Put 10 equal coins on a table to make a filled triangle with a side of 4 coins.

geometry/coins.triangle New

New entry, edited from material collected for the archive and not yet in a released version.

Problem

Put 10 equal coins on a table to make a filled triangle with a side of
4 coins.  The task is to remove the least possible number of coins to
ensure that no triple of coin centres makes an equilateral triangle.

[ Editor: posted by Pertsel Vladimir. ]

Show solution

   A      Removing three coins won't work.  There are fifteen equilateral
  B B     triangles altogether.  Removing an A kills three triangles;
 B C B    removing a B kills five and removing a C kills six.  We note that
A B B A   we have to remove an A to kill the outer triangle, but that
          leaves 12 triangles left to kill with only two removals.
   O
  . O     But it's easy to kill all the equilateral triangles by removing
 . . O    four coins.
O O O .

-- Gareth Rees <gdr11@cl.cam.ac.uk>

[ Editor: in the second diagram, O marks a coin that is kept and . a
  coin that is removed. ]