Visitors to my Starmaze exhibit often asked where this puzzle came from. It's a question I asked myself and spent several years trying to answer.

I first came across this puzzle in 1979. It was included in a demo program for a speech recognition board that plugged in to an Apple II computer.

The system was fairly primitive and could really only recognize the spoken numbers from 1 to 9 (and then only on a good day). So to demo it, they included a program which put up a simple 3 x 3 grid on the screen with an asterisk (or "star") in the center cell. If you said "5" into the microphone, the pattern would change according to a set of listed rules (the cells were numbered 1 to 9 starting in the upper left corner). The challenge was to invert the pattern. The instructions did not indicate how many steps this would take or whether it was even possible.

That's what got me started, as I described on my intro page. This speech recognition card belonged to one of my psychology professors and I soon lost track of it - and haven't seen it since.

In the early '90s, after having spent more than a decade exploring the mathematical structures behind this puzzle, I began to wonder myself where the original idea really came from. Was it invented by the programmer who created this speech recognition board? Or did he get it from somewhere else? In the days before search engines and the web, this was not an easy question to answer.

I spent a lot of time looking through old puzzle books and references to computer games of that era and writing letters to book editors and game authors. I eventually discovered what I believe to be the original source of the what became my starmaze puzzle. To my surprise, it was part of a hand-held electronic game. I was unaware that any hand-held games had even been invented before 1979.

In fact, one of the very first widely-available hand-held electronic games had been introduced the year before, in 1978. It was called "Merlin" and was manufactured by the Parker Brothers game company. It was big by today's standards, about 9 inches long and 3 inches wide, shaped like a telephone receiver of the era. It used 6 AA batteries or could be plugged into the wall.

The center part of the device featured a panel of eleven buttons which could light up, one on top, one on the bottom, and a 3 x 3 grid in between. It was capable of playing six different games, Tic-Tac-Toe, some memory games, a number-guessing game, and another game called "Magic Square".

I was later able to acquire a working Merlin game complete with its original handbook. After so many years of wandering the starmaze, it was a thrill to hold this strange device in my hand.

Here are the instructions to "Magic Square":

Click to Enlarge


To form a square of eight lights by lighting buttons 1, 2, 3, 4, 6, 7, 8 and 9. At the same time you will have to turn off button #5 - the middle button.


Press NEW GAME. Then press #5 and watch the numbered buttons closely. MERLIN will *very briefly* show you the Magic Square. Then you'll see a random display of one or more blinking lights.

(Note: At this point, or at other times during the game, buttons 1-9 may all go dark. If this happens, button #10 will blink, indicating that the game is still on.)


Press buttons 1-9, in any order, until you've formed the Magic Square. When you succeed, MERLIN will flash the Magic Square and will make the WIN sound.

MERLIN's Secret Code

As you play Magic Square, you'll see that the nine numbered buttons affect each others' lights in certain predictable patterns. Each time you press one of the buttons, certain lights always come on and other lights always go off. To form the Magic Square, you'll find it helpful to understand these patterns - MERLIN's secret code...

The instructions go on to give examples and describe a two-person variation. (You solve the puzzle, then press 2 or 3 buttons to scramble the pattern and challenge your opponent to deduce and press those same buttons in any order to restore the "Magic Square".)

This game is very similar to the demo game I saw a year later with two exceptions:

  1. Merlin started at a random position
  2. You could press ANY button, not just buttons that were lit.
The second point is important, as it significantly simplifies the puzzle. It eliminates the one-way movement so important to what I call the starmaze; it means there is no longer a source and a sink in the network and any one position is never more than 9 moves away from any other (as opposed to 13 for the starmaze). In geometric terms, "Magic Square" is based on the same 9-cube as the starmaze, but with undirected edges.

In later years, another very similar puzzle appeared called "Lights Out", but it also allowed players to push any button, not just the lit buttons. I found one reference to the "Magic Square" puzzle in a math textbook written in the early 80s and a few other papers in mathematical journals, but all of these analyze the undirected version of the puzzle. A number of other "binary light grid" puzzles emerged in the 80s and 90s with names like Shooting Stars, Flip, and Orbix, but I have found nothing that predates Merlin's Magic Square.

My belief is that the inventors of this game (a Harvard PhD astrophysicist named Bob Doyle and his wife Holly) started out to create a hand-held Tic-Tac-Toe game. But once they had a 3x3 grid of buttons that also lit up, they began to create other games using the same basic apparatus. I understand that they coded many such games, only six of which made it to the final version.

