This is a problem on an infinite chessboard with pieces called lolcatz. They can move like queens or knightriders, but have a strange disadvantage…
… you’ve never heard of knightriders? Well, they’re long-range knights. They make an odd number of regular knight moves in the same direction at once. Knightriders can jump over pieces, except those on the intervening squares that it could have landed on. (So, b1-e7 only requires c3 to be vacant, d5 can be occupied.) Like regular knights, knightriders land on a square of opposite colour to their starting square.
There are no knightriders in this puzzle — just lolcatz, which have rook, bishop, or knightrider moves available. The squares that can be reached by a lolcat in one move are illustrated here.
Other than that, the puzzle involves standard chess pieces with their usual moves: pawns, knights, bishops, rooks, and queens (no kings).
Anyway, as I was saying… lolcatz have a strange disadvantage: if a piece captures a lolcat, or lands where it could be captured by an enemy lolcat in one move, then, the piece gets to make an extra move. Though it has to transmogrify first, naturally…
…how does transmogrification work, you ask? Well, if you’ve ever seen a transdimensional kitten playing with a hypersphere of cosmic yarn immersed in de Sitter space, then it happens pretty much they way you think it does:
If multiple cases apply (due to multiple enemy lolcatz), then the moving player chooses which case applies.
The puzzle is this: There is an initially empty chessboard that is infinite in all directions. First white chooses any square to place a white bishop, then black chooses any (empty) squares to place any (finite) number of black lolcatz. Then white moves first. Win by capturing all enemy pieces. Does white have a winning strategy? Or can black place lolcatz so as to evade white indefinitely, or force a win?
No that’s not a mistake: white, with a single bishop, versus as many lolcatz as black wants, in any position black wants.
Still have questions? Here’s some fine print which tries to anticipate them:
The fine print covers all the relevant questions that I could think of, but if anything is still unclear please ask!
Unless I'm mistaken, the result is
Reading through the wall of text, the rules seemed a bit too complicated for it to be "just some random game", so figuring out the magic seemed interesting.
To figure out if the bishop can capture all the lolcats on his first move, we need a "hitbox" for the lolcat; that is, the set of all those squares from which there is an extra move path to capturing the lolcat.
Since a bishop, rook, queen or a lolcat can always reach the same rank as any lolcat, we can start on that rank without loss of generality. We'll also assume that the lolcat is on a white square as in the example picture.
If we end up on a white square, our distance from the lolcat is even, and since we are now a bishop, we can halve the distance by moving to the diagonal, turning into a rook, and moving back to the lolcat's rank. Notice that if (and only if) we ended up at an even distance, there won't be a knightrider square away from lolcat we could reach with a rook move. This will turn out to be important.
If we are on a black square, we must find a knightrider square (in the appropriate direction so that the pawn move is away from the lolcat. This is always possible because of the symmetry of the board: going aroung the lolcat with bishop moves rotates the pawn move direction, but doesn't otherwise affect the position), then transmogrify to a queen, and return to the lolcat's rank diagonally away. Because of how the knightrider squares are spaced, this always exactly triples our distance, and adds one square to it from the pawn move. (If we arrived to this diagram with a rook move, we could also have reached a knightrider square with that move. If there's a path forward from there, it will always be the same knightrider square as in the diagram, so that path will converge with the diagram.)
Notice that these are the only move sequences that change the bishop's distance from the lolcat while still providing extra moves. In particular, if the queen moves to another square threatened by a knightrider move (away from the lolcat, as dictated by the rules for a queen move), the following pawn move won't be away from the lolcat, so the pawn cannot promote.
All this changes in the near vicinity of the lolcat: if we can get the distance to 1, then we can capture the lolcat:
So to recap: Our horizontal distance is N. If N is even, we can halve it, and if N is odd we go to 3N+1. If we get N to 1, we capture the lolcat, and get to hunt the next one with the same strategy.
So if we could always get from any N to 1, then the bishop would always win on the first turn: every lolcat would then have an infinitely long continuous hitbox. So all we have to know is: can we always get to 1 using this process?
In addition to that, a very strong reason I believe this has to be the intended solution is that
As for the strategy of the lolcatz player:
Correct answer by Bass on December 4, 2020
(Note: I'll be using regular chessboard diagrams to illustrate this.)
Here's a nice diagram:
Alright, now we can attack the puzzle.
Answered by Excited Raichu on December 4, 2020
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