Wednesday, June 29, 2022

Arranging Your Balls - Pyradox

Pyradox by George Bell
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 Packaging
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.


  1. Don Knuth looked into this puzzle and even found a piece which can build FOUR different pyramids, the fourth one being the 7x2 "roof". I had thought this was impossible, but the piece he found is non-planar, which I hadn't considered. He mentions this puzzle in his latest volume of The Art of Computer Programming.

    1. Thank you for the update George! Perhaps we'll see a Don Knuth inspired exchange puzzle in the future.

    2. For the Knuth work, the Wiki shows a multi-volume work with some complete and the rest is planned. Which chapter, please? And do I need this in my library?
      For the puzzle, I can also attest that the three challenges are fun to solve. It is a good one to leave on the coffee table, for guests.

    3. You want The Art of Computer Programming, Volume 4, Fascicle 5: Mathematical Preliminaries Redux; Introduction to Backtracking; Dancing Links. It has Dancing Links in the title, the algorithm used by Burr Tools.

    4. I was afraid that the post would discourage people from leaving the puzzle on a coffee table.

  2. Actually, all are subsets of face-centered cubic, surprisingly!