Mathematical_Recreations-Kraitchik-2e

Mathematical Recreations
The route symmetric with respect to the vertical median
belongs to the domain a if k is odd, to {3 if k is even. On the
other hand the route symmetric with respect to the center
belongs to a if k is even, and to {3 if k is odd. Hence for every
k there is just one route symmetric to a given one and belonging
to the same domain.
Every route can be transformed by permuting rows 1 and
1 5 9 6 10 4 5 1 7
6 10 4 1 5 9 10 6 2
2 8 7 3 4 8
7 3 2 8 9 3
9 3 7 13
2 8 12 6
10 4 14
1 11 5
FIGURE 136.
2, and rows 3 and 4. The new route, called the derivative
route, begins on an inside cell and ends on an outside cell, so
that we must reverse such a route to get a route from an
outside to an inside cell. The derivative route may be identical
with the original.
In order to go over every cell we must first cover one domain
completely. Suppose that we use the route of Figure
137 to go from a4 to a3. To complete the tour we need a
route in domain {3 beginning on c2. This may be obtained
by taking the route symmetric (with respect to the horizon-