Poisoned drink

FROM the diary of Dr John H. Watson, MD: By January, 1896, I had moved out of the Bakers Street apartment after spending some of my most memorable years in the company of Sherlock Holmes which had given me the opportunity to study the methods of this foremost champion of the science of deduction. As I had not seen him for the past few months, I was surprised to receive Inspector Gregson and him in my clinic today. They said there had been a murder and my services were urgently required.

A hansom took us to a Central London house that belonged to a person named Professor Moriarity, where we found three dead men around a table that had an empty wine glass on it. I examined the bodies and deduced that all those men had died of poisoning. The owner of the house, Professor Moriarity was ill, so I examined him as well.

The professor was as pale as his friends in the other room, except that he was alive. The old man could barely speak, but he told us that there had been a party at the house the other night, where four of his childhood friends, besides him, had sipped wine from the same glass, which was a custom they had followed from high school.

"Mr Holmes, my view is that we should look for this fifth guest, who I am sure, is the murderer," said Gregson. I could not contradict his theory, as it looked to be obvious from the evidence that we had. "There is a probability that we are proceeding in the wrong direction," said Sherlock Holmes, "and probability here is the key word." "We haven’t followed you Holmes," Gregson and I said. Holmes said, "Suppose, there are ‘n’ number of friends around a table with a glass of wine in front of them that no one has yet tasted. You take a swig of it and pass it to the person on your left or right with the probability of 1/2. Suppose your neighbour does the same and each guest continues in this fashion. Because the glass is moving randomly around the table, it may be a while before some of the guests get to taste the wine for the first time. Which person around the table is most likely to be the last one to try the wine?"

"It has to be the person who is farthest away from the first one who begins the round," said I. "I am afraid we I don’t share your opinion Watson," said Holmes, "Calculate the probability for some specific cases, assuming ‘n’ to be 3, 4, 5 or anything. The general case can be proved by considering any fixed player and the time when the wine first reaches one of his neighbours. You will find that all guests, except the first one, are equally likely to be the last with the probability of 1/(n-1)." I said, "It means..." "That there was no fifth man and the murderer is right here in this house," Holmes interrupted me. "My dear Holmes! Who could it be?" I said. "None other than the foremost champion of crime in England — Professor Moriarity," he said. "You can’t be serious Holmes. I have examined the old man. He has been poisoned as well." I said. "You have examined the patient, but I have examined the glass. It is arsenic," said Holmes. He picked up the empty wine glass in his hand and said, "Imagine, if this man plans to kill his friends, there is no better weapon than arsenic. Three sips of its solution in wine can kill a man, but two sips only make you sick for a while." Inspector Gregson, who had been quiet all this while, said, "How did he ensure that the others took three sips of the solution, while he took only two?"

Holmes said, "Moriarity is also a brilliant mathematician. We have examined how all guests, except the first one, have an equal probability of being the last one to receive the glass. Now examine this old entry in his diary that explains this wine ritual, where the host determines the rule to pass on the glass to others. There were two ways in which he could make all three guests take three sips from the glass before he himself took the third one. One was that the host would pass on the glass to his right. The second person should also pass it to his right, following which, the third person should pass it to his left. Then, the cycle is repeated. The other way is to pass the glass to the left. The second person should also pass it to his left, following which, the third person would pass it to his right. The cycle is repeated after this. Both ways, Moriarity, the first person to pass the glass, is the last person to take the third and the fatal sip." Thus the foremost champion of deduction defeated his most hated rival and saved London from the wrath of Moriarity.

— Aditya Rishi