There are $n$ fleas on an infinite sheet of triangulated paper. Initially the fleas are in different small triangles, all of which are inside some equilateral triangle consisting of $n^{2}$ small triangles (see Figure 1 for a possible initial configuration with $n=5$). Once a second each flea jumps from its triangle to one of the three small triangles as indicated in the figure. For which positive integers $n$ does there exist an initial configuration such that after a finite number of jumps all the $n$ fleas can meet in a single small triangle?

![](https://cdn.mathpix.com/cropped/2024_04_17_e0eeca76de2a70e8352dg-3.jpg?height=571&width=751&top_left_y=2076&top_left_x=664)

Figure 1
The small triangles can be coloured in four colours as shown in Figure 2. Then each flea can only reach triangles of a single colour. Moreover, number the horizontal rows as in Figure 2, and note that with each move a flea jumps from a triangle in an even-numbered row to a triangle in an odd-numbered row, or vice versa. Hence, if all the fleas are to meet in one small triangle, then they must initially be located in triangles of the same colour and in rows of the same parity. On the other hand, if these conditions are met, then the fleas can end up all in some designated triangle (of the right colour and parity). When a flea reaches this triangle, it can jump back and forth between the designated triangle and one of its neighbours until the other fleas arrive.

It remains to find the values of \( n \) for which the big triangle contains at least \( n \) small triangles of one colour, in rows of the same parity. For any odd \( n \) there are at least \( 1 + 2 + \cdots + \frac{n+1}{2} = \frac{1}{8} \left( n^2 + 4n + 3 \right) \geq n \) such triangles. For even \( n \geq 6 \) we also have at least \( 1 + 2 + \cdots + \frac{n}{2} = \frac{1}{8} \left( n^2 + 2n \right) \geq n \) triangles of the required kind. Finally, it is easy to check that for \( n = 2 \) and \( n = 4 \) the necessary set of small triangles cannot be found.

Hence it is possible for the fleas to meet in one small triangle for all \( n \) except 2 and 4.

![](https://cdn.mathpix.com/cropped/2024_04_17_e0eeca76de2a70e8352dg-4.jpg?height=371&width=448&top_left_y=894&top_left_x=818)

Figure 2