Portland, Oregon

I corresponded with Martin Gardner from 1968 to 2002 mainly about Langford's Problem (below).


I attended G4G10 (my first Gathering 4 Gardner conference), where I presented a some interesting facts about Geom-e-Trees and met some very interesting people who have MG interests in common.


My son Gus and I attended G4G11 in 2014. I presented a 12-minute update on Langford's Problem and a 6 minute demonstration of PolygonJazz (below).


I did not attend G4G12.


I attended G4G13. My wait-listed talk was 13 Parallels between Martin Gardner and Stan Freberg.

G4G10 - What Shape is a Tree? (2012)

This short talk was based on some of the 150 Geom-e-Trees on a poster designed by yours truly. Click for more about the Geom-e-Trees Poster.

Abstract: What shapes can geometric trees make? By varying the common ratio between levels, the angle between branches, and the branching factor, one can generate a surprising number of shapes, the simplest being regular polygons and various grids. Patterns with fractal-like margins expectedly appear. Stars appear! The seemingly solid patterns have complex inner structures. We end with a challenge for the audience.

Download PDF of What Shape is a Tree?. This six page paper was contributed to the G4G10 Gift Exchange Book.

Only audio recordings were made at G4G10.

G4G11 - Langford's Problem (2014)

In November 1967, Martin Gardner challenged readers to arrange 4 pairs of colored blocks in a certain way. He told readers that no solutions were possible with 5 or 6 pairs, but that there were 25 unique arrangements for 7 pairs (no references cited).

Early in 1968, as a freshman at Gonzaga University, I programmed Langford's Problem and found 26 (not 25!) solutions for n=7 and 150 solutions for n=8. Three or four others did likewise. E.J.Groth cracked n=11 and n=12. Martin published these results in March, 1968, thus beginning decades of correspondence as solutions for higher values of n were computed by myself and others.

Click on the colored blocks above to land on my highly-referenced page on Langford's Problem. (Sorry, no interactive gizmos, YET.)

Here is a video of my 12 minute G4G11 talk on Langford's Problem:

Langford's Problem, Remixed

Download PDF of On Langford's Problem. This paper was contributed to the G4G11 Gift Exchange Book, March 2014.

G4G11 - The Music of the Polygons (2014)

I gave a seven minute demonstration and talk about PolygonJazz. This was noted in Magic, Puzzles Delight Math Fans at G4G as: John Miller showed us how sound can be used to understand caroming billiard balls within polygons. Here is a video of my G4G11 talk:

The Music of the Polygons

G4G13 - 13 Parallels between Martin Gardner and Stan Freberg (2018)

My Paper: [PDF (draft)]. PDF does not contain the 30+ images used during my talk but does have links to some of them. [Link to video of talk here when/if available]

I also wanted to give an update on Langford's Problem, but talks were limited to one per person. The conference was sold out (n=?). I registered late, and submitted my talk proposal even later. I was on a waitlist with a dozen others. Happily I gave the last talk on Saturday, so that was fun. But it was stressful not knowing when I might be called!

My Ternary Tree Star, Version 1.618 was on display at the Different Trains gallery in Decatur during the Gathering's excursion there on Friday the 13th!

How Mathematical Games Enriched my Life

In 1961, my 7th grade math teacher, Ray LaLonde, shared Martin's June 1961 Brain Teasers column with the class. I was hooked. After years of checking out previous columns in the library, I finally subscribed to Scientific American beginning in May 1964. when I was 14 years old! While on Sabbatical 1988-89, I went back through all previous MG columns and indexed them.

Over the years, Martin wrote about:

  • philosophy, logic, paradox
  • forms of infinity
  • fairness in games
  • history of mathematics and thought
  • graph theory, finite state machines
  • geometry, parity, topology, puzzles, fractals, origami
  • magic tricks (eg. Kruskal's principle)
  • the not-so-long arm of co-incidence
  • and so on.

Martin would often combine several of these into a month's column, e.g. The Cycloid: Helen of Geometry.

I was interested in the popular culture of the early 60's, and had a very impressionable mind, so my first Dr Matrix column was practically traumatic. I thought Martin was sometimes too skeptical, but now I believe his writings helped me become a rational adult. The current anti-science movement is disturbing, with social media being used to spread ignorance. We need more voices like Martin.

A side benefit of that SciAm subscription was being exposed to the Amateur Scientist column, conducted by C.L.Strong. I was so interested in harmonography, I built the Double-Elliptical Pendulum Harmonograph.

TO be fair, other writers continued the column (under various names). In my mind, I associate some of what they covered with Martin. But it was Hofstadter that covered Rubik's Cube (Mar 81), and AK Dewdney who covered the Mandelbrot Set and attractors (Aug 85), both of which were featured on the magazine cover.

Math & Art

  • Ternary Tree Star, The story of discovering and making Ternary Tree Star (version 1.618), a 24″x24″ digital print of a 12 level ternary tree consisting of 265,720 lines.
  • Squaring the Square, was written in 2014 as a COLUMN on a prototype Celebration of Mind website.
  • The Immersive Bridge Between Math and Art, presented at Bridges in Baltimore, July 2012.
  • D-ART Digital Art Gallery, 2012 Edition. (Not pleased with their website.)
  • Stained Glass Exhibit

Immersive iPad Apps

I like to think Martin would've enjoyed these iPad apps I've published. Icons all by Gus Miller!

Geom-e-Tree App icon Geom-e-Twee App icon PolygonFlux App icon PolygonTrix App icon PolygonJazz App icon

Geom-e-Tree lets you explore the n-ary tree form immersively. You control the angle between branches, the common ratio between levels, and the number of branches all with multi-touch gestures on the display.

Geom-e-Twee is a free children's version of Tree. It only handles two or three branches and has just a few themes, but they are designed so that kids won't stall the device with wild gestures while having 10,000 nodes on a tree.

PolygonFlux lets you experience the flow of energy inside a polygon. In simplest terms, it traces the path of a point bouncing around inside a polygon of your choice, generating a very pretty pattern. You control the various parameters via multi-touch gestures.

PolygonTrix is a gameful design based on PolygonFlux. PolygonTrix rewards you with a geodesic flux for each target vertex hit. You begin with straight World Lines, then progress through world lines with ten deflections. You can switch among seven polygons (n=3..9) at any time.

PolygonJazz combines music and geometry. A ball bouncing around inside a polygon makes a sound whenever it hits a side. Or does the side make the sound when the ball hits it?

You can click on any of the icons to visit that app's commercial website.

A Mathematical Games Index

In the 70's I started an index of MG on 80-column computer punch cards, so some titles were shortened. While on Sabbatical in 1988, I completed a Subject Lines index of all the years of Mathematical Games and subsequent incarnations, such as Metamagical Themas, I made an attempt to categorize. I got Martin's permission to place my index on line. On October 23, 2014, I transferred the index to the web site, so that it would have a more permanent home:

Mathematical Games Index on

In the meantime, Wikipedia editors used the index to bootstrap a wikipedia page covering the MG columns, crediting my index as the source. They later figured out how to scrape the data directly from the publisher's website, and so wikipedia no longer referenced my earlier work. But Wikipedia doesn't have the subject categories that my page has. They do reference other wikipedia pages that cover some of the concepts and problems, and that is good. (Also complicating things, my personal site moved from to approximately during this same period!)

One final note on the index. Sci Am has since closed direct access to the MG columns, but you can see the titles of each column, one at a time. Some of their titles are wrong, and some my titles are shortened. I don't think that either table (list) is definitive, including WikiP!

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