WakeUp wrote: each flea jumps from its original triangle to one of the three small triangles having a common vertex but no common side with it. There are nine such triangles, not three. Do you only mean the three which are reflections of the original triangle w.r.t. its vertices?

From what I understand from the wording of the problem, yes, the flea can pass through vertices but not though edges, so to speak.

Paint all triangles which point north, i.e. $\Delta$, red and the other triangles blue. Note that if a flea is on a red triangle, it will jump to a blue triangle, and viceversa. So a necessary condition is to have all the fleas on triangles of the same color. Since clearly any flea can go from any triangle to any other triangle, just pick any triangle and have all fleas go to it, and once a flea reaches the triangle, make it jump back and forth. Since the number of fleas is finite, eventually all the fleas will be on the same triangle.

jestrada wrote: Note that if a flea is on a red triangle, it will jump to a blue triangle, and viceversa. So you make the assumption raised in post #2. jestrada wrote: Since clearly any flea can go from any triangle to any other triangle ... I very much doubt that.

All positive integers $n$ works. Assume the triangle grid faces up and down (that is, there are horizontal gridlines). Let the fleas be on the up-facing triangles on the bottom-most row of the triangle, and pick the top triangle in this big triangle. Now, let each flea jumps to one of the two small up-facing triangles above that doesn't go out to the top triangle. It's obvious that each up-facing triangle has at least one move available, thus all fleas can jump. Eventually they will meet at the top triangle. Notes: The wording of the problem itself is pretty vague.

That is assuming the contrary of that raised in post #2, and it's pretty trivial, so I doubt this was meant to be taken this way.

chaotic_iak wrote: The wording of the problem itself is pretty vague. Then we should agree on the wording of the problem, once and for all. In particular: 1- spanferkel wrote: WakeUp wrote: each flea jumps from its original triangle to one of the three small triangles having a common vertex but no common side with it. There are nine such triangles, not three. Do you only mean the three which are reflections of the original triangle w.r.t. its vertices? 2-The triangle with side n. Can the fleas jump to a triangle outside this triangle? 3-The problem says "For which n does there exist a configuration", not "prove for all possible configurations". But this is too trivial, right? I mean, you can pick a triangle T in the middle that has two different triangles A, B you can jump to, put some fleas in A, the rest in B and the next second they all jump to T. mavropnevma wrote: jestrada wrote: Since clearly any flea can go from any triangle to any other triangle ... I very much doubt that. Mavropnevma: Even assuming you can't jump out of the triangle, the only counterexample I can find is when n=2 - a flea on the middle can't jump anywhere. I believe for n big enough you can go from any triangle to any other triangle, even assuming "no" in question 1 and "yes" in question 2.

Just found this problem in an old thread: http://www.artofproblemsolving.com/Forum/viewtopic.php?t=27447 Apologies, I have no idea how this didn't come up when I searched for it..