You’re told of a curious island:

There are:

  • 13 red chameleons
  • 15 green chameleons
  • 17 blue chameleons

Rule: When two chameleons of different colors meet, they both change to the third color.

Question: Is it possible—through repeated pairwise meetings—for all chameleons to eventually become the same color?

Step 1: Model the Rule

When two chameleons of different colors meet:

  • (Red, Green) → both become Blue
  • (Green, Blue) → both become Red
  • (Blue, Red) → both become Green

Each meeting reduces two chameleons of different colors and adds two of the third.

So the total number of chameleons remains constant: 13 + 15 + 17 = 45.

Step 2: Look for an Invariant

We look for a quantity that doesn’t change—an invariant.

Let \( (R, G, B) \) denote the number of red, green, and blue chameleons.

Key idea: Examine differences mod 3

Let’s track:

  • \( G - R \mod 3 \)
  • \( B - G \mod 3 \)

Let’s calculate the initial state:

  • \( G - R = 15 - 13 = 2 \mod 3 \)
  • \( B - G = 17 - 15 = 2 \mod 3 \)

Let’s try some examples of a color change:

Suppose Red and Green meet → both become Blue:

  • R ↓ by 1
  • G ↓ by 1
  • B ↑ by 2

So:

  • \( G - R \) is unchanged (both ↓1 → difference unchanged)
  • \( B - G \) increases by 3 → same mod 3

So both differences mod 3 remain constant.

Step 3: Use the Invariant

Currently:

  • \( G - R \equiv 2 \mod 3 \)
  • \( B - G \equiv 2 \mod 3 \)

If all chameleons were to become the same color (say all red), then:

  • G = 0, B = 0, R = 45 → \( G - R = -45 \equiv 0 \mod 3 \), contradiction.

In fact, for any same-color state:

  • The differences G−R and B−G would be zero → \( \equiv 0 \mod 3 \)

But our current values are \( \equiv 2 \mod 3 \), and they never change.

Final Answer

No, it’s not possible for all chameleons to become the same color.
The values \( G - R \mod 3 \) and \( B - G \mod 3 \) are invariant, and they don’t equal 0 initially—so total unification is impossible.

Reference