Three spiders and a fly are moving along the edges of a regular tetrahedron. The three spiders want to catch the fly. Problem: The fly is invisible, hyper intelligent and psychic.
The rules are as follows:
1. The spiders have to choose a strategy beforehand, i.e. where they start and a fixed path that each will follow.
2. The fly (being psychic) knows the spiders' strategy and will choose it's starting point after the spiders, and (being hyper intelligent) will always choose the best path to avoid the spiders.
3. The spiders are minimally faster than the fly. Say, when the spiders have moved along 1 edge, the fly could have moved only 999/1000 of an edge.
4. The spiders have no way to "see" the fly (being invisible), but if a spider runs over the fly, it is caught.
Is there a way the spiders can eventually catch the fly for sure? If so, give a strategy/path. If not, prove that they can't.
The rules are as follows:
1. The spiders have to choose a strategy beforehand, i.e. where they start and a fixed path that each will follow.
2. The fly (being psychic) knows the spiders' strategy and will choose it's starting point after the spiders, and (being hyper intelligent) will always choose the best path to avoid the spiders.
3. The spiders are minimally faster than the fly. Say, when the spiders have moved along 1 edge, the fly could have moved only 999/1000 of an edge.
4. The spiders have no way to "see" the fly (being invisible), but if a spider runs over the fly, it is caught.
Is there a way the spiders can eventually catch the fly for sure? If so, give a strategy/path. If not, prove that they can't.
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Re: Three spiders and a fly
Fri, June 9, 2006 - 5:08 AMNo one seems to have made any progress on this one.
I know that if one spider would be able to catch the fly on just one triangle simply by going round the triangle one way.
With multiple triangles, there are junctions and so the fly can escape by taking the path at the junction that the spider will not take. This may be worthwhile doing even if there is another spider in the path coming towards the fly. The fly can occupy the path just long enough for the first spider to pass then exit the path the at the end it entered.
Karl
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Four spiders and a fly Re: Three spiders and a fly
Fri, June 9, 2006 - 6:36 AMI've found a way that four spiders can catch the fly within the time it takes a spider to travel four edges of the tetrahedron.
Karl -
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Re: Four spiders and a fly (answer) Re: Three spiders and a fly
Tue, June 13, 2006 - 5:05 AMHere is how four spiders could catch the fly
Three spiders start on one vertex and the fourth anywhere.
The three spiders go along the three different edges leading from their vertex to the vertex at the other end. The fourth spider meets one of the other three at a vertex.
Now either the fly has been caught or is confined to one of the three edges between the spiders
The fourth spider then traverses each of these three edges in turn and catches the fly on one of them.
All this takes place in the time it takes a spider to traverse four edges.
Karl
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The Spiders are getting hungry!
Mon, June 19, 2006 - 4:55 AMThe spiders are getting hungry.
Can you tell them how to catch the fly
or why thay are doomed to starve?
Karl
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A Spider has died!
Fri, June 30, 2006 - 6:00 AMOne of the three spiders on the tetrahedron has died of starvation and the other two are not very well. Can you get use three spiders to replace the two surviving spiders on the tetrahedron. Then the two surviving spiders can retire on a fly abundant triangle.
Karl -
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Re: A Spider has died!
Fri, July 7, 2006 - 5:05 AMI found a spider to replace to one who has died. The other two spiders are half-starved so can only go at half speed. They can catch the fly.
Karl
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Re: Three spiders and a fly
Thu, July 6, 2006 - 5:26 AM -
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Re: Three spiders and a fly
Fri, July 7, 2006 - 4:59 AMI did figure out that the spiders could catch the fly, if two of the spiders were able to force the fly to go round the tetrahedron on a fixed route. The third spider would then chase and eventually catch up with the fly. But how can you confine the fly to one route with just two spiders?
The web link has a solution hidden in it. The spiders follow the above strategy, but allow the fly to leave the route forcing it back onto the route where it left the route, exactly when the chasing spider gets there.
Karl
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