01
Fix one colour without losing the other cases
Begin by assigning red to A. Since B is connected to A, B must be blue. Since C is connected to B, C must then be red if only those two colours are allowed. Now inspect the remaining edge from C to A: both endpoints are red, so that edge violates the rule. The first choice of red does not hide a solution starting with blue. Swapping the two colour names would produce the same pattern of equality and difference around the three edges, with the same final conflict. Those are the two possible initial colour choices.
02
State precisely what the contradiction shows
The argument establishes that this three-node cycle has no valid assignment using only the two stated colours. It does not say that every connected diagram needs three colours. A chain A to B to C, with no edge from C back to A, accepts red, blue, red because its two edges have different-coloured endpoints. Comparing the cycle with that chain identifies the relationship that creates the conflict. Keep the missing edge visible in the comparison so the conclusion is tied to the actual structure rather than to the number of nodes alone.
03
Change the palette and verify every edge
Allow a third colour, green, and assign A red, B blue and C green. Check A–B, B–C and C–A individually: each pair differs, so the assignment is valid. The worksheet can finish with fields for available colours, node assignments and edge checks. This makes a solution reviewable without requiring a reader to trust a colourful picture. The two-colour impossibility and the three-colour example answer related but different questions; keep both results rather than replacing the failed attempt with the successful drawing.