# Jigsaw puzzle: packing pentominoes into a rectangle

I've got this jigsaw puzzle that I can't figure out. The major problem is that there are no signposts on whether a piece is in the right place. How does one get all the pieces into the 6x10 container? Is there a way to approach the puzzle other than trial and error?

• Can you post a more clear picture of the pieces, by taking a picture more from above? Dec 24, 2021 at 15:51
• @riskymysteries I don't have access to the puzzle anymore sadly (it belongs to my cousin). Which piece are you looking at? If it helps, each piece is 5 unit squares big. Dec 24, 2021 at 15:53
• Is there any indication which side goes up on all the pieces? Dec 24, 2021 at 15:57
• @riskymysteries no :( Dec 24, 2021 at 15:58
• en.wikipedia.org/wiki/… Dec 24, 2021 at 16:11

By the look of it, this puzzle comprises one of each of the 12 different pentomino shapes. In which case, an example packing can be found on the Wikipedia page for 'Pentomino', as follows:

If you would like a little history, the 6x10 case was first solved as far back as 1960 by Colin Brian Haselgrove and Jenifer Haselgrove, and there are apparently 2,339 potential solutions in total (see the previously linked Wikipedia page for more details), excluding trivial reflections and rotations of the whole 6x10 rectangle.

Considering how many possibilities exist, trial and error is probably your safest least-effort approach to finding a solution!

• There is also a nice solution consisting of two 5x6 rectangles which you can slide together to make either a 6x10 or 5x12 rectangle. Dec 24, 2021 at 16:48

I wrote a solver for this (and related polycubes) using or-tools. You can see it here https://github.com/MrBenGriffin/or-tools-fun

There are many related puzzles!

A standard pentomino puzzle is to tile a rectangular box with the pentominoes, i.e. cover it without overlap and without gaps. Each of the 12 pentominoes has an area of 5 unit squares, so the box must have an area of 60 units. Possible sizes that use just one of each pentomino are 6×10, 5×12, 4×15 and 3×20. The 6×10 case, first solved in 1960, has exactly 2339 solutions. The 5×12 box has 1010 solutions, the 4×15 box has 368 solutions, and the 3×20 box has just 2 solutions The 8×8 rectangle with a 2×2 hole in the center has 65 solutions.

Here one solution for 3x20 (using the solver)

 ╔═══╦═╦═════╦═══════╦═╦═════╦═══╦═╦═════╗
║ ╔═╝ ╚═╗   ║ ╔═══╦═╝ ╚═╗ ╔═╝ ╔═╣ ╚═══╗ ║
║ ╚═╗ ╔═╩═══╩═╩═╗ ╚═══╗ ║ ║ ╔═╝ ╚═══╗ ║ ║
╚═══╩═╩═════════╩═════╩═╩═╩═╩═══════╩═╩═╝


and here is one of the 2339 solutions (using the solver)

 ╔═════╦═════╦═╦═════╗
╠═╗ ╔═╝ ╔═══╝ ╚═╦═╗ ║
║ ║ ╠═══╬═╦═════╝ ║ ║
║ ╠═╝ ╔═╝ ╚═╦═╦═══╬═╣
║ ╠═╗ ╠═╗ ╔═╝ ║ ╔═╝ ║
║ ║ ╚═╝ ╠═╝ ╔═╝ ║   ║
╚═╩═════╩═══╩═══╩═══╝


• While this link may answer the question, it is better to include the essential parts of the answer here and provide the link for reference. Link-only answers can become invalid if the linked page changes. - From Review Dec 24, 2021 at 19:57
• @BeastlyGerbil, thanks for your timely reminder. Dec 24, 2021 at 20:57
• . . . and now you know. Dec 24, 2021 at 21:25