I can see how a circuit designer working with binary on/off switches and what amounts to AND and OR gates, could become bored with Tic-Tac-Toe and start applying those AND and OR gates to create a "pattern game". It's the kind of puzzle that could arise fairly naturally under those conditions, but which would be unlikely to be invented in an earlier era. It is essentially at the confluence of Tic-Tac-Toe and binary circuit logic. So my guess is that the Doyles originated the basic idea and that it was probably not based on any earlier puzzle.

Apparently, the anonymous developer of the Apple speech recognition program decided to use the recently released "Magic Square" puzzle from MERLIN as the basis for his own demo puzzle. In the process of doing this, he decided to always start from the center-cell-on pattern, and added the crucial restriction of only allowing players to choose cells already "on" in the current displayed pattern.

My guess is that he may have added these elements to reduce the chance of false recognitions. As I recall, the speech recognition was pretty unreliable and often guessed wrong. Without the added restriction, the program would always have to match against all nine possible numbers and would be wrong often enough to produce a frustrating demonstration. Reducing the number of legal moves was a clever way of improving the apparent reliability. And always starting from a pattern with only one possible input ("five") would have further improved the demo.

These additions to the original MERLIN puzzle added just enough of an extra twist to capture my attention. I now see that what I found so fascinating was the combination of simplicity and complexity: the simple yet dazzling symmetries that come with a 9D hypercube combined with the complex and somewhat arbitrary pattern of DIRECTED edges implicit in the modified demo puzzle.

I took it from there and added more changes to create the puzzle you now see on my website. I eventually figured out that there was a one-to-one mapping from each of the 512 possible patterns displayed in the original puzzle to a different set of 512 patterns representing the underlying binary coordinates of corners in the 9D hypercube.

It is this mapping which forms the basis for the circles and squares of my current design. Each of the shapes in the nine cells are either filled or empty in a way that corresponds to lights being on or off in the MERLIN puzzle. But the SHAPE, circle or square, is derived from the binary address: circle for 0, square for 1.

I made this change as a convenience to further my own research. It allowed me to see both the classic pattern and the underlying address as the same time. But the combination of these two schemes changes the "flavor" of the puzzle in an interesting way. It makes the goal of finding the solution pattern much easier, but makes it harder to deduce the rules which govern the changes from pattern to pattern.

This represents my best guess about the sequence of invention that produced my starmaze puzzle. But if you loosen the definition of the basic puzzle to include "any system comprised of patterns formed from an array of two-state cells which shift according to a fixed set of rules", then this puzzle has far more ancient origins.

Just such a system is described by the I Ching, or "Book of Changes", thought by some to be the oldest book in the world. The system consists of 64 patterns called hexagrams formed by an array of six cells or "lines", each of which can be drawn in one of two states: broken or solid.

Each of the six lines can also be either moving or fixed as defined by the rules of divination, essentially a table of probabilities. Each moving line has the potential of changing into its opposite; taken together, the six lines describe how the current hexagram will change into the next hexagram. Thus each pattern represents a momentary state in a process of continual change.

Today we could use on or off LEDs or pixels on a screen to represent each line and electronic circuitry or software to calculate each new pattern. 3000 years ago they would be drawn on parchment with ink or just scratched into the dirt and the calculations would be done using yarrow sticks or tossed coins. But the basic idea is the same.

The I Ching has fascinated philosophers and poets, kings and commoners, mathematicians and mystics, for thousands of years. Leibniz was drawn to it because it exposes the beauty and power of binary numbers. Jung devoted decades to it because of the dreamlike way it taps the collective unconscious. It is at once utterly simple and kaleidoscopically complex, deeply symmetrical and endlessly chaotic. I believe that the starmaze, and the other shifting binary pattern puzzles which have appeared in the last few decades starting with Merlin's Magic Square, possess this same quality. Over the years I have come to think of the starmaze as nothing less than an extension of the I Ching.

For this reason I worked hard to create a way of presenting the puzzle that had the same simple yet mystical feel as the I Ching hexagrams. I went through countless designs before settling on the elemental circles and squares motif. Following in the footsteps of Confucius, I have started to use the 512 patterns of the starmaze as a kind of moving skeleton on which to hang meaningful images and ideas - a kind of dream-catcher or memory palace or nine-dimensional mandala.

This is true of the origins of any idea: skim the surface and you can find a clear chain of successive invention. But dig beneath the surface and you will soon discover that everything is connected to everything else.