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Posers and Puzzles

Posers and Puzzles

  1. 17 Aug '06 09:26
    Planet X is a ball with centre O.

    Three spaceships A, B, C land at random positions on the surface of Planet X, these positions being independent and uniformly distributed.

    Once landed, a pair A, B of spaceships can communicate provided that the angle AOB is less than 90 degrees. Similarly for B, C and C, A.

    What is the probability that all three spaceships can stay in touch (with, for example, A communicating with C via B if necessary)?
  2. 17 Aug '06 09:47
    Zero, if they are uniformly distributed. I think. Wouldn't any two make an angle of 120 from center?
  3. Standard member XanthosNZ
    Cancerous Bus Crash
    17 Aug '06 10:07
    Originally posted by smomofo
    Zero, if they are uniformly distributed. I think. Wouldn't any two make an angle of 120 from center?
    In stats a uniform distribution is a distribution where all possibilities are equally possible. So A is just as likely to land at a1 as a2 no matter where a1 and a2 are on Planet X.
  4. 17 Aug '06 10:17
    Originally posted by XanthosNZ
    In stats a uniform distribution is a distribution where all possibilities are equally possible. So A is just as likely to land at a1 as a2 no matter where a1 and a2 are on Planet X.
    Well then I change my answer to a probability of 0.125. How 'bout that?
  5. 17 Aug '06 21:46
    Originally posted by SPMars
    Planet X is a ball with centre O.

    Three spaceships A, B, C land at random positions on the surface of Planet X, these positions being independent and uniformly distributed.

    Once landed, a pair A, B of spaceships can communicate provided that the angle AOB is less than 90 degrees. Similarly for B, C and C, A.

    What is the probability that all three spaceships can stay in touch (with, for example, A communicating with C via B if necessary)?
    Would I be right in thinking that if just *two* ships were involved the probability would be 0.5?

    My reasoning being that one ship landing would define a hemisphere in which communication was possible - with the ship at the pole. The other ship would then have a 50/50 chance of landing in the same hemisphere.

    I'm not sure this helps with the final answer. Intriguing problem though.
  6. 17 Aug '06 22:54
    Originally posted by howardbradley
    Would I be right in thinking that if just *two* ships were involved the probability would be 0.5?

    My reasoning being that one ship landing would define a hemisphere in which communication was possible - with the ship at the pole. The other ship would then have a 50/50 chance of landing in the same hemisphere.

    I'm not sure this helps with the final answer. Intriguing problem though.
    Yes that's right for two ships: without loss of generality we can suppose ship A has landed at the north pole. Then there is probability 1/2 that ship B lands in the northern hemisphere, and that's the region where the two can communicate.
  7. 17 Aug '06 22:58
    Originally posted by smomofo
    Well then I change my answer to a probability of 0.125. How 'bout that?
    Alas, it is not 1/8.

    XanthosNZ is right. Here 'uniform' means that each of the ships is equally likely to land in one place as any other. Also, 'independent' means the place where a given ship lands is not affected by where the other ships have landed.
  8. Standard member XanthosNZ
    Cancerous Bus Crash
    18 Aug '06 00:08
    In approaching this problem you may find it easier to look at the case of a circle instead of a sphere first. Think of the two cases (B lands connected to A and B lands unconnected to A) as seperate and work from there.

    I hope this helps someone.
  9. 18 Aug '06 02:12
    Ugh. This going to require integration isn't it? It's been waaay too long for that. Plus, I never dealt with polar coordinates well, and I suspect this might be a problem where they are appropriate. I was on to the whole 0.5 thing, at least!

    My new answer is: I don't know. For me, it would be fastest to do it by experiment.
  10. 18 Aug '06 09:30
    Originally posted by XanthosNZ
    In approaching this problem you may find it easier to look at the case of a circle instead of a sphere first. Think of the two cases (B lands connected to A and B lands unconnected to A) as seperate and work from there.

    I hope this helps someone.
    Certainly that is a good idea, and would get you a feel for the sphere case. And the integration is easier over a circle.
  11. 18 Aug '06 09:36 / 2 edits
    Originally posted by smomofo
    Ugh. This going to require integration isn't it? It's been waaay too long for that. Plus, I never dealt with polar coordinates well, and I suspect this might be a problem where they are appropriate. I was on to the whole 0.5 thing, at least!

    My new answer is: I don't know. For me, it would be fastest to do it by experiment.
    Yep integration is one way. And spherical polar coordinates are probably helpful.

    On the other hand, we all know the surface area of a unit sphere is 4*pi, right (we choose units where the radius of Planet X is 1).

    And we know the area of a 'lune' (intersection of two hemispheres) is 2*alpha where alpha is the angle (in radians) between the two great circles forming the lune.

    Perhaps this will help...?