Some notation: I use tuples \( (r, g, b) \) to represent the numbers
of chameleons of each colour at any given time. The transformation
\( (r', g', b') \to (r, g, b) \) denotes that the number of red,
blue, and green chameleons changed from \( (r', g', b') \) to \( (r,
g, b). \) Then I work backwards in time. So, if there are \( (r,
g, b) \) chameleons now, the previous transformation must have been
one of the following:

\( (r - 2x, g + x, b + x) \to (r, g, b) \)

\( (r + x, g - 2x, b + x) \to (r, g, b) \)

\( (r + x, g + x, b - 2x) \to (r, g, b) \)

Let \( d \) be the number of blue chameleons when the zoologists
first arrived.

We know that when the zoologists arrived for the first time, they
observed \( (2000, 3000, d). \) We also know that when they arrived
for the second time, they observed \( (5000 + d, 0, 0). \) The
question is: what are the possible values of \( d? \) I answer this
in two parts.

Part 1.The integer \( d \) must be a multiple
of 3.

Proof. Let us look at how each number in the tuple change
modulo 3 for each possible transformation. For example, without
loss of generality, consider the transformation \( (r - 2x, g + x, b
+ 2x) \to (r, g, b). \) Note that \( -2x \equiv x \pmod{3}. \) So
every transformation just adds the same number modulo 3 to each
number in the tuple.

We know that we start with \( (2000, 3000, d) \) and reach \( (5000
+ d, 0, 0). \) Since the values of green and blue are congruent
modulo 3 in the end (both are 0), therefore their values must be
congruent modulo 3 at the start. Hence \( d \equiv 0 \pmod{3}. \)

Part 2.If \( d \) is any multiple of 3 we can
find a sequence of transformations to go from \( (2000, 3000, d) \)
to \( (5000 + d, 0, 0). \)

Proof. Consider \( (r, g, b) \) where \( r \ge 1 \) and \(
g \ge 3. \) We can get
\[
(r, g, b) \to (r - 1, g - 1, b + 2) \to (r + 3, g - 3, b).
\]
This means that we can keep taking 3 from the value of green and
keep adding it to the value of red. So if we start with \( (2000,
3000, d, \) we can can get \( (5000, 0, d). \)

Similarly for \( r \ge 1 \) and \( b \ge 3, \) we can get
\[
(r, g, b) \to (r - 1, g + 2, b - 1) \to (r + 3, g, b - 3).
\]
So we can keep taking 3 from the value of blue and keep adding it to
the value of red. If \( d \) is a multiple of 3, we can use this to
go from \( (5000, 0, d) \) to \( (5000 + d, 0, 0). \)

## Joshua Tobin said:

Some notation: I use tuples \( (r, g, b) \) to represent the numbers of chameleons of each colour at any given time. The transformation \( (r', g', b') \to (r, g, b) \) denotes that the number of red, blue, and green chameleons changed from \( (r', g', b') \) to \( (r, g, b). \) Then I work backwards in time. So, if there are \( (r, g, b) \) chameleons now, the previous transformation must have been one of the following:

Let \( d \) be the number of blue chameleons when the zoologists first arrived.

We know that when the zoologists arrived for the first time, they observed \( (2000, 3000, d). \) We also know that when they arrived for the second time, they observed \( (5000 + d, 0, 0). \) The question is: what are the possible values of \( d? \) I answer this in two parts.

Part 1.The integer \( d \) must be a multiple of 3.Proof.Let us look at how each number in the tuple change modulo 3 for each possible transformation. For example, without loss of generality, consider the transformation \( (r - 2x, g + x, b + 2x) \to (r, g, b). \) Note that \( -2x \equiv x \pmod{3}. \) So every transformation just adds the same number modulo 3 to each number in the tuple.We know that we start with \( (2000, 3000, d) \) and reach \( (5000 + d, 0, 0). \) Since the values of green and blue are congruent modulo 3 in the end (both are 0), therefore their values must be congruent modulo 3 at the start. Hence \( d \equiv 0 \pmod{3}. \)

Part 2.If \( d \) is any multiple of 3 we can find a sequence of transformations to go from \( (2000, 3000, d) \) to \( (5000 + d, 0, 0). \)Proof. Consider \( (r, g, b) \) where \( r \ge 1 \) and \( g \ge 3. \) We can get \[ (r, g, b) \to (r - 1, g - 1, b + 2) \to (r + 3, g - 3, b). \] This means that we can keep taking 3 from the value of green and keep adding it to the value of red. So if we start with \( (2000, 3000, d, \) we can can get \( (5000, 0, d). \)Similarly for \( r \ge 1 \) and \( b \ge 3, \) we can get \[ (r, g, b) \to (r - 1, g + 2, b - 1) \to (r + 3, g, b - 3). \] So we can keep taking 3 from the value of blue and keep adding it to the value of red. If \( d \) is a multiple of 3, we can use this to go from \( (5000, 0, d) \) to \( (5000 + d, 0, 0). \)