I cannot seem to understand something, it has been puzzling me for days. There exist $304$ tours from a corner in a $5 \times 5$ board based on an exhaustive search algorithm. $304$ is a multiple of $19!!$ How is that possible?
I assume that we are talking about a tree-like search, when moving the knight several possibilities appear to move it further so you multiply the numbers, and then select the tours out of all possibilities.
If a knight can only move to seven squares normally (if it is not the first move or by lack of space). How can the $304$ tours be multiple of $19$? Where does it come from?
Thanks in advance.
A 5x5 board is an extremely restricted space for knights. Only the centre square gives them their full move set (of $8$ options, not $7$).
The corners have a separate mini-tour which goes first or last (assume first for now). For the tour around the rest of the squares, there are options to use the centre square to reverse the remainder of the tour, plus the option to finish on the centre square. This gives the odd multiplier.
The two main tours:
These two circuits are linked via one of the non-corner squares on the red circuit, and the centre square can be inserted into the blue circuit at a number of points.