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:

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

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

  • G = 0, B = 0, R = 45 → \( G - R = -45 \equiv 0 \pmod 3 \), which is a contradiction!

In fact, for any same-color state, two of the color counts will be exactly 0, and the third will be 45. This means the differences between any two color counts will always be 0, 45, or -45.

Notice that all of these possible differences are multiples of 3 (i.e., \( \equiv 0 \pmod 3 \)).

But our current values are \( \equiv 2 \pmod 3 \), 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