182 UNICURSAL PROBLEMS [CH. IX 



Mazes. Everyone has read of the labyrinth of Minos in 

 Crete and of Rosamund's Bower. A few modern mazes exist 

 here and there — notably one, a very poor specimen of its 

 kind, at Hampton Court — and in one of these, or at any 

 rate on a drawing of one, most people have at some time 

 threaded their way to the interior. I proceed now to consider 

 the manner in which any such construction may be completely 

 traversed even by one who is ignorant of its plan. 



The theory of the description of mazes is included in 

 Euler's theorems given above. The paths in the maze are 

 what previously we have termed branches, and the places 

 where two or more paths meet are nodes. Th8 entrance to 

 the maze, the end of a blind alley, and the centre of the maze 

 are free ends and therefore odd nodes. 



If the only odd nodes are the entrance to the maze and the 

 centre of it — which will necessitate the absence of all blind 

 alleys — the maze can be described unicursally. This follows 

 from Euler's third proposition. Again, no matter how many 

 odd nodes there may be in a maze, we can always find a 

 route which will take us from the entrance to the centre 

 without retracing our steps, though such a route will take us 

 through only a part of the maze. But in neither of the cases 

 mentioned in this paragraph can the route be determined 

 without a plan of the maze. 



A plan is not necessary, however, if we make use of Euler's 

 suggestion, and suppose that every path in the maze is dupli- 

 cated. In this case we can give definite rules for the complete 

 description of the whole of any maze, even if we are entirely 

 ignorant of its plan. Of course to walk twice over every path 

 in a labyrinth is not the shortest way of arriving at the centre, 

 but, if it is performed correctly, the whole maze is traversed, 

 the arrival at the centre at some point in the course of the 

 route is certain, and it is impossible to lose one's way. 



I need hardly explain why the complete description of 

 such a duplicated maze is possible, for now every node is even, 

 and hence, by Euler's second proposition, if we begin at the 

 entrance we can traverse the whole maze; in so doing we 



