# What approach can I use to solve this wooden packing puzzle?

I am looking for an approach to solve a wooden packing puzzle which my three-year-old got as a present. We enthusiastically unpacked and disassembled the puzzle when she got it. (It came put-together in the correct configuration) Since then, we've been unable to put all the pieces back together. Occasionally I get it out and attempt to solve it again but fail, although it seems trivial at first. It is frustrating.

This is a polyomino puzzle with twelve pieces. Every piece is painted the same on the front and the back. Thus, they can be turned around (double-sided). The box is a square with twelve cells to the side. It turns out we misplaced a piece of size six (so that the pieces completely fill the area).

Pictured are:

• Elephant (gray): 21
• Ape (violet): 19
• Polar bear (white): 11
• Meerkat (yellow): 13
• Snake (light green): 7
• Chameleon (dark green): 12
• Macaw (red): 7
• Shark (light blue): 5
• Toucan (dark violet): 4
• Zebra (black/white): 10
• Leopard (orange): 14
• Octopus (pink): 15
• Unknown 2×3: 6

(After having found a solution with @JaapScherphuis' help, I remembered that there used to be a 2×3 sized piece. I don't remember (yet) what animal it was.)

I am unable to find the solution online. I found visually and conceptually very similar products, but none with the same shapes.

At this point, I would even take the solution so that I can put all the pieces back together. However, I'd most appreciate an explanation of how to go about solving such a puzzle.

Edit: We found the missing piece. It is a 3×2 rectangle and painted as a penguin. See the spoiler below for the solved puzzle, including the missing piece.

• "There is no other way of filling that space." I think the pieces have a total of 138 cells, which is 6 short of the 144 cells of the board. So the space in the chameleon need not be filled completely, and the head of the octopus or the yellow piece could be sufficient. Commented Oct 8, 2022 at 8:28
• @JaapScherphuis, that is amazing. I hadn't even noticed that. I just assumed that the pieces would fill the entire area. Thanks for pointing that out. (It's really weird, though, isn't it?) Edit: Or maybe we misplaced a piece of size 6. Commented Oct 8, 2022 at 8:58
• When I counted & got 138 Squares ( out of 144 Squares ) , I thought that either [[1]] there were holes in the Puzzle or [[2]] I had miscounted or [[3]] all Squares are not Same Size, where the minor changes add up to 6 Squares. @JaapScherphuis
– Prem
Commented Oct 8, 2022 at 9:03
• You said your three-year-old daughter got this as a present. The animals are cute toys for a three-year-old but the puzzle as such is clearly grownup difficulty level, as you noticed yourself. Commented Oct 9, 2022 at 7:42
• Yippie, we found the missing piece. It is indeed a 3 times 2 rectangle and painted as a penguin. Commented May 21, 2023 at 17:12

This is a rather difficult puzzle. Not only are the pieces double-sided, their total area is only 138, six short of the base area of 144. This means that there can be a few gaps, and you can no longer assume every nook and cranny of the pieces needs to be filled.

General advice for packing puzzles is to leave the smaller pieces last whenever possible, because they have a bigger chance of fitting into whatever gap you have left over near the end.

I used a computer to find solutions, and there are remarkably few. Before giving away a solution, I'll provide a few hints.

Hint 1:

The gaps are all bunched together at the final corner, so you can solve the majority of the puzzle as if there are no gaps at all.

Hint 2:

The space inside the chameleon can be completely filled without using the toucan.

Hint 3:

The elephant and the meerkat fit together.

Here is a solution:

As the gaps are all together, you can move the toucan and shark pieces around, and flip over the octopus, to give many minor variations of the same solution.

• Thanks a ton. I am going to reveal only one hint at a time and try step by step. I'll come back with feedback later. "This is a rather difficult puzzle." Thank you for this confirmation. I am relieved. Do you mind me asking what tool you used for finding the solution? I've entered it in cemulate.github.io/polyomino-solver, and it has been running for a while without a solution yet. Commented Oct 8, 2022 at 10:38
• @justfortherec I used my own solver, and just now it finished searching the whole space. It took 2 hours and 20 minutes, and found the 30 solutions (all variations of the one above) after 95 minutes. Commented Oct 8, 2022 at 10:43
• So, the first and second hint together were sufficient for me to quickly find a mirrored solution of the one you also gave. Thank you so much. Now I only need to find the missing piece (because I am fairly certain that there was no gap when we got the puzzle). Commented Oct 8, 2022 at 10:46
• @justfortherec Ah, that makes sense given that the solution has all the extra space bunched together. Commented Oct 8, 2022 at 11:48
• The white goose on the 3rd link of similar games would be just perfect to complete the board. Commented Oct 8, 2022 at 16:56

General strategy

• look at the most difficult piece(s) first, large irregular pieces are typically difficult
• avoid using the easier pieces as long as possible
• create and extend corners, then combine corners

In this case,

The chameleon is a very difficult piece. the toucan is very easy.

If the toucan is not to be used, the zebra is needed to fill the chameleon's 'interior'
This leaves a small opening where apart from the toucan, only the snake fits.

(some) potential corners combinations:

Chameleon, zebra, snake can use the shark to form a corner

The leopard can be combined with either the octopus or bear to form a corner. The latter option allows the gorilla to be added to extend the corner further.

The remaining big pieces: The elephant can form a corner on its own and combines well with the meerkat as corner extension.

combining corners:

The elephant + meerkat corner is a perfect neighbor for the leopard + bear + ape corner as well as the chameleon + zebra + snake+ shark corner.

The rest (starting with the octopus) is now easily filled

Note that it was lucky that the toucan was not needed early

Not sure if anyone has already noticed, but in the complete solution provided above, if the dark blue/violet piece (toucan) in the lower right hand corner is rotated 90 degrees clockwise, and placed along the bottom edge in the lower right corner, this would provide perfectly for the missing 2x3 piece. ( Perhaps, an oval shaped turtle painted on that missing 2x3 piece would be nice ... )

With puzzles that you have put together where some of the pieces are complicated and other pieces are simple, I try to place the complicated pieces first and save the simple ones for the end. In your puzzle, the two simple pieces are the dark blue one at the top (looks like an L shape with a bird design on it) and the light blue one underneath it (looks like a T shape with a shark design on it). Good luck!

• Thanks for the advice. This is more or less what I tried already. The issue is: Even after starting this way, I quickly end up with way too many possible combinations. Also, I found out that the toucan (what you call dark blue L shape) must go into the gap of the dark green chameleon (which is in the top left corner of the box). Commented Oct 8, 2022 at 8:17
• As you have discovered now, the toucan didn't have to go in the gap of the chameleon. The fact that you though it should is probably why you didn't make any progress with the puzzle. Commented Oct 10, 2022 at 19:39