# Combinatorial puzzle concerning labelled equilateral triangles

Consider equilateral triangles $\Delta$ of fixed size and in a fixed position with each side labelled by a label $l \in \{1,\dots,k\}$. Obviously there are $k^3$ such labelled triangles.

Let $\operatorname{refl}(\Delta)$ be the labelled triangle which results from the labelled triangle $\Delta$ by reflecting it at its vertical symmetry axis.

Let $\operatorname{rot}_\alpha(\Delta)$ be the labelled triangle which results from the labelled triangle $\Delta$ by rotating it by the angle $\alpha \in \{ 0°, 120°, 240°\}$ around its center.

Now define the following natural equivalences between two such triangles $\Delta_1$ and $\Delta_2$:

1. $\Delta_1 \simeq_{\operatorname{refl}} \Delta_2\quad$ iff $\quad\Delta_1 = \Delta_2\quad \vee\quad \Delta_1 = \operatorname{refl}(\Delta_2)$

2. $\Delta_1 \simeq_{\operatorname{rot}} \Delta_2\quad$ iff $\quad(\exists \alpha)\ \Delta_1 = \operatorname{rot}_\alpha(\Delta_2)$

3. $\Delta_1 \simeq \Delta_2\quad$ iff $\quad(\exists \alpha)\ \Delta_1 = \operatorname{rot}_\alpha(\Delta_2)\quad \vee\quad \Delta_1 = \operatorname{rot}_\alpha(\operatorname{refl}(\Delta_2))$

I'd like to know

1. the size of the equivalence classes $\Delta_{\simeq_{\operatorname{refl}}}$, $\Delta_{\simeq_{\operatorname{rot}}}$, $\Delta_{\simeq}$ (with respect to $k^3$)

2. canonical representatives

• What's wrong with the question? Trivial? Home-work? Ill-posed? – Hans-Peter Stricker Dec 2 '14 at 15:46
• I don't know, mabye someone wanted his reputation to go down so he could reach a certain amount of rep, like 42 or 1729. – Jorge Fernández Hidalgo Dec 2 '14 at 16:35

Hint: For the first, if the two top sides are the same you do not have a different triangle to pair it with, so count them all. If the top two sides are different, you have another triangle to pair it with, so divide these by $2$. How many of the $k^3$ have the top two sides the same? Similarly for the second, if all three sides are the same you don't have any triangles to match it with. How many of the $k^3$ is that? Then the rest are in groups of ?????