### Geometry Puzzle: Crush the Rebellion

**Source:**CMU Puzzle Toad

**Problem:**

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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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?

ReplyDeleteLikewise, 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.

Thanks for your comment Stephen.

DeleteRegarding 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

This comment has been removed by the author.

ReplyDeleteIt looks like this.

ReplyDeleteA 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.

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

Delete