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Neurobics

Mazes

Russell Dear

Mazes have fascinated people for thousands of years. Greek legend has it that Daedalus built one at Knossos for the Cretan king Minos. At the centre of the maze resided the Minotaur, a monster with human body and bull's head. After Crete's defeat of Athens, Minos decreed that the citizens of that city sacrifice seven young men and seven young women to the monster every nine years. None of those who entered the maze ever found their way out. The Athenian prince Theseus volunteered to fight the Minotaur. Minos's daughter Ariadne gave Theseus a silken thread which he unwound as he traversed the maze. He killed the Minotaur, used the thread to retrace his steps and escape from the maze, and so ended the tribute.

The latest stage of maze development, called "The Modern Maze", was originated and refined by Stuart Landsborough in New Zealand during the 1970s. Not only has he built a maze popular with tourists in his home town of Wanaka, but he has designed a whole chain of "Landsborough Mazes" throughout Japan and is extending the idea to South Africa and elsewhere.

Stuart's mazes typically have between one and two kilometres of passageways bounded by two-metre-high wooden fences. To add interest and complexity, maze travellers are able to ascend stairways to overhead bridges that take them to totally new parts of the maze. According to Stuart, these overhead walkways help to reduce any feeling of claustrophobia people might have on the lower level. They also provide vantage points from which to chat to friends below, besides giving a feeling of achievement so necessary to alleviate frustration when trying to solve this type of large puzzle. The advantage of wooden barriers is that at strategic locations they can be changed regularly, to provide alternative challenges for visitors returning another day.

Compared to the present one, the first maze at Wanaka was very primitive. The average time taken for a visitor to solve it was about ten minutes. The current maze takes over four times as long and holds enough interest for people to accept the challenge and enjoy themselves at the same time.

There are a number of algorithms for solving or traversing mazes, none of them very efficient, and all involve investigating blind alleys. A more specific problem concerning mazes is one relating to what are called "simple closed curves". Imagine a circle deformed in a plane without crossing over or touching itself. In the language of topology all such curves are said to be simply closed and equivalent.

Given that the maze shown below is a simple closed curve, can you see from which of the positions marked A, B, C, D and E you are able to escape. Having done that, given any starting point within the maze can you work out a very quick way of deciding whether it is possible to escape from it?

Russell Dear is a Mathematician living in Invercargill