps to complete the argument of this excellent puzzle.
In case you did not want to bother watching, here is the problem statement from the video:
As the presenter mentions, you could clearly do it with an uncountable continuum of points.
A circle is such a set and no matter how close the enemy is to us, we can center ourselves in a circle of bodyguards with radius half the distance between ourselves and our enemy.
(If you know some Topology you trivially recognize that this comes from the fact that with the Euclidean topology is Hausdorff, and in particular a Kolmogorov space).
The presenter however tells us that only 16 bodyguards are necessary and sufficient to block all possible shots!
What is counter-intuitive about this problem is that there are infinitely many dense trajectories in the rectangle, meaning we can get hit from an infinite number of angles, and yet the claim is that only 16 bodyguards suffice to block all shots – amazing, no?
In case you wish to attempt the problem yourself here is the customary SPOILER ALERT!
It might be interesting to look at other shapes and try to answer the same question.
Here is an interesting example I came to think of:
Perhaps even more interesting would be to look at other n-sided polygons.
In fact this could make an interesting mini-project.
Clearly there is something funny going on when angles are not rational multiples of .
I will let you work out what it is.
It may therefore be easiest to start with regular n-gons.
For example in a room shaped like an isosceles triangle we would get a picture that looks something like the below:
Perhaps it may be tempting to draw some kind of conclusion from this, but inaccurate pictures have fooled many!
This needs a mathematical argument.