Norm and Ray were about to sit down to a game of chess when Sally walked in.
"Still playing the same boring game?" she asked, "Let me make it more interesting. I'll set up the pieces on the board a little differently."
"OK," they said, glad to let someone else take care of the chore of setting up the pieces. When they sat down to play, it seemed that only a few pieces were in an unusual position.
"Come on, make your move," said Norm. But when Ray picked up a piece to move, Sally said, "No, that would be illegal."
Ray's brows furrowed for a minute, and then he saw that she was right. He pondered the board for a few more minutes.
"I can't move," he said, "and what's more, neither can you."
How had Sally set up the board?
"OK, this is getting us nowhere", said Norm, "Sit down, Sally, and play against Ray. I know you hate pawns, so I'm taking all the pawns away. And I know the standard chess board seems too big, so I'm making you a smaller one, that's 5 by 5. And I know you hate aggression, so I'm going to place the remaining pieces on the small board in a very nonaggressive way. OK?"
Sally was too polite to refuse to get a taste of her own medicine, and sat down. Norm took away the big chess board and the pawns, placed a smaller 5 by 5 board, and set the remaining 16 pieces down on the 25 squares. Surprisingly, he had arranged them in such a way that there was no attacking going on, that is, no white piece was attacking a black piece, and vice versa.
How had Norm set up the board?
After Norm showed his solution to the "Play Nice" puzzle, Ray was quite annoyed and got up and stared out the window, muttering quietly.
"Don't worry, Sally," said Norm, "he never stays mad long."
Sally noticed that Norm had started to trace out patterns on the panes of the window. Suddenly, he wiped out whatever he had been working on, and came back to the chess table.
"OK, your solution was very nice," said Ray, "but you know, in a real chess game, each player's bishops are on different colors. That's not true in your solution. But I just worked out a solution on the windowpanes over there that satisfies this extra condition. Can you?"
After half an hour of moving pieces on the board, neither Norm and Sally gave up.
How had Ray set up the board?
(The "Play Nicer" puzzle appeared in the problem section of Math Horizons, for April 2006, and was contributed by Martin Gardner).
I give up, show me the solution.