What is the longest possible Cats & Boxes game?

Question

I have enjoyed playing a puzzle game called "Cats & Boxes" published by Smart Games.

Briefly, the way the game is played is:

1. Place $5$ cat pieces and $4$ box pieces (one of which has $2$ boxes) onto a $5\times5$ grid. The game comes with $60$ initial puzzle setups, ranging in difficulty. The cats are always unboxed in each initial configuration.
2. Move the box pieces one at a time until each cat is in a box. See the images below for an example setup and eventual solution.
3. (An edit for clarity) The box pieces are rigid, each consisting of $1$ or $2$ boxes that may or may not be placed over a cat, as well as $3$ flat "carpet" squares. The carpet squares must be placed flat on the board and cannot go onto a cat or any part of another box piece. See this video that shows an example.

Some actions are specifically not allowed.

• You may not move any of the cats.
• You may not overlap any pieces, i.e. each box piece needs to be placed flat on the grid (whether or not it holds a cat) before moving another box piece. The cats are $3$ dimensional, so the only part of a box piece that can go on a cat square is a box itself.

The minimum number of moves required to solve a puzzle is $4$, since there are $4$ box pieces. The easiest puzzles in the game can be solved in $4$ moves.

Harder puzzles require more moves. The fun of the game comes from figuring out how exactly to move the pieces around until all the cats are boxed.

The puzzle booklet tells the minimum number of moves needed to solve each puzzle. Easy puzzles can be done in about $4$ to $7$ moves at a minimum. Mid-level puzzles take about $8$ to $13$ moves at a minimum. The "hardest" puzzle in the booklet is the last one, puzzle $60$. It takes a minimum of $33$ moves to complete, as indicated in the booklet.

$33$ moves seems to me like an outlier, when compared with the rest of the puzzles. The next highest is $26$ moves, and those are the only two puzzles requiring $>20$ moves.

I expect that computers were very much involved in coming up with initial puzzle configurations, and their optimal solutions. My own programming skills are not up to the task of writing code to answer my question.

I know that many initial configurations of cats are impossible to cover with the box pieces. One restriction comes from the piece with $2$ boxes that are a chess knight's move apart. So, having all $5$ cats in a single row, column, or diagonal will be impossible to cover. I know there are other restrictions due to the shapes of the box pieces.

My specific question is the one in the title. It could be that $33$ moves is the greatest minimum possible, but I'd like to know for sure!

Answer 1

Since the cats don't move, we can start there. There are ${{25}\choose{5}}=53130$ ways to place the cats on the board. Because the entire puzzle is invariant under rotations (though not reflections), we can treat cat placements related by a rotation as equivalent. This leaves us with $13302$ (equivalence classes of) cat placements. Call this set $\mathcal{C}$.

We can also enumerate all the legal ways to place the four box pieces on the board, such that they don't overlap each other. Call this set $\mathcal{P}$. I find $|\mathcal{P}|=8020$ (which is slightly fewer than the $8256$ implied by one of the comments).

Now, each cat placement $c \in \mathcal{C}$ is consistent with some subset of the legal placements of the four box pieces, $\mathcal{P}(c) \subseteq \mathcal{P}$. Among these placements (and contingent on $c$), some may be legal start states for a puzzle (because no cats are in boxes), and some may be legal end states (because all the cats are in boxes) -- call these sets $S(c)$ and $E(c)$, respectively. We can find $S(c)$ and $E(c)$ for each $c$, and then eliminate cat placements for which either set is empty. This leaves us with $556$ cat placements $c$ that might be used for solvable puzzles. The number of potentially solvable puzzles (again, modulo rotations of the cat placements) is $\sum_c |S(c)| = 3048$.

Finally, for each potentially solvable puzzle $\langle c, p_s \in S(c)\rangle$, we form an undirected graph whose nodes are $\mathcal{P}(c)$, and where there is an edge $\langle q, q' \rangle$ whenever box piece placements $q$ and $q'$ differ by exactly one box piece's location. (Note that these graphs are all subgraphs of a much larger graph whose nodes are all of $\mathcal{P}$.) We then find the minimum graph distance from $p_s$ to $E(c)$ in the usual way. (If no node in $E(c)$ is reachable from $p_s$, then in fact this puzzle is not solvable.)

Carrying out this procedure finds $1727$ solvable puzzles, with difficulty (= minimum path length) ranging from $4$ to $33$. A single puzzle has the maximum difficulty of $33$, and a single puzzle has the next-highest difficulty of $26$. There are only six puzzles with difficulty greater than $20$, so the choice to include only two of them may just have been a design preference.

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Update:

Here is a link to a file containing all 1727 solvable puzzles. The first column is the initial layout (print the five strings in order to get an ASCII representation -- '#' is a cat, and the four box pieces are 'A' through 'D'); the second column is the difficulty; and the third column lists the shortest distances to all reachable end states from the given start state. In particular, the first entry in the file, which corresponds to the most difficult puzzle, is:

('#BbB-', '-D#B#', 'DdaA-', '#DCAa', 'CCcA#') 33 [33]

So the ASCII representation of the puzzle's initial layout is:

#BbB-
-D#B#
DdaA-
#DCAa
CCcA#

This corresponds exactly to the card image posted in the question.