1996 OIM Problems/Problem 5
Problem
Three pieces 
, 
and 
are located one at each vertex of an equilateral triangle of side 
. The triangle has been divided into equilateral triangles with side 1, as shown in the figure in the case n=3.
![\[Triangles\]](http://latex.artofproblemsolving.com/5/a/1/5a1b9abaab9ae01ef853b3bfecaf06dc07fbdcd5.png)
Initially all the lines of the figure are painted blue. The pieces move along the lines, painting their path red, according to the following two rules:
i. First, moves, then , then , then and so on in turns. On each turn each checker travels exactly one side of a triangle from one end to the other.
ii. No piece can travel along one side of a triangle that is already painted red; but it can rest on a painted end, even if there is already another piece waiting there for its turn.
Show that for all integers 
it is possible to paint all the sides of the triangles red.
~translated into English by Tomas Diaz. ~orders@tomasdiaz.com
Solution
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