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imagine you have a square-shaped room and inside there is an assassin and a target and suppose that any shot that the assassin takes can ricochet off the walls of the room just like a ball on a billiard table is it possible to position a finite number of security guards inside the square so that they block every possible shot from the assassin to the target let's walk through this puzzle a little more precisely first instead of thinking of a physical room with actual people inside I really want you to think of a square and the xy-plane pick any two points in the square and let's call one of those points a four assassin and the other point T for target now a shot from the assassin is really just a ray emanating out of the point a which can like a ball on a billiard table bounce it back and forth between the size of the square but unlike an actual game of pool let's assume that trajectory has constant speed and that it can bounce back and forth for forever we'll also assume that the law of reflection holds in other words the angle of incidence and the angle of reflection are always equal and if the trajectory ends up in one of the four vertices then we'll just say it stops that's the end of it now if one of these billiard trajectories intersects with the point T then that means the assassin has hit the target of course an obvious hit is just a straight line shot from a 2t but don't forget the assassin can also aim away from the target and still hit it like this so the puzzle is can you block all possible hits but what do I mean by block if we can place another point in this square call it B so that one of those trajectories from a to T actually passes through B first then we'll say that the point B has blocked the assassin shot okay that's the puzzle let's recap start with a square in the XY plane whose sides you can think of as the size of a billiard table pick any two points in the square call them a and T is it possible to place a finite number of points in the square so that they will block any possible shot from a Tootsie what do you think I won't reveal the solution just yet I'm gonna leave it as a challenge however and the rest of the episode I will tell you about some mathematics that could help you arrive at a solution by the way this puzzle is sometimes called the secure polygon problem because you can ask this same question for other polygons not just squares it's closely related to the illumination problem that was featured on numberphile a while ago they're both examples of billiard problems which fall under the mathematical study of dynamical systems in fact here's a little backstory that I think you'll enjoy on august 13 2014 the late miriam mirza connie became the first woman and the first iranian to win the Fields Medal later in that same year she gave a talk at Harvard University in which she described her research in fact it's right here on YouTube if you pause me and go watch this video then you can listen to an explanation of the secure polygon problem around 25 minutes and 50 seconds in now as it turns out emily real a category theorist and professor of mathematics at johns hopkins university was in the audience during Marion's talk she thought about the puzzle and came up with a wonderful solution and in the next few minutes I'll present a couple of mathematical ideas that we'll need to understand that particular solution at first they may not seem related to the puzzle at all but finding the connection is all part of the fun ok what are those ideas we'll start with the torus in particular I want to talk about the flat torus both Gabe and I have talked about this in previous episodes in fact now's a great time to if you haven't already go watch Gabe's episode telling time on a torus there he showed how a torus can be viewed as a ocean space obtained by identifying opposite sides of a square for the sake of illustration I'm going to use green and blue to denote that these two sides are identified and these two sides are identified so if we have a path going this way on a torus once it reaches the right edge it will reappear at that same spot but coming through on the left edge and similarly when the path moves between top and bottom edges but all that's been covered in a previous episode now I want to expand on that idea just a little by sharing with you a common technique that mathematicians use namely if you want to understand a path on the torus like this one well it can be tricky even though it's just one path we have to chop it up into six smaller pieces and then remember to keep track of which ends are glued together and that's fine but it can get messy so to make things easier we'd like to represent that path is one single line how so by tiling the plane with square tour I we've actually done this before in a previous episode the geometry of the card game set but let's recap again how can we see that a path on the torus is really a straight line in the plane start at the beginning of the path as it moves upwards you can imagine it re-entering the bottom edge of an identical copy of a flat torus on top then as the path continues towards the right edge it'll pass through the left edge of another identical copy of the torus and so on keep doing this until the path ends and you end up with a straight line in the plane so this path on the torus is exactly the same as this line segment in the XY plane because we can think of the plane as a tiling of infinitely many flat touring and let Miriam beside each of these squares is an identical copy of the original one that means for example that this point is the same as or equivalent to this point which is the same as this point which is the same as this point all of these other points and for the sake of simplicity if we assume that our original square is just a 1 by 1 tile then each of these points are separated by exactly one unit in the x-direction and one unit in the y-direction so if this point has coordinates a B then this point is a plus 1b this point is a B minus 1 this point is a plus 2 B plus 3 and so on the collection of dots that you get is called a lattice and you might find that using one or more lattices may be helpful when trying to solve the puzzle anyways the upshot is that any path on a torus corresponds to a line in the plane and any point on the torus gives rise to a lattice in the plane so let's think back to the puzzle is the square billiard table actually a flat torus well no that's not suppose that the assassin is here if this trajectory moves up towards the top edge then it does not reinter through the bottom instead it reflects back so unlike the opposite sides of a square torus which are identified the sides of our square billiard table are all different no two of them are identified all right so the square billiard table is not a flat torus but can you use one or more of them to make a torus and if so can you solve the puzzle by using the fact that paths on a torus correspond to lines in a plane and points on the torus give rise to lattices here's the puzzle again is it possible to block any shot from the assassin to the target to arbitrary but fixed points in the square using only a finite number of carefully plotted points and if it is possible then and this is really the key what is that finite number lots to think about I'll let you work out a solution feel free to discuss in the comments and after you felt long and hard about the answer you can find dr. Emily real solution in a link in the description below there I've typed up a full solution to this secure polygon problem using the ideas we've discussed in today's episode so enjoy the mathematics and see you soon