A sphere is painted black and white. Show that there are 3 equidistant points of the same color.

Assume not.

Take a regular icosahedron.

All 12 vertices lie on the surface of a sphere

call one of the white vertices A.  There are 5 vertices adjacent to A, call them a1,a2,a3,a4,a5 (with a1 adjacent to a2, a2 adjacent to a3, etc).

There are at least two of those that are adjacent and are black, wlog a1 and a2 (why? each triangle needs one black.  if no two adjacent are black make a1 black then a2 white then a3 black then a4 white then a5 black so two adjacvent are black)

Now, a1 and a2 are also adjacent to another vertex b1 which must be white.
call b2 the other vertex adjacent to a2 and a3
b3 the other vertex adjacent to a3 and a4
...
now, a, b1, and b3 are equidistant as are a, b1, and b4
so, b3 and b4 must be black.  Thus,  a4 is white
thus, a5 and a3 are black.  Thus, b5 and b2 are white.

but this won't work because now b (the point directly opposite a and adjacent to b1..5 must be both black and white.

so, there must be at least three equidistant points (of twelve that are vertices of a regular icoshedron) that are  of the same color.

Posted by Joel on 2007-02-13 23:48:43