User:Simanelix
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[edit]Okay, I'm quite confused about how the non-jumping knight / mao works.
The idea is simple, a mao is a knight that can be blocked. But its unclear on which tiles you can block the mao, since the movement pattern could be defined in many ways. So, first, let's define which tiles we are considering:
| 1 | 2 | 3 |
| 4 | 5 | 6 |
Let's say the mao is on tile 1, and the tile it is going to jump to is tile 6. The other 7 potential moves the mao could make are symmetric to this setup, so we don't need to label them.
Now, theoretically speaking, any subset of the tiles could block the mao. And there could be multiple such subsets that block the mao. Let the set of the tiles be A = {tile 2, tile 3, tile 4, tile 5}. Now for any version of the mao, there could be some set P, a subset of A. P is the set tiles which have pieces on them. Now, we will say a set B is a subset of A, such that B is also a superset of P, then having pieces on all of the tiles in B blocks the mao from moving. Each set P is effectively a "blocking rule", since by definition, if all the tiles in P have pieces on them, the mao is blocked. Finally, we can define a version of the mao by a set S, where each item in S is a valid P. Notice that every P in S should not be superset or subset of another P in S. Finally, there is T, which is the set of all possible S. i.e. all of the possible "blocking rulesets".
Gosh, this is complicated. And there doesn't seem to be an official rule on which S is used.
We could look at this mathematically.
So, here for T is a function of A. Notice that the size of T is a simple function of the size of A. So you could graph that as a normal f(x). Also notice that one of the items in T(A) is the empty set, which in this context denotes a fully leaping knight. Now, in my personal opinion, the best definition is S = {{2,5},{3,4},{3,5},{4,5}}. This occurs if we strictly define the mao as: moving 2 squres in one direction, and then 1 square in the other direction, or vice versa. So in Parlett notaton: 2<>1=, 2=1<>. And in Betza 2.0 notation: (sW2-vW)(sW-vW2). I find this one to be pretty intuitive. Though perhaps it makes the mao too strong, since you must place a pair pieces to block it, and any lone pair only blocks 1 of the 8 possible moves. Though 3 pieces can block 3 or 4 of its possible moves.