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To solve the problem of finding the number of distinct ways to paint the three edges of a triangle with three colors (adjacent edges different, rotations same, reflections different), follow these steps:

Step 1: Understand Constraints

• Adjacent edges cannot share the same color.
• Rotations are equivalent (so we can fix one edge to eliminate rotation).
• Reflections are distinct (so mirror images count as separate).

Step 2: Count Valid Colorings Without Equivalence

For a triangle (cycle of 3 edges):

• First edge: 3 choices.
• Second edge: 2 choices (different from first).
• Third edge: 1 choice (different from first and second, since it’s adjacent to both).

Total: $3×2×1=6$ colorings, all permutations of the three colors.

Step 3: Group by Rotations

Rotations collapse permutations into two equivalence classes:

• Clockwise permutations: {RGB, GBR, BRG}
• Counterclockwise permutations: {RBG, BGR, GRB}

Step 4: Reflections Are Distinct

These two classes are reflections of each other, so they count as separate.

Answer: 2

$\boxed{2}$

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