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Topic: 3 Spiders and a Psychic Fly (Read 8793 times) |
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Rezyk
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Re: 3 Spiders and a Psychic Fly
« Reply #25 on: Oct 17th, 2007, 5:07pm » |
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Yeah, your solution is much more efficient and elegant.
For your 1 spider solution, I will challenge it with this fly: At all times, it dances next to the top vertex, always moving toward or along the edge that the spider will (from that point in time) be on latest among the 3 edges of that vertex, and never straying so far that it cannot return to the vertex before the spider intends to.
The problem is equivalent to having moss covering all edges with spiders eating up the moss as they move -- but the moss regrows from its boundary points at a constant rate. A single spider can never stop the moss from extending out from each junction along at least 2 edges, no matter how fast it moves.
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Hippo
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Re: 3 Spiders and a Psychic Fly
« Reply #26 on: Oct 17th, 2007, 11:08pm » |
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Of course I think in the moss variant model.
You are not correct in the 1 spider analysis.
The spider can catch the "moss", spider's path has infinite number of steps but finite length.
It's simillar to Achilles and the turtle argument ...
Other thing would be if turning is penalised by a small amount of time. In that case there does not exist solution.
... but if the spider has nonzero size ... the solution again exists ... but who would be interested in computing such complicated scenario  ?
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| « Last Edit: Oct 18th, 2007, 4:49am by Hippo » |
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rmsgrey
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Re: 3 Spiders and a Psychic Fly
« Reply #27 on: Oct 18th, 2007, 5:49am » |
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on Oct 17th, 2007, 11:08pm, Hippo wrote:Of course I think in the moss variant model.
You are not correct in the 1 spider analysis.
The spider can catch the "moss", spider's path has infinite number of steps but finite length.
It's simillar to Achilles and the turtle argument ...
Other thing would be if turning is penalised by a small amount of time. In that case there does not exist solution.
... but if the spider has nonzero size ... the solution again exists ... but who would be interested in computing such complicated scenario ? |
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I don't see how it's possible for a lone spider to catch a fly given just three lines meeting at a point - each time the spider enters and leaves the junction point, the fly is an arbitrarily short distance down the third line, where it will take less than the spider's next return time to get back to the junction point.
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Hippo
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Re: 3 Spiders and a Psychic Fly
« Reply #28 on: Oct 18th, 2007, 6:19am » |
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For speed ratio 1:7, the spider can do trips to distances d,d/3,d/9,d/27,... from the junction point.
It's total length is 3d/2.
The fly cannot run away from the junction. Therefore fly can live only under the junction during the iterations (and at the iteration end, too).
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| « Last Edit: Oct 18th, 2007, 2:14pm by Hippo » |
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rmsgrey
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Re: 3 Spiders and a Psychic Fly
« Reply #29 on: Oct 19th, 2007, 9:04am » |
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on Oct 18th, 2007, 6:19am, Hippo wrote:For speed ratio 1:7, the spider can do trips to distances d,d/3,d/9,d/27,... from the junction point.
It's total length is 3d/2.
The fly cannot run away from the junction. Therefore fly can live only under the junction during the iterations (and at the iteration end, too). |
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Yeah, I see it now...
The next trick is trapping the fly at the junction to start with, but it's obvious that there's some speed below which the fly can't get away from one junction in the time it takes the spider to eliminate the other three and sweep the edges...
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Hippo
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Re: 3 Spiders and a Psychic Fly
« Reply #30 on: Oct 30th, 2007, 9:36am » |
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As noone replies here are my results may be it can be improved:
1 spider:Should be at least 7times faster than fly. Trick after cleaning a vertex is to make a "loop" around to clean the opposite sides of edges. This defines the ratio 7
2 spiders:Should be at least 4 times faster than fly. The original soulution mimicking 3 spiders (1 playing role of two guards) works well, but there is very fast solution ... start on opposite vertices of an edge, go to the same empty vertex, continue on nontraversed edge and divide to the original vertices. Now only the original edge and the 3/4 of first traversed edges can be occupied by fly. Stay with one spider and run the first traversed edge again with the other. It arrives to the vertex exactly et the time 4 times slower fly can. Therefore the fly cannot escape through the vertex. Continue with the spider to the vertex where the other waits. Now the fly can be only on original edge or on first 1/2 of edges leading from vertex where the running spider started. Go with both spiders to the opposite vertex of the original edge and divide them to go the last two edges where the fly cen resist in first 3/4. A spider will catch the fly till the time it enters the vertex.
3,4 spiders ... already discussed.
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temporary
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Re: 3 Spiders and a Psychic Fly
« Reply #31 on: Jan 23rd, 2008, 8:34pm » |
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Need verification:size of surface they are on and difference of speed. In general, fly would have advantage, but it may vary.
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Hippo
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Re: 3 Spiders and a Psychic Fly
« Reply #32 on: Jan 23rd, 2008, 10:58pm » |
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on Jan 23rd, 2008, 8:34pm, temporary wrote:| Need verification:size of surface they are on and difference of speed. In general, fly would have advantage, but it may vary. |
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Sorry, I do not understand your post.
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temporary
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Re: 3 Spiders and a Psychic Fly
« Reply #33 on: Jan 24th, 2008, 9:17pm » |
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I do not understand what there is not to understand. I am asking for specific verification.
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Hippo
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Re: 3 Spiders and a Psychic Fly
« Reply #34 on: Jan 25th, 2008, 2:43am » |
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OK, sorry: Now I understand:
If spider moves with speed s times fly speed, the difference in speed is (s-1) times fly speed.
Length of the tetrahedron edge is 1 unit.
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temporary
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Re: 3 Spiders and a Psychic Fly
« Reply #35 on: Jan 25th, 2008, 6:52am » |
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In that case, if the fly's speed is less than 1, he is safe, exactly 1 I'm unsure of, and greater than 1 he is caught.
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Hippo
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Re: 3 Spiders and a Psychic Fly
« Reply #36 on: Jan 25th, 2008, 10:00am » |
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on Jan 25th, 2008, 6:52am, temporary wrote:| In that case, if the fly's speed is less than 1, he is safe, exactly 1 I'm unsure of, and greater than 1 he is caught. |
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   I would expect it depends on ratio s more than on the speed of fly, but it seems you are sure the speed of spider is not important.
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temporary
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Re: 3 Spiders and a Psychic Fly
« Reply #37 on: Jan 25th, 2008, 5:58pm » |
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Actually, I am not sure. For some reason I had just plugged in the fly's speed as
"s". No cuss word intended.
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