Arranging Your Balls - Pyradox

So, how many different ways are there to arrange your balls.  It turns out that it’s the same as the number of licks it takes to get to the tootsie roll center of a tootsie pop as demonstrated by Mr. Owl – Three.

The summer of 2018 was a time of many fine things, the foremost being the Edward Hordern Puzzle Exchange at IPP 38 in San Diego.  Amongst the many enticing entries was Pyradox exchanged by George Bell, designed by George Bell, made by George Bell, and packaged by George Bell.   I’m specifically calling out the packaging since it’s nice and compact and well laid out within a transparent box.

Packing balls in tight spaces is a niche area of spatial mathematics and ball packing master George Bell has written several articles concerning this topic.  CFF Issue 94, July 2014, includes George’s article on the development and analysis behind Pyradox – Pyradox: A Pyramid Packing Puzzle.  The article also describes how George paradoxically accomplished the impossible when creating this puzzle.

Pyradox consists of 5 plane looking pieces and 3 coasters.  The objective is to fulfill your prayers by fully filling the holey coasters with a pyramid.  Each polysphere piece consists of 4 wooden balls glued together with colored bands.  Each of the 3 colors used, red, blue, and green, identifies a unique piece shape.  On careful inspection, you will notice that one of the 3 types of shapes is different, which may be useful when solving the puzzle.  The 3 base plates are made from laser cut wood with Pyradox engraved on each and I don’t really recommend that you use them as coasters.

Each base plate provides a different pyramid building challenge.  To make it easier for you, one utilizes hexagonal close packing, one a face-centered cubic packing, and the last a warped face-centered cubic packing.  You’re welcome!

George did an amazing job discovering how the same 5 pieces can be used to construct 3 different pyramids with different packing geometries.  I found each to be fun, non-trivial, and not too difficult.  Copies are occasionally available on George’s PolyPuzzles Etsy shop.  Get one and have yourself a ball.

Some Puzzles Give You The Shutters – W-Windows

Provided: Box with 2 large windows and 3 escaped pieces.  Your Mission (should you choose to accept it): Contain the pieces and shutter the box.  That’s shUtter the box, not shAtter the box, no matter how tempted you may be mid-solve.  As always, should you or any or your puzzling buds be caught shAttering the box, we will disavow any knowledge of your actions.

W-Windows is an apparent cube packing puzzle designed by Osanori Yamamoto and made by Pelikan Puzzles.  In this case the box has 2 large 2x2 windows that need to be shuttered (i.e., filled) and 2 of the 3 zig-zaggy pieces have a W theme going on.  The box is made from Apple and the pieces are made from Ovangkol.  Usually, I don’t talk about the types of woods used to make puzzles, but since I had to look up Ovangkol in my wood book (Wood! Identifying and Using Hundreds of Woods Worldwide by Eric Meier from the online The Wood Database), it’s worth mentioning that it’s another name for Shedua.  However, Ovangkol is the title for the wood description and Shedua is only mentioned in the comments.  Apparently Ovangkol is used by guitar makers who have better lobbyists than puzzle craftsmen.

After having done many of these types of puzzles and looking at the pieces for W-Windows, I expected:

• All pieces will be used to fill the windows.
• The W pieces will be added with some variation of an insert-shift-insert movement.  Sorry to state the obvious
• One of the Ws has a 2x2 face, which obviously fills one of the box windows.  It’s so obvious that it can’t possibly be part of the solution.  Or could it?
• The tetracube piece will be in the center somewhere moving around to allow the W pieces to slowly emerge from the box.
• Each W piece will be associated with its own window.
• The tetracube piece will have the highest move count.

I freely share these expectations with you because they are of no help whatsoever when solving the puzzle.

When I first sat down with the puzzle, I took the approach of looking for an assembly and then determining whether it could be placed in the box.  I learned 2 important things from this approach: 1) There are too many assemblies, and 2) There are a lot of ways that the pieces cannot be oriented within the box.  It’s definitely worth the time to determine how the pieces can and can’t be oriented within the box so that you can quickly recognize a possible assembly from an impossible one.

When looking for the solution, I try to find an assembly that looks like it has a few good moves.  These types of puzzles usually don’t have deep false paths, and once you find something that looks interesting, it’s frequently the solution, which was the case with W-Windows for me.

It may be the eternal optimist in me, perhaps the stubbornness, or more likely the short-mindedness, but I looked at the 2 big windows of the box and the 3 measly pieces and thought that this would be a quick score.  However, those simple pieces kept me entertained for about an hour and I did enjoy the solution once I found it.  You really can’t go wrong with these apparent cube packing puzzles from Osanori Yamamoto.  Thankfully, Osanori creates new ones faster than I can acquire and solve them.

Puzzle Philosophy - Yin Yang

I like puzzles but they aggravate me.  When I’m working on a puzzle, I can’t wait to find the solution.  When I find the solution, I’m disappointed that it’s over.  I expected all these reoccurring conflicting emotions to surface when I pulled (pushed?) Yin Yang out to work on.

Yin Yang was developed by Volker Latussek and made by Pelikan Puzzles.  The box was made using Cherry and has a Maple (Yang) top to contrast nicely with the 6 Wenge (Yin) pieces.  When all the pieces are packed in the box, you end up with a nice digitized taijitu symbol.

Upon inspection, 4 of the Yin pieces are symmetric and the other 2 are not.  These asymmetric pieces are the key Yin and Yang pieces.  I know that I stated that the Wenge pieces were the Yin to the box’s Yang, but even when you separate them, you still have both in each.  Yeah, it doesn’t make a lot of sense, so let’s just agree to call it a principle and keep building on it.   Now that we have our key Yin Yang Yin pieces, as long as they aren't symmetrically situated within the assembly, these mystical keys can be used to affect a transformation between Yin and Yang assemblies.  You’ll also notice that extra effort went into highlighting that each piece was constructed by combining a 2x3 block with a 1x2 block.  I suppose one of those would be the Yin and the other the Yang requiring the key pieces to be formally referred to as the Yin Yang Yin Yang Yin pieces.

Yin and Yang

Since the Yang box encompasses a 4x4x3 space, the objective is to 1) Find a way to make a 4x4x3 shape with the Yin pieces and 2) discover a method for jamming those Yins into the Yang.  And the answers to all your questions are Yes – Are rotations allowed/required, Are there uncrammable assemblies, If I Yin instead of Yang will the pieces get stuck?

I’m embarrassed to admit it took longer than expected to find a 4x4x3 assembly.  Of course once I found it, it had to be the required assembly and I was never going to let it go.  I could see how the first 3 pieces would come out and only had to figure out how to release the remaining 3.  Surely, taking out (Yang?) / putting in (Yin?) 3 pieces with all that space couldn’t be that hard (With all my experience with 3D apparent packing cubes, it’s a wonder I can still think that).  Needless to say, I spent a considerable amount of time experimenting with some amazing cramming techniques with those obstinate pieces until I remembered the principle of the whole thing and immediately solved it.