🚀 go-pugleaf

In article <1993Jan9.030824.13624@Csli.Stanford.EDU> hiraga@Csli.Stanford.EDU (Yuzuru Hiraga) writes:
>In article <1993Jan8.193314.3289@zip.eecs.umich.edu> kanad@quip.eecs.umich.edu (Kanad Chakraborty) writes:
>>Here's a very *easy*, and nice, puzzle....
>
>Along a similar vein	# no, it's not original.
>
>A plane is totally colored with two colors.
>* Show that there exist three points of same color that
>  (when connected) form an equilateral triangle.

Proof. WLOG, let the colors be red and blue.
Choose a point A in the colored plane. Suppose, WLOG, that 
A is colored blue. Find the (or a) point colored red _nearest_ to A 
(Euclidean distance-wise). (This is always possible because both
colors have been used.) Call this point B. Suppose the
distance AB = x units.  Then any equilateral triangle with A
as one vertex and side < x units has all three vertices colored
blue, for obvious reasons.

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

If x > 1 cm in the previous proof, the above is possible. Otherwise,
I don't see right off hand a way to construct such an equilateral triangle.

>-YH


Kanad