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Caboodle is of the same genre of puzzle as the classics, the 15 Puzzle
or Rubik's Cube, although a lot closer to the former than the latter
in complexity.

#### Lemma 1:

#### Lemma 2:

#### Theorem:

Colors of any two spaces can be swapped, leaving everything else in place.
#### Proof:

Figures 1-a and 1-b show how to use a triangle

(Lemma 1a).

When the desired pair isn't on a line, Figure 2 shows how to use a common intermediary to do 3 swaps (9 moves), leaving blue and red interchanged.

The intermediary/ies need not be adjacent to the elements of the pair, but can be anywhere in line with them.

Figure 3 shows the same thing using two intermediaries,

and, again, 9 moves:

You want to swap blue with red.

In fact, it is in one sense simpler than the 15 Puzzle, because while in that other case half the configurations are unreachable, in Caboodle, any configuration is reachable from any other.

We demonstrate that below.

For any two spaces, there is another space which

is "in line" (possibly separately) with each of them.

There are two cases:

a)

the two spaces are also in line with each other, forming a "Caboodle
triangle".

Caboodle triangles can be flat (Fig 1-a) or normal (Fig 1-b).

The corners of the triangle don't have to be adjacent,

they can have one or two disks between them, forming

sides of length 3 or 4 as well.

b)

they are not on one of the lines. (Fig 2, Fig 3)

By Lemma 1, the empty space can be moved anywhere in one or two steps.

If the empty space lines up with the desired space, just move it there in one step.

If it is not, find some suitable intermediary, and use two steps.

- If necessary, using Lemma 2, move the empty space so that Lemma 1a or 1b applies.

- Then swap the desired pair using one of these procedures.
- Reverse Step 1 to put the empty space back where it was.

Figures 1-a and 1-b show how to use a triangle

(Lemma 1a).

When the desired pair isn't on a line, Figure 2 shows how to use a common intermediary to do 3 swaps (9 moves), leaving blue and red interchanged.

The intermediary/ies need not be adjacent to the elements of the pair, but can be anywhere in line with them.

Figure 3 shows the same thing using two intermediaries,

and, again, 9 moves:

You want to swap blue with red.

Step 1: Find a space which lines up with both of them (green).

Step 2: Swap blue and green using 'x' as the intermediary. (3 moves)

Step 3: Swap blue and red using 'z' as the intermediary. (3 moves)

Step 4: Swap green and red using 'x' again. (3 moves)

Note that you must have the empty space at 'x' or 'z' when needed,
you might have to make more moves to get it there,
which you then reverse at the end to get it back where it was
before you started.

You usually have several choices for the intermediary besides the one shown as green above (Fig 4-a), with various choices then for space that can be used for the swaps. (marked by * in Fig 4-b). Choose one where the empty space is already in a usable place to speed things up.

You usually have several choices for the intermediary besides the one shown as green above (Fig 4-a), with various choices then for space that can be used for the swaps. (marked by * in Fig 4-b). Choose one where the empty space is already in a usable place to speed things up.

Now, all the above would work if all the 36 disks were distinct.

The fact that there 6 x 6 colors just makes it easier.