This month we investiate fairy chess endgames. In chess, it is well known that under normal circumstances, White needs (in addition to his King of course) a Queen, a Rook, 2 Bishops, or a Bishop and Knight to mate a lone Black King. But what about other possible pieces? Can White mate with an additional piece that moves like a king? How about a Rook that moves no more than 2 squares at a time? How many Rooks that only move 1 square at a time are needed? Feel free to invent your own pieces.
A Rook has 28 different moves (7 translations up, 7 down, 7 left, and 7 right). What is the fewest number of moves that a piece can have and still mate a lone King?
Here are some miscellaneous chess quickies:
Quickie #1:
Find a legal chess position in which we can conclude that at least one of the players has castled, but we don't know which.
Quickie #2:
Place a White King, White Pawn, and Black King on an n x n chess board at random. Let p(n) be the probability that if White moves first, White can win (that is, White can safely promote the pawn). As n→∞, p(n) converges to what value?
Quickie #3:
Who wins this 5x5 mini-chess game? Can you prove it is a tie?
ANSWERS
There weren't many responses this month. White can mate with an additional King, or with a Rook that moves no more than 2 squares at a time. The strategy is fairly complicated. I can also mate with 3 additional Rooks that move only 1 square at a time. Can any one do it with 2 such pieces?
The smallest number of moves that a piece can have and mate is probably 6. One piece that works is a King that cannot move diagonally backward. To mate, get the King and modified King to the bottom of the board, then use the 2 Kings vs. 1 King strategy referred to above.
Each side has a promoted Rook. Since the Pawns did not pass each other on the c-file, at least one Pawn made three captures: off the c-file, back on to the c-file, promote on the d-file. Now each side's Queen's Rook and King's Bishop were captured on the back rank and not on the d-file. Therefore the three pieces captured must have been the Queen, a Knight, and the King's Rook. If A captured B's Rook on the d-file, then B must have castled, for his King and Rook have no other way to switch places.
Quickie #2:
John Hoffman thinks the answer is larger than 3/4, but smaller than 7/8. He came up with a very complicated integral for computing the exact probability.
Alan Williams, who suggested this problem, thinks the answer is 149/192. A Monte Carlo simulation with billions of trials supports this conjecture. And finally Ulrich Schimke has proved it! Here is his proof, written up as a text file.
Quickie #3:
I'm pretty sure this game is a tie, and the state space is just small enough that someone might prove it with brute force. But no one has yet.
If you can extend any of these results, please
e-mail me.
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