Mindbenders # 20
Remember your table manners and the rules of etiquette you learnt at your grandmas knee? Among the more arcane was the dictum that it was rude to answer a question with another question. This usually followed close upon the heels of being taught to say, How do you do? in response to the same enquiry. It confused the infernal regions out of me, which is why I like this set of problems almost as much as I liked the Marx brothers crosstalk acts.
The Island of Questioners: On a remote island somewhere in the reaches of a childs subconscious, there dwells a race of pathological smartalecks. They never make statements; they only ask questions. And they only ask questions to which they know the answers.
Some (Type A) ask only questions to which the correct answer is yes. Others (Type B) ask questions which can only be answered in the negative. Communication isnt easy and it behoves a visitor to figure out whether his companions are type A or Type B as fast as possible. Instead of dwelling on postilions struck by lightning, a typical visitors phrasebook contains encounters like the following with the status updates:
1. If someone you meet on this island asks, Am I type B?, can you immediately deduce his status? Similarly if someone asks, Am I type A?, can he be immediately classified?
2. If you meet two islanders and they ask, Is at least one of us Type B?, what can you deduce about their status?
3. In a similar encounter, one native asks the other, Are we of different types? Can you place both?
4. Similarly, if you meet a native who asks, Am I of the type that can ask whether I am Type B?, what conclusions can you draw?
5. More complicatedly, you meet three natives A, B, & C. A asks B: Are you of a type that can ask C whether she is of the type that could ask you whether you are of different types? You ought to be able to work out something about their respective status.
Answers to last weeks puzzles:
By the standards of Indian coalition politics these are all simple real-life real-time problems. Still you might as well brush up on your political theory.
1. Political Accounting: Five MPs must defect from one party to another for the second party to own 10 more MPs. The key is remembering that one MP defecting means a double advantage. The same principle holds when calculating vote swings.
2. Utopian Politics: Given that one politician is honest, the logician can find an unique solution. He can form 99 pairs involving that honest MP and each of the other 99 MPs in turn. According to the terms of the problem, the other politician in each of those 99 pairings must be corrupt. Hence there is one and only one honest politician in Utopia.
3. Coalition accounting: If the independents vote is considered a commodity, then two separate complete transactions have taken place. In the first, the broking party gains Rs 1 lakh by buying a vote for Rs 7 lakh and selling it for Rs 8 lakh. In the second transaction, the broking party gains another Rs 1 lakh by buying the vote for Rs 9 lakh and selling it for Rs 10 lakh. Thus the party makes Rs 2 lakhs on the deals.
4. Allocation of posts: The larger parties get six ministerial posts each while the smaller get five. To make the total add up to 56, there is only one split possible. There must be six larger parties which pick up equal shares of 36 posts. There must be four smaller parties which get a total of 20 posts each.