# How long can you survive at the devil's playground?

The devil has trapped you in his playground.

The devil knows that you can't cross over the burning boundary of his circle, so he allows you to choose a position within the circle before he starts to chase you down. You know that

• You and the devil move at speeds $$V$$ and $$1$$ respectively.
• Both move simultaneously and continuously, in any choice of direction.
• Radius of the circle $$R=1$$.
• The devil leaves an uncrossable burning track along his trajectory:

You're caught by the devil if the distance between you is $$0$$. The devil will try to catch you as quickly as possible. You know that an angel is en route to save you, so you move to survive for as long as possible.

Question 1: How long can you manage to survive if $$V=1$$? How should you move?

Question 2: Suppose now you move twice as fast as the devil, i.e. $$V=2$$. How long can you manage to survive?

Question 3: As your speed $$V$$ approaches infinity, how long can you manage to survive?

Hint

Notice that you can survive for at least $$T=2$$ by choosing to stay at the opposite side of the devil. On the other hand, you can't survive indefinitely no matter how fast you move, because the devil can carve the disk into patches of exponentially decreasing areas with you inside, shrinking that area to $$0$$ in finite time.

• But what is the devil's strategy? While he can move in any direction, which would he choose at any time? A deterministic devil strategy seems important for anyone else to develop a counter-strategy. Sep 5 at 15:29
• @bobble The devil tries to catch you as quickly as possible. He's current optimal velocity depends on your relative positions and your current velocity.
– Eric
Sep 5 at 15:36
• Regarding the hint: if I'm understanding correctly, I don't think it's correct. Can the devil really shrink the area to 0? The area can certainly approach 0, but the line of fire has no width. Sep 5 at 18:08
• @Alira That is essentially like Zeno's paradox. The fact that each subdivision takes the devil a proportionally smaller amount of time means that this is a supertask. In my opinion this is a supertask that does make sense and can be completed so that a zero area is reached in a finite time, but opinions can differ. Sep 6 at 6:45
• @Alira, I was confused at first, but check out this answer for a similar dilemma. Sep 6 at 7:39

If the devil always chases straight after me, following my movements tropistically, then

it seems like I should be able to always stay one step ahead of him at any speed, and lead him on a convoluted meditation-labyrinth-style path almost indefinitely.

But if he's smarter than that,

he'll probably ignore my location altogether and just keep hemming me into smaller and smaller patches of the disk, as the hint suggests.

In that case, my instinct would be to cower on the other side of the circle (or circle portion) from him (i.e., first near point 1, then 2, then 3, then 4, etc. -- where he aims each time to cut the portion I'm in again in half), and dodge to one side or the other at the last moment. (I use the word "cower" even though the question says I move continuously -- I'm assuming little jittery back-and-forth movements would amount to the same thing as staying still for all intents and purposes. [Also the hint uses the word "stay".)

But I must be missing something, because variations in my speed wouldn't seem to matter to this solution, as long as I can move at least as fast as him.

• Of course you are missing something big. D does not have to follow the halving algorithm, so you cannot assume that in escaping. Nov 3 at 20:53