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Two of your sister's former friends have gotten themselves lost in a storage room in some abandoned building. While your girlfriend managed to get one of them out, the other one apparently got turned around in the maze and is still lost

The lost individual happens to be a fugitive, and you've decided to head over to the building to capture them. As you reach the storage room in question, you notice a small map scribbled on the wall, which looks like this:

Maze Map

Luckily, it seems the fugitive didn't see the map, as she can only say that she's sitting at a T-junction

The fugitive, from what you've seen of them, seems rather unintelligent, so you feel you can anticipate her behaviour: Specifically, when you move the fugitive will run to the path closest to going directly away from your current position (based on her immediate surrounding when she starts running) and follow the path until she reaches another junction. She should be able to pinpoint where you are in the maze based on your footsteps, but you cannot determine where she is, where she started, or how far she travels. Based on previous experiences, you know that the fugitive can run 50% faster than you, and you also believe they are unlikely to stop or slow down to think. If the fugitive makes it to the exit, she will be able to escape and run away

Assuming these rules, is there a strategy to guarantee that you can catch her before she can escape?

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  • $\begingroup$ Are there any restrictions on how fast/slow she can move relative to "me"? And can she change speed? $\endgroup$
    – fljx
    Commented Jul 7, 2023 at 15:47
  • $\begingroup$ (I believe I have solutions for any constant speed, and for any variable speed above a known minimum. But not for the case where she can move faster than me some of the time, but can also choose to move arbitrarily slowly sometimes.) $\endgroup$
    – fljx
    Commented Jul 7, 2023 at 15:50
  • $\begingroup$ @fljx I edited the puzzle $\endgroup$ Commented Jul 7, 2023 at 16:09
  • $\begingroup$ Does 'directly away from you' take into account your own direction? Because otherwise, that could result in ambiguity $\endgroup$ Commented Jul 13, 2023 at 17:19
  • $\begingroup$ @newQOpenWid no $\endgroup$ Commented Jul 13, 2023 at 17:40

1 Answer 1

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Is there a strategy?

Yes

As long as the fugitive really does keep going until the next junction, even if you are standing there. If she turns round when she sees you, she can always run away faster than you can chase, and will never be caught.

Why/how?

Start by loitering around the maze entrance (walking on the spot if necessary to ensure that the fugitive knows where you are and moves away).

Wherever she starts, she will move in the directions of the arrows, and eventually end up moving back-and forth or round in circles on one of the marked paths.
(The dark blue arrows and loop are not actually possible, as they start from crossroads, and we know that the fugitive starts from a T-junction. I marked them up before realising they are redundant.) enter image description here

Now start moving west towards the junction on the red loop (A).

If the fugitive in on the light blue, red or pink paths, she will stay on that path.
If she is on the orange path. At some point when you get far enough west, she will start going east when she cannot go north. She will then hit the pink arrows and end up on the pink loop.

When you get to A, wait long enough for the fugitive to travel the entire red loop.
If she turns up, you catch her.
If she doesn't appear, you know she must be on the light blue or pink paths.

Head back east towards the entrance. Whichever path the fugitive is on, she will stay there.
Now start heading north up the eastern edge of the maze toward D.

If the fugitive in on the pink path, she will stay there.

If she is on the light blue path things get a bit more complicated. As you move north from the entrance, she will start preferring west to north. She will at some point move from her original path to the new one in the north-west corner.
enter image description here
When you get to B, she could go north or south from the westermost junction (both are equally "away"), but when you get to C you know that she must go south to the new purple path.
Then when get to D, there is again a junction with two equally "away" directions, so she might stay on the purple path, or head south to the grey path.

In either case, when you get to D you must wait long enough for the fugitive to travel the entire pink loop.
If she turns up, you catch her.
If she doesn't appear, you know she must have started on the light blue path, and is now on the purple or grey.

Now, move west one square from D and wait. This ensures that if the fugitive was on the purple path, she will (when she reaches its eastern end) head south to the grey path.

Retrace your steps to the entrance. Once you have passed B the fugitive will prefer going west from the western end of the grey path and end up in the original red loop.

So finally, when you reach the entrance, you head back to A and wait.

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