Every engineer and scientist knows that when facing a particularly complicated problem, it’s well-advised to reduce its complexity by reducing the problem to a smaller dimension. Often, it simplifies the problem and provides valuable insights into how to solve the bigger and more complicated challenge. If the smaller in-dimension problem is complicated as well, it is, at least, interesting on its own.
Chess is a reach and complicated game, played on a two-dimensional board, but what about a one-dimensional variation? Will it be complicated as well? And how do you even reduce chess to a one-dimensional board?
There are multiple one-dimensional chess variations, and the following one was introduced by Martin Gardner, American popular mathematics and popular science writer, in his column in the July 1980 issue of Scientific American. Gardner gives the following one-dimensional chess variant and adds a question:
From left to right: White King, Knight, Rook, two empty squares, Black Rook, Knight, King.
The King and Rook move as usual, and the knight moves exactly two squares, and may jump over a piece doing that. White plays first: Can he win?
It is easy to see that white can cause a stalemate (a draw) in his first step by moving the rook from c to f. Winning, on the other hand, is more challenging.
In fact, there are many stalemate situations in the game, and it’s also possible for the black to win if the white does a mistake. However, if the white plays correctly, he can ensure a win, regardless of black’s moves. We suggest you to play a few games to see for yourself how this variation of chess behaves, when stalemates are reached, and when each player wins. You can play it with someone else, or simulate the whole game by yourself. Since the game is limited in dimension, it’s much easier to check all possible game positions.
Now, buckle up, we are starting a full analysis: To win, the white player has to start with moving the knight from b to d, and continue the game with the following sequences, based on the black’s counter-moves:
The black has three options to respond: Rf-d, Rf-e, or Ng-e.
Option A:
Nb-d Rf-d
Rc-d Ng-e
Rd-e
Checkmate, white wins.
Option B:
Nb-d Rf-e
Here the game splits to options B.1 and B.2, based on the black’s move.
Option B.1:
Ka-b Re-d
Rc-d Ng-e
Rd-e
Checkmate, white wins.
Option B.2:
Ka-b Re-f
Nd-f
Checkmate, white wins.
Option C (the longest game for the black):
Nb-d Ng-e
Nd-f Kh-g
Here the game splits to options C.1 and C.2, based on the black’s move.
Option C.1:
Rc-d Kg-f
Rd-b Ne-c (or Rd-b Kf-g. White can win in the next step anyway)
Rb-c
Option C.2:
Rc-d Ne-c
Here the game splits again, to options C.2.I and C.2.II, based on the black’s move.
Option C.2.I:
Ka-b Nc-a
Nf-h
Checkmate, white wins.
Option C.2.II:
Ka-b Nc-e
Nf-h Kg-h
Rd-e
Checkmate, white wins.
Finally, we analyzed every move that ensures the white’s win. While practicing by yourself, did you find more stalemate situations? Or even some situations where the black wins?
Of course, this is not the only one-dimensional variation of chess, but the story of the other one-dimensional chess variations will be told some other time, in one of our future articles.
Featured image: https://pixabay.com/vectors/chess-game-board-strategy-knight-2938267/