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If n is an even number, then each n-level interesting maze gives rise to eight distinct (n+4)-level interesting mazes, which are different from the mazes constructed by this process from any other n-level interesting maze. Their paths are illustrated in this figure, where the part of the path inside the box is the original n-level maze.
To see why, let s be the permutation of 0,...,n that gives the level sequence of the original maze as
Then the illustrated mazes have level sequences:
a. 0, n+3, 2, s_{1}+2, . . . , s_{n-1}+2, n+2, 1, n+4b. 0, n+3, n+2, s_{n-1}+2, . . . , s_{1}+2, 2, 1, n+4
c. 0, 3, s_{1}+3, . . . , s_{n-1}+3, n+3, 2, 1, n+4
d. 0, n+3, n+2, 1, s_{1}+1, . . . , s_{n-1}+1, n+1, n+4
e. 0, n+3, n+2, 1, 2, s_{1}+2, . . . , s_{n-1}+2, n+4
f. 0, s_{1}+2, . . . , s_{n-1}+2, n+2, n+3, 2, 1, n+4
g. 0, n+3, s_{n-1}+3, . . . , s_{1}+3, 3, 2, 1, n+4
h. 0, n+3, n+2, n+1, s_{n-1}+1, . . . , s_{1}+1, 1, n+4
The verification that these sequences are all distinct can be done in a table, where the (x,y) entry gives one place where (at least) the level sequences of mazes x and y must differ. * means that the ``interesting'' condition is invoked (it means here that s_{1} is not 1, nor is s_{n-1} equal to n-1.)
a b c d e f g h a 2 1 2 2 2 2 2 b 1 3 3 1 2* 3* c 1 1 1* 1 1 d 4* 1 3 3 e 3 3 3 f 1 1 g 2*
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