Paterson's Worms - Discussion

Discussion

There are many different types of worm depending on which direction they turn when encountering a new type of intersection. The different varieties of worm can be classified systematically by assigning every direction a number and listing the choice made every time a new type of intersection is encountered.

The six directions are numbered as follows:

So direction 0 indicates the worm continues to travel straight ahead, direction 1 indicates the worm will make a right turn of 60° and similarly for the other directions. The worm cannot travel in direction 3 because that is the gridline it has just traversed. Thus a worm with rule {1,0,5,1} decides to travel in direction 1 the first time it has to make a choice, in direction 0 the next time it has to make a choice and so on. If there is only one available gridline, the worm has no choice but to take it and this is usually not explicitly listed.

A worm whose ruleset begins with 0 continues in a straight line forever. This is a trivial case, so it is usually stipulated that the worm must turn when it encounters a point with only uneaten gridlines. Furthermore, to avoid mirror-image symmetrical duplicates, the worm's first turn must be a right hand turn. A worm dies if it returns to its origin a third time, because there are then no untraversed edges available. Only the origin can be lethal to the worm.

There are 1,296 possible combinations of worm rules. This can be seen by the following argument:

1. If the worm encounters a node with no eaten segments, other than the one it has just eaten, it can either make a sharp turn or a gentle one. This is the situation shown in the figure above.
2. If it encounters a node with one eaten segment, it can leave along any of the remaining four. Only the worm's first return to the origin has this character.
3. For two eaten segments, the location of the eaten segments is important. There are four distinct combinations of eaten segments and approach directions, each of which offer a choice of three departure directions, making 81 different alternatives.
4. If the worm encounters three eaten segments, it must choose between the two remaining uneaten ones regardless of their distribution.
5. For four eaten segments, there is only one uneaten segment left and the worm must take it.

There are therefore 2×4×81×2=1,296 different combinations of rules.

Many of these are mirror-image duplicates of others, and others die before having to make all the choices in their ruleset, leaving 411 distinct species (412 if the infinite straight-line worm is included). 336 of these species eventually die. 73 patterns exhibit infinite behaviour, that is, they settle into a repeating pattern that does not return to the origin. A further two are strongly believed to be infinite and one remains unsolved. Eleven of the rules exhibit complicated behaviour. They do not die even after many billions of iterations, nor do they adopt an obviously infinite pattern. Their ultimate fate was unknown until 2003 when Benjamin Chaffin developed new methods of solving them. After many hours of computer time, nine of the eleven rules were solved, leaving the worms with rules {1,0,4,2,0,2,0} and {1,0,4,2,0,1,5}. The first of these was solved by Tomas Rokicki, who determined that it halts after 57 trillion timesteps, leaving only {1,0,4,2,0,1,5} unsolved. According to Rokicki, the worm is still active after 5.2×1019 timesteps. He used an algorithm based on Bill Gosper's Hashlife to simulate the worms at extraordinary speeds. This behaviour is considerably more complex than the related rectangular grid worm, which has a longest path of only 16 segments.

It is possible for two different species of worm to produce the same path, though they do not necessarily traverse it in the same order. The most common path is also the shortest: the seven point "radioactivity symbol". One example of this path is shown in the animated figure above. In total there are 299 different paths, and 209 of these are produced by just one species.