Solution

Assume the colors are A, B, C and the balls labeled A₁, A₂, B₁, B₂, C₁, C₂. We have to replace the labels with more informative A_L, A_H, B_L, B_H, C_L, C_H, where "L" stands for "light" and "H" stands for "heavy".

Weigh A₁ + B₁ against B₂ + C₁.

1. If A₁ + B₁ < B₂ + C₁ then

   • B₁ < B₂ because even if A₁ < C₁, B₁ > B₂ would lead to A₁ + B₁ = B₂ + C₁, at best. Thus B₁ = B_L, B₂ = B_H.

   • If A₁ = A_H then C₁ = C_H and

   • if C₁ = C_L then A₁ = A_L.

   With this in mind, weigh A₁ + C₁ against B₁ + B₂.

   • A₁ + C₁ = B₁ + B₂ implies A₁ = A_L and C₁ = C_H.

   • A₁ + C₁ < B₁ + B₂ implies A₁ = A_L and C₁ = C_L.

   • A₁ + C₁ > B₁ + B₂ implies A₁ = A_H and C₁ = C_H.

2. If A₁ + B₁ = B₂ + C₁ then weigh B₁ against B₂.

   • B₁ = B_L implies A₁ = A_H and C₁ = C_L.

   • B₁ = B_H implies A₁ = A_L and C₁ = C_H.

3. The case where A₁ + B₁ > B₂ + C₁ is similar to the first one.