Neurobics

Of Murder and Knaves

Russell Dear

Usually Arnie didn't bother to look at the daily orders board, after all, his name was rarely there. It didn't matter today either because again his name wasn't on it. Bernie checked the board just about every day but on this occasion forgot. As a consequence he didn't report for duty and a person died. Who was the most guilty? Arnie who rarely read the board or Bernie who did, except for this occasion when someone died.

I won't attempt to answer this dilemma but Raymond Smullyan in his book of logical puzzles called What Is The Name Of This Book? (Simon & Schuster, 1986) poses a similar problem:

Alan and Bob both want to kill Colin. They take their chance while on a safari through the desert. Down to their last flask of water each, Alan secretes poison into Colin's. Not knowing about the poison, Bob drills a hole in Colin's flask causing the poisoned water to drain away. Colin subsequently dies of thirst. Who is guilty of Colin's murder?

What Is The Name Of This Book? has been described by Martin Gardner as "the most original, most profound, and most humerous collection of recreational logic and math problems ever written".

Smullyan begins the book on an old theme, that of liars and truth tellers. He imagines an island peopled by knights, who always tell the truth, and knaves, who always lie. You'll know the idea. You might like to have a think about four of the introductory problems:

1. There are two people, A and B, each of whom is either a knight or a knave. A makes the following statement: "At least one of us is a knave."

What are A and B?

2. Suppose A says, "Either I'm a knave or B is a knight."

What are A and B?

3. Suppose A says, "I'm a knave, but B isn't."

What are A and B?

4. I came across A and B and asked one of them, "Is either of you a knight?" He replied, and I knew the answer to my question.

What are A and B?

As an added clue, the four answers include all four possible pairs of permutations of knight and knave.

Russell Dear is a Mathematician living in Invercargill