🚀 go-pugleaf

In article <1993Jan9.030824.13624@Csli.Stanford.EDU> hiraga@Csli.Stanford.EDU (Yuzuru Hiraga) writes:
>A plane is totally colored with two colors.
>* Show that there exist three points of same color that
>  (when connected) form an equilateral triangle.

This was shown by numerous persons so I will not discuss it.

>* How about one with a 1cm (no, not inch :-) edge?

Try the following coloring of the plane:  Set w = sqrt(3)/2.  For each
point P=(x,y), find the integer n such that w*n <= y < w*(n+1).  Color
P red if n is even and blue if n is odd.

This way, the entire plane is colored in alternating w-cm-wide red
and blue strips, with the border between two strips colored the same
as the strip above.

Now let T be any triangle of side 1 cm with all vertices the same
color, WLOG they are all red.  The "highest" and "lowest" vertices
must differ by at least w in y-coordinates, so they cannot lie in the
same strip but must lie in consecutive red strips.  But then if the
third vertex lies in either red strip it must have a y-coordinate
difference greater than w from one of the first two points, which
cannot occur in any orientation of the triangle.  So the third point
is blue, a contradiction.

Ergo, the answer is "no" if the side of the triangle must be 1 cm.

-- David Karr (karr@cs.cornell.edu)