Worms: Description & Rules

Playing Field

    One characteristic that distinguishes WORMS from most other recreations is the base on which it is played.  Whereas standard graph paper is easily obtained, that required for WORMS may not be so readily available.  Graph paper, being ruled both vertically and horizontally describes a “rectangular grid” (see Fig.1).  Here every point has four immediate neighbours: up, down, left and right.  In a WORMS “isometric grid” however, every point has six closest neighbours (see Fig.2).

Rectangular gridIsometric grid
Fig.1 Fig.2

Small grids are easily drawn by hand.  Larger ones are best generated by computer on a laser printer.  If you want a page-sized sample, click on the Fig.2 image which can then be printed.

Pattern Making

     A pattern is formed by a path consisting of contiguous line segments drawn between the grid points.  We start with an empty grid.  Draw a line segment between two points (see Fig.3).  This is the first and only move that can be made without consulting rules.  Now we must decide which of the possible paths to take for the next segment.  Retracing occupied paths is illegal.  The direction is not an absolute one like north, south, etc.  Instead, we must consider the orientation of the segment just drawn for a reference.  We can take one of five paths: straight, right 60° turn, right 120° turn, left 60° turn, or left 120° turn.

first moveavailable
direction 1,
Fig.3 Fig.4 Fig.5

The directions are numbered (see Fig.4).  Zero is always straight ahead.  There is no number three since that path is already taken.


     When faced with a choice where there is no precedent for action, we must “invent” a rule.  A rule states which path to follow given the circumstance at hand, i.e. choice amongst the possibilities.  When we choose, we must adhere to this rule for subsequent situations meeting similar criteria.

     For example, take path zero (straight ahead).  Rules, once defined, are in effect for the duration of the pattern that is emerging.  It’s easy to see that the straight ahead rule will lead to an infinitely long line to the right as it is reapplied at each subsequent grid point.  Clearly, this rule at this time will not result in an interesting pattern.

     If we choose direction one (a 60° right turn) then we will create a path that curves right, a hexagon, until it ends up at the starting point (see Fig.5).  The arrow shows where we are currently.  Here is a new situation we haven’t yet encountered.  From our point of view, one path is already taken (direction 1).  We can only go straight (zero), or two, four, or five.  Remember the numbering is relative to the orientation of last segment.  Zero is always straight ahead.  We must establish a new rule which accounts for the cases where path one is taken.     

This process is repeated – existing rules used wherever applicable and new rules created as required.


     The goal of WORMS is to come up with a set of rules that will yield the largest pattern.  Pattern size is defined only as the population of line segments (units) and not by the grid area it occupies.  Growth stops when we arrive at a grid point which has no more free paths.  Thus we are looking for the largest pattern which terminates.  This qualification is necessary because it’s very common to encounter patterns which apparently go on forever and have a very regular, predictable shape.  Our infinitely long line was a trivial example.  We’re looking for patterns which aren’t predictable within limits of grid size and computing time.

     The smallest pattern has nine units:   It starts and ends at the same point.  Be prepared to see patterns exceeding many thousands of units.

     What this site offers is a description of WORMS sufficient for you to write your own program.  You will be able to delight in the discovery of many patterns.  Many are beautiful not only in their finished state, but in their intricate growth behaviour along the way.  You may even be able to answer the question “What is the largest pattern?”

     If you’re just curious or have no desire to explore WORMS on your own, the remainder of this site is dedicated to the results of intense study by this author.  All possible patterns are catalogued and illustrated.  Various recurring textures are shown.  Finally, some possible algorithms are reviewed.

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