Three guards are chasing a ninja in a structure that is shaped like a big tetrahedron where each edge, rather than side, is a passable corridor. The guards are a little bit faster than the ninja, but the ninja can anticipate their moves perfectly and is invisible unless a guard is very, very close. So, the ninja can't pass a guard in a corridor, but other than that the guards have no idea where the ninja is at any given time.

Do the guards have a strategy by which they can capture the ninja in a finite amount of time?

if memory serves, a tetrahedron is a triangular based pyramid. there are three side thus three corridors. and three guards. how large are the sides compared to the guards?

Indeed it is. Equilateral triangles in all sides. In roleplayer terms, a four-sided die or "D4".

Compared with the guards, the corridors are very long.. say, a mile each.. and so narrow that no one can pass anyone else in them. Rephrasing without referring to triangles or pyramids one could consider a system of crawl-ways that join points A, B, C, and D to each other so that the distance between all six possible pairs is the same.

They can certainly make it very difficult for him by all running to one point, running out on different paths to the corners of a triangle, then all running clockwise or all anticlockwise (randomly. Then running back to the start point again and repeating. He has to keep on guessing correctly which way they will run.

Or to give him 1/3rd chance each time, two run guards run to a random stationary guard, all guards run out to the corners of a triangle, then two guards run to a random stationary guard and repeat.

Originally posted by iamatiger They can certainly make it very difficult for him by all running to one point, running out on different paths to the corners of a triangle, then all running clockwise or all anticlockwise (randomly. Then running back to the start point again and repeating. He has to keep on guessing correctly which way they will run.

Or to give him 1/3rd chance each tim ...[text shortened]... n out to the corners of a triangle, then two guards run to a random stationary guard and repeat.

I agree, if the ninja is not omnipotent, this could work probabilistically. It leaves only one of the corridors unchecked.

However, the OP specifies that the ninja can anticipate all of the guards' moves, which means this strategy cannot work.

Originally posted by forkedknight I agree, if the ninja is not omnipotent, this could work probabilistically. It leaves only one of the corridors unchecked.

However, the OP specifies that the ninja can anticipate all of the guards' moves, which means this strategy cannot work.

It is as if it is all laid out in advance.This puzzle is thus a sort of maze, depending on the tetrahedron to define the six allowable turns of the paths. Each possible maze is entirely known to the ninja as he plans his route.

After much thought, I think he can always escape. The guards can pin him down to one of three paths, but they can only get it so that those three paths are in a triangle, (with them at each corner) and therefore can only check two of them at once; the ninja is in the unchecked path and escapes.

If the guards could pin the ninja down to three paths that all met at a single point then they could get him, but that seems to be impossible as far as I can tell.

Originally posted by iamatiger After much thought, I think he can always escape. The guards can pin him down to one of three paths, but they can only get it so that those three paths are in a triangle, (with them at each corner) and therefore can only check two of them at once; the ninja is in the unchecked path and escapes.

If the guards could pin the ninja down to three paths that ...[text shortened]... at a single point then they could get him, but that seems to be impossible as far as I can tell.

I'm trying to think how the guards could use their speed advantage to assist them. With a "slight" speed advantage, I don't really see how that helps, but was if the speed advantage was more extreme?

In this puzzle, being a ninja simply means that the guards can't see her. No throwing stars, kuzari-gama, or wall-climbing.. and the slight speed difference simply means that if the ninja goes along some closed path and a guard chases her, say A-B-C-A-.. forever, the guard will eventually catch her.

Originally posted by talzamir In this puzzle, being a ninja simply means that the guards can't see her. No throwing stars, kuzari-gama, or wall-climbing.. and the slight speed difference simply means that if the ninja goes along some closed path and a guard chases her, say A-B-C-A-.. forever, the guard will eventually catch her.

4 guards and it's game over, but I really don't see how it's solvable with 3.

I don't see any circuits that you can form w/ 2 guards.

Originally posted by forkedknight I'm trying to think how the guards could use their speed advantage to assist them. With a "slight" speed advantage, I don't really see how that helps, but was if the speed advantage was more extreme?

Why would the speed advantage need to be more extreme? He only limits the amount of time you have to finite. I would think that if the guards knew that the ninja could predict their layout then they could catch him with an advantage in speed because they could then predict the ninja and minimize the time it takes them to change their strategy continually limiting the ninja untill they had him in a small enough area in a corridor where they covered all possible exits.

eidt: if you can leave one corridor unchecked then you would know the ninja is taking this corridor... even if you didn't a perfect plan would assume the ninja is taking the unchecked corridor and you could then leave the next unchecked corridor one that is not connected to the previous corridor or some such and you need not stop moving because with a speed advantage you do not need to see the ninja until you bump into him.

edit 2: basically you need not worry about the checked corridors because if the ninja is dumb enough to take one then you will bump into him... but maybe this could only work if you knew how much faster you were than the ninja?

Let the intersections be A, B, C, and D. There are passages leading from each to all others.

Time 1: Guard1 A -> B Guard2 D->B Guard3 A->D
Time 2: Guard1 B -> C Guard2 B->D Guard3 D->C
Time 3: Guard1 C -> A Guard2 D->A Guard3 C->D
Time 4: Guard1 A -> B Guard2 A->D Guard3 D->B
Time 5: Guard1 B -> C Guard2 D->C Guard3 B->D
Time 6: Guard1 C -> A Guard2 C->D Guard3 D->A

That strategy eventually traps any ninja starting in corridors A-B or D-B by forcing her to the path A-B-C where guard 1 will eventually catch her. Not sure if it suffices for other starting locations.