Star Lord

[puzzlereview]

Hello guys, this puzzle is created by me, I hope you like it. If something is unclear about it, let me know.

Star Lord has landed on a deserted planet with one space policeman. He is moving around the planet, painting a line along his path, claiming any land which is surrounded by paint (the part containing less unclaimed area). The policeman is trying to restrict the total land claimed by Star Lord as much as possible. If he encounters him, Star Lord gets arrested and can not continue painting anymore. Can you prove that the policeman has a strategy, which prevents Star Lord from claiming more than 50% of the planet's surface? If there are two policemen on the planet, can you prove that they have a strategy, which prevents Star Lord from claiming more than 25% of the surface?

Remark: We assume that Star Lord and the policemen are moving with the same speed, take decisions in real time and are fully aware of everybody's locations. Their initial positions are arbitrary and the planet is a perfect sphere.

Recently I started developing a website about various puzzles - www.puzzlereview.com, you can check it out if you are interested. I keep updating it regularly and will be happy to hear some fresh ideas.

[Molly Mae]

Clarification question: Does the paint merely need to cross any previous paint, or must it intersect the beginning of the paint?

[puzzlereview]

4 hours ago, Molly Mae said:

Clarification question: Does the paint merely need to cross any previous paint, or must it intersect the beginning of the paint?

It can intersect the previous paint, at the end you can end up with several disjoint painted areas.

[DejMar]

What is the divisibility rule?

Spoiler

Except for the unclarified manner in how disjointed areas are to be divided up, the policeman has no need to chase down Star Lord in order to restrict the ravager in claiming over 50% of any planet. The largest area that could be claimed in a non-disjointed division is 50%.

 

[CaptainEd]

@Dejmar, If I understand it right, a stationary policeman will do no good. The Star Lord can circumnavigate, grabbing 50%. Then, the Star Lord can turn 90 degrees and cut off an additional quarter, then turn 135 degrees and cut off an additional eighth, and so on. 

Here's my guaranteed bad answer (because it, too, allows SL to get more than half).

Spoiler

Policeman's strategy is to always move directly toward the SL. SL's strategy could be to move directly away P. Once SL arrives at original starting point, he changes course by an acute angle (to be calculated based on the ratio of the circumference of the planet and the distance between SL and P). This allows P to get closer to SL, but if SL has calculated it properly, SL can hit the other side of the circle before P captures him. 

Unless I calculate the appropriate angle, I can't say whether SL is better off choosing smaller angles and thereby allowing himself several wedges, or picking the largest angle all at once that prevents capture.

As I understand it, P restricts SL not only by threat of capture, but by driving SL into SL's already-captured territory. This doesn't stop SL, but at least he isn't capturing additional territory. But, since they have the same speed, is it even possible to prevent him from moving to the other side of his already captured territory and adding to it? 

This a tough puzzle. 

[CaptainEd]

Better answers:

1 Policeman 

Spoiler

When starting, P imagines a mirror bisecting the planet exactly halfway between P and SL. Consider that plane to define the Equator, with SL's hemisphere the Northern Hemisphere. Ps strategy is to mirror SLs move: stay on the same longitude and opposite latitude. In other words, if SL moves toward the equator, so does P. If SL moves West, so does P.

The result is, SL can never touch the equator, because P would capture, so SL can never surround as much as half the planet. Interesting thing is, the game may never end, but SL can't win.

2 Policemen

Spoiler

We need two orthogonal planes, an Equator and a Prime Meridian. Each Policeman establishes an Equator; which should they choose? I think the Policeman who starts farther from SL should define the Equator, and follow the 1 Policeman strategy. Let's call the hemisphere that contains SL the Northern Hemisphere. Policeman 2 should move directly North or South to reach the same latitude as SL. At that moment, draw the plane bisecting the planet between SL and P2, it is orthogonal to the Equator; call it the Prime Meridian. P2's strategy is to mirror SL against the Prime Meridian; if SL moves North, P2 moves North; if SL moves West, P2 moves the same amount East.

SL cannot approach the Prime Meridian or the international date line, or P2 will capture. So, from that moment onward, SL is trapped in one quadrant of the planet The question I haven't resolved is: can SL's movement before this moment result in area captured outside of the quadrant?

 

 

[puzzlereview]

CaptainEd, your solution for 1 policeman is 100% correct. For two policemen I am not sure this strategy would work though. My solution is a bit different, following similar type of reasoning.

Edited April 25, 2016 by puzzlereview

[CaptainEd]

Yes, I expect my 2 Policeman strategy to fail. For example, suppose they landed on a spherical Earth. StarLord could start by walking around a circle enclosing 1/2 sq mile, then walk to another quadrant (while P2 is hurrying to get to SL's latitude). Once SL is trapped in that quadrant, walking one inch from the equator and Prime Meridian (thereby giving up a little less than 1/2 square mile), SL can capture slightly more than the quarter in total. i'm floundering with finding a 2 P strategy: The 1 P strategy is applicable and effective as soon as the game begins, so SL has no time to grab something outside the trap, but my 2 P strategy takes time before the quadrant can be nailed down. 

[CaptainEd]

The solution given on your website is fascinating. I learned something basic about spheres today! Thanks!

Edited April 28, 2016 by CaptainEd
my ignorance of geometry, had to remove my objection

[puzzlereview]

45 minutes ago, CaptainEd said:

The solution given on your website is fascinating. I learned something basic about spheres today! Thanks!

Glad you liked it, you are welcome!