Mazes and Labyrinths: A General Account of Their History and Development

by W. H. Matthews

There are one or two points which are of general application and should be borne in mind. For instance, if the object of the designer is to provide a maze which shall offer a fair amount of puzzledom without imposing undue fatigue on the visitor, he must take care that the nearest route from the centre to the exterior be neither too long nor too short. If the space to be covered by the maze is large the tendency to over-elaboration of the design must be avoided.

Another feature which is likely to spoil an otherwise good design is the inclusion of long stretches of path without bend or branching; these are tedious and annoying, especially when they have to be retraced by reason of their leading into a cul-de-sac.

In a large maze it is well to relieve monotony by means of occasional variations in the mode of treatment of the hedge, the introduction of arbours, statues, etc.; but these should not be of such a character as to defeat one of the main objects of the design by providing easy clues.

If the maze is intended to be seen at all from above, some attempt should be made to introduce a symmetrical and artistic element into its design. Usually some vantage-point is available from which an attendant or expert can observe and direct over-bewildered visitors, but if this point be accessible to the visitors themselves the hedges should be provided with pinnacles or balks, here and there, to prevent the observer from solving the puzzle by unfair means. This is the case with, for instance, the Saffron Walden maze; at Hampton Court, where there are no balks, only the attendant is permitted to mount the rostrum.

The "solution" of mazes means the discovery of a route to their "goal." (This word is preferable to "centre," as the object of quest is not necessarily at the geometrical centre of the maze, but may be considerably removed from it.)

It would be going too far to say the shortest route, as this would be discoverable only from the plan or by prolonged experience, but the goal in any maze will on the average be reached more certainly and quickly by observing a little method than by fortuitous wandering.

The subject of the solution of mazes has been examined by various mathematicians, in their lighter moods, but we need not burden ourselves with more than a few simple considerations.

In most cases it is not practicable to adopt a system of marking the various paths as we reach them, but if this be permitted we can so arrange our marks that we need never traverse any portion of the path more than twice—i.e., once in each direction—so that in any finite maze we must eventually arrive at the goal, though not necessarily by the shortest route.

Using the word node to signify a point of branching, and the terms odd and even to describe respectively those nodes at which odd or even numbers of paths are to be found, we see that there must be at least three paths meeting at a point to form a node, for two paths meeting at a point constitute only a change of direction of the path without formation of branches, whilst the arrival of one path only at a point also precludes the idea of "branching" at that point, and can only occur at the end of a blind alley, at the entrance of the maze, or at the goal. We find it convenient, however, to regard the latter arrangement as an odd node of the lowest order, the lowest possible order of even nodes being, of course, that at the meeting of four paths.

It will be clear that if the entrance and the goal are the only odd nodes the maze will either be unicursal, in the sense in which we have been using the term, or any branches must form loops on the main route; in either case it will be possible to traverse the maze unicursally, i.e., to thread every portion of the path without going over any part twice.