Geometry Puzzle: Crush the Rebellion

Source: CMU Puzzle Toad

Problem:

Ten rebel encampments have sprung up on the plane of Usyan. The Martian Federation plans to send flying saucers to deal with them. They are pretty ruthless. They will simply land on top of the encampments. The encampments are small and the saucers are huge. It must be done simultaneously, or the rebels will flee. Also, the saucers must not overlap when they land. Can the Martians prevail?

Mathematically, the encampments are points in the plane and the saucers are non-overlapping disks of equal radius.

So, The problem is 
Given Radius r > 0 of circle, is it possible to arrange 10 points on the plane such that no number of non-overlapping circles of radius r would be such that all 10 points lie in a circle.

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Comments

  1. Stephen Jones8:54 PM, November 12, 2012

    I'm not sure you've defined your problem carefully enough. As stated, there's no limit to the radius of the circles (just that the be equal) so one circle is always sufficient to cover all 10 points, assuming that the Martians know the placement of all points. Is there a limit to the number of circles?

    Likewise, you did not constrain the geometry of the plane such as max dimensions or whether circle centers must lie inside the rectangular field, etc.

    OK, so the problem is only interesting if you assume that there's a limit to the size of the circles and that size is smaller than the field. Also, the aliens must know the locations of the points. Otherwise, it becomes a probability problem.

    If all those assumptions are true, then the answer is for the colonists to place points at corners and the centers of isosceles triangles whose sides are 2R where R is the radius of the saucers. That should allow some colonies to hide between adjacent saucers.

    ReplyDelete
    Replies
    1. Pratik Poddar5:30 PM, February 04, 2013

      Thanks for your comment Stephen.

      Regarding the assumptions:
      1) Its a standard problem. I have not made it up.
      2) "Given R", means that the radius is given, and its finite.
      3) If not given, plane is assumed to be 2-dimensional, I have been told.
      4) Since I am arranging the points first, its assumed that aliens would know the locations of the points.

      Regarding the solution:
      Please provide a formal proof that your construction will work. Thanks

      Delete
  2. JDGM10:26 AM, November 28, 2012

    This comment has been removed by the author.

    ReplyDelete
  3. JDGM8:14 PM, December 05, 2012

    It looks like this.

    A formal written proof accompany the picture but it's a lot of words for something intuitively gleaned from staring at the diagram and visualising the circles.

    ReplyDelete
    Replies
    1. Pratik Poddar5:24 PM, February 04, 2013

      can you please give the outline of a formal proof. Thanks a ton

      Delete

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