Explanation
Since obviously the order of objects in the row is not important, any configuration of objects of three different shapes can be represented by a triple of numbers (s, c, t) where s is the number of squares, c the number of circles, and t is the number of triangles.
- Move with elimination
Given a configuration (s, c, t). A move may result in one of the three possible configurations: (s - 1, c - 1, t + 1), (s - 1, c + 1, t - 1), or (s + 1, c - 1, t - 1), depending on what shapes have been selected. From here it follows that, modulo 2, the differences of any two of the numbers s,c, or t remain invariant under all eligible moves. A necessary condition for getting a single triangle is s = c (mod 2). Is it sufficient?
- Move without elimination
As before, but now a move may convert (s, c, t) into one of (s - 1, c - 1, t + 2), (s - 1, c + 2, t - 1), or (s + 2, c - 1, t - 1). Therefore, the difference of any two numbers remain invariant modulo 3. In this case, a necessary condition to get a row of triangles is s = c (mod 3). Is it a sufficient condition?
The latter problem appears in a different guise in [Cofman, p. 97] with a reference to Kvant (1985). Chameleons on an island come in three colors. They wonder and meet in pairs. When two chameleons of different colors meet, they both change to the third color. Given initial amounts of the lizards of each color are 13, 15, and 17, may this happen that, after a while, all of them acquire the same color? [Tao, p. 83], too, discusses this variant with a reference the Tournaments of Towns competition (1989). We consider two solutions to that problem elsewhere.
References
- J. Cofman, What To Solve?, Oxford Science Publications, 1996.
- T. Tao, Solving Mathematical Problems, Oxford University Press
