Replace the King

by John P. Robertson

We have avoided chess problems in this column, but the following is such an interesting problem that we can’t resist. The wind had died, so I was cruising the clubhouse at the Berwyn Yacht Club looking for some action. I spotted Walt Wright looking perplexed at a chessboard. He had accidentally knocked his king off the board, and wasn’t quite sure where it had been. He asked for help figuring out, if possible, where to put the white king. The diagram of the remaining pieces is at right (the black king is on d1, a black rook is on b5, a black bishop is on d5, and a white bishop is on a4). Can you help? Where was the white king? You will need to know the rules regarding how chess pieces move to solve the puzzle. A hint is at the end of the solution to the Dividing the Rashers puzzle.

Dividing the Rashers

The puzzle gave rules for the number of rashers that each of Jones, Brown, Wag, and Robinson would take, noted that the Oldest Boarder found half a rasher when he came to breakfast, and asked how many rashers Wag had taken.

Bob Hallstrom sent in the most complete analysis. First he showed, by considering various cases, that if Mrs. Crabbe had set out 10 or more rashers, then there would have been more than half a rasher left for the Oldest Boarder. Next he rules out Wag arriving first, second, or fourth. Had Wag arrived first, he would have taken 3 and left 2. But then Jones, Brown, and Robinson, regardless of order of arrival, would have taken more than 11/2 among them, leaving less than half a rasher for the Oldest Boarder. Consideration of the cases where Jones or Robinson arrive first and Wag arrives second, and where Brown arrives first and Wag arrives second, rules out Wag arriving second. If Wag had arrived fourth, Jones, Brown, and Robinson would have taken at most 31/2 rashers among them, and Wag would have left one rasher. So Wag arrived third. It is not hard to show that Jones and Brown were the first two to arrive, in either order, and took 21/2 between them, Wag was third and took 11/2, and Robinson was fourth, taking half a rasher.

Solutions were also sent in by John Herder, Kevin Hobbs, Peter Sachs, Dave Skurnick, and Walter Wright.

Hint for the chess problem: Consider the possibility that there might have been other pieces on the board recently.