I had a bit of spare time late Sunday afternoon so I thought I would separate out from my four years worth of IPP Exchange Puzzles those 3D packing ones that I thought would make a good article featuring “impossible” 3D packing puzzles. Not because they are impossible to solve; physically or otherwise, obviously they can of course…but such puzzles which on first glance looks so impossibly difficult that you won’t even know where to start. In this small list of mine, such puzzles would be those that have an extraordinary number of pieces to fill a box…which makes it seem humanly impossible!

I narrowed the list down to about four to five puzzles and as I was going through each of them, I decided to give Formula a try. A couple of the others came in the solved state already but my Formula came unassembled. A large number of pieces were outside of its container and loosely strewn about inside the cardboard packaging, while the rest were jumbled up inside the wooden cube.

First off, Formula is Tim Udall’s IPP37 Exchange Puzzle. I had played with one of his other exchange puzzles previously, the **Cubic Burr**. Formula was made by **Vinco Obsivac** out of a mixture of different woods not identified. Very well made and finished with all the pieces fitting nicely with pretty tight tolerances. The puzzle has an incredible 27 pieces of varying shapes and sizes (as you can see from the photo)! The goal is to place all 27 pieces flush inside the box with none sticking out. Scary for the uninitiated! The 27 pieces come in ten different sizes, with 7 of them consisting of more than one piece. Size wise, its about 6cm all cube all round.

Formula was designed by a Mr. Justin Math (I have never heard of him tho’ in the puzzle community, but with a name like that, he must obviously be good at math or a good designer or both). Strangely, aside from the usual information listed on the packaging about the puzzle, there is that extra bit; * “pieces: (a+b=c)3 = 27 bricks”*. Not being a maths kind of guy myself, I didn’t quite get what the formula meant and/or its relationship with the puzzle or solution. If anyone can shed some light on what this formula means, please PM me, thanks!

**Update 29 Jan 2018** – Paco Molina, a puzzler from Spain has offered the following explanation and analysis of the formula above-

*“(a+b+c)^3 = a^3+b^3+c^3+3a^2b+3a^2c+3b^2a+3b^2c+3c^2a+3c^2b+6abc*

*The formula describes how many pieces are for each type (and their size). **And I guess the solution can be obtained by placing the pieces that represents a^ 3, b ^ 3 and c ^ 3 diagonally (bottom-up right-left and forward-back).* The solution could be drawn on a piece of paper before solving it or at least part of it and figure the rest. (Well, I guess, because I do not have the puzzle). It is related to this one that I posted here (L’s Tri-Ls from Vinco)** ****http://puzzlesab.blogspot.com.es/2014/02/ls-tri-ls.html**“

Typically of such 3D packing puzzles, I usually try to figure if there was any logical or systematic way to find the solution, without all that mathematics mumbo-jumbo. My first random placing of the pieces inside the box to check out how the pieces would fit resulted in the third photo here. The last small piece sticking out a bit ie wrong solution!

Logic (and my puzzling experience) would indicate that the largest pieces should be at the bottom and allow the smaller pieces to fill the gaps in the middle and towards the top. So I tried this method and surprise surprise, my approach actually worked! On the third attempt to fill the cube, everything went in nicely. A nice a-ha moment for me to end the weekend. But in all that I also realised that I needed to keep a number of pieces with similar dimensions to fill the final layer on top. So my solve I guess was combination of a bit of luck, logic and experience. But more than that, I also think that the puzzle has multiple solutions; otherwise I would not have been able to solve it so quickly!

Bill Cutler has done some work on the number of solutions. He used a lot of computer time on it and found more than 9 billion solutions!

Thanks Tim….with so many possible solutions, no wonder I managed to solve it.

I assembled the puzzle in 3 layers of height a, b, then c. The final assembly is then a large cube sliced by 6 planes, 2 in each direction. Very nice!