# Fitting rectangles into square (optimal/perfect rectangle packing)

I gave the puzzle you can see on the image below to a friend of mine for christmas last year. I thought it would be fun to dump it out in front of him so he would not know the solution. Unfortunately I did not write down the solution myself.

Until now multiple people have put countless hours into this puzzle and still nobody has managed to solve it. So please help me!

The puzzle is complete and definitely solvable, the tiles were all inside the frame when I bought it.

I have also started implementing a program to brute force the solution, but I thought I would post here as well.

I have taken pixel dimensions for all the tiles from the image (not 100% accurate, but should suffice):

• Large wrench: 164x655
• Blank: 164x234
• Pliers: 491x422
• Hammers: 750x234
• Pipe wrenches: 422x491
• Small screwdriver: 327x94
• Tenon saws: 327x491
• Screw wrench: 258x750
• Screwdriver: 140x655
• Hacksaw: 491x327
• Carpenter's rule: 397x327
• Wrenches: 327x655

I hope I chose the right terms for the tools, English is not my native language.

Does anybody have any tips for solving puzzles like this? If you think you can solve it, I would love to see your attempt!

Cheers and happy puzzling!

• Can we have the dimensions of the pieces? – TroyAndAbed Jun 18 '15 at 20:31
• @TroyAndAbed I only have the pixel dimensions I took from the image. I'll add them to the question. – muenchdo Jun 18 '15 at 20:34
• Any chance you can measure the actual pieces? I expect that most of them are a multiple of the same size. Or at least measure the pixels of the inside of the frame? – Bobson Jun 18 '15 at 20:50
• @Bobson Sorry, I don't have the actual pieces here. For the measurements I have taken into account that e.g. the hacksaw and tenon saws are of the same size. Also, since the puzzle is laser-cut, I am afraid the dimensions might be completely arbitrary. I'll ask my friend to get the dimensions, might take a while though. – muenchdo Jun 18 '15 at 20:55
• This looks like an incarnation of the Calibron 12-Block Puzzle. The internet is full of pages of people who talk about how hard it is and how they haven't solved it, but solutions seem pretty sparse. This page claims to have some solutions in Python, but I haven't checked them out: puzzles.bostonpython.com/blockparty.html (It also lists the dimensions of the blocks, for those following along at home...) – GentlePurpleRain Jun 18 '15 at 22:00

After GentlePurpleRain gave the hint that this is a variation of the Calibron 12-Block puzzle I did some research and I'm pretty sure I have found a (or even the only) solution. Since the original puzzle is laser-cut and therefore fits quite precisely into the frame I will still have to verify with my friend if the solution is actually correct. However, with my DIY print-out version of the puzzle it works!

Thanks to everyone who put time into this!

• So, was this the solution? :) – Rubio May 24 '18 at 3:55

## This is not an answer - just reorganizing the information

Small screwdriver     94 x 327
Screwdriver          140 x 655
Blank                164 x 234
Large wrench         164 x 655 (the one in the top-left corner)
Hammers              234 x 750
Screw wrench         258 x 750 (the really big one on the right side)
Carpenter's rule     327 x 397
Tenon saws           327 x 491
Hacksaw              327 x 491
Wrenches             327 x 655 (the set of many on the left)
Pliers               422 x 491
Pipe wrenches        422 x 491 (the pair in the middle)


By working with these numbers, I get a width of 1309-1311 and a height of 1310-1311, depending on where I measure.

It looks like we can work with a unit of $1 \approx 165$ pixels and get some reasonable numbers.

Using that, and rounding each pixel value to one decimal point of new units produces the new grid:

Small screwdriver     94 x 327  ->  0.6 x 2.0
Screwdriver          140 x 655  ->  0.8 x 4.0
Blank                164 x 234  ->  1.0 x 1.4
Large wrench         164 x 655  ->  1.0 x 4.0
Hammers              234 x 750  ->  1.4 x 4.5
Screw wrench         258 x 750  ->  1.6 x 4.5
Carpenter's rule     327 x 397  ->  2.0 x 2.4
Tenon saws           327 x 491  ->  2.0 x 3.0
Hacksaw              327 x 491  ->  2.0 x 3.0
Wrenches             327 x 655  ->  2.0 x 4.0
Pliers               422 x 491  ->  2.6 x 3.0
Pipe wrenches        422 x 491  ->  2.6 x 3.0


As you can see, most of these numbers work out surprisingly cleanly. This also produces a grid of 8.0 x 8.0 using the previous measurements.

The actual dimensions of the original Calibron 12-block puzzle (published in 1933 by Theodore Edison, son of Thomas) were apparently a little tricky to work out. A 56x56 version was recreated by Pavel Curtis in 2010, and in 2014 Jean-Claude Constantin sold a 40x40 version, Toolbox, which appears to be your version.

The 40x40 version appears to be the original intended dimensions, as you can add a 1x20, a 2x10 or a 4x5 block to tile a 36x45 rectangle (non-uniquely in each case), and this appears to have been part of the original design. I don't believe this works with the 56x56 version.

Both versions with dimensions shown here. The tiling in both versions is unique as you suggest. The 56x56 version is not simply a scaled version of the 40x40, the aspect ratio of some rectangles is significantly different.