A group of N logicians is having dinner at the same table where they can all talk to each other at a restaurant. They have finished the main course, but none of them have any idea which of the others want to have dessert. The waiter stops by their table and asks them, “Do you all want to have dessert?” N – 1 of the logicians each answer in succession, “I don’t know.” How might the Nth logician then answer?
Know the answer? Send your solution to ar@casact.org.
Getting to the Root of Things
Can you simplify the following expression into a rational number? If you can, then do it.
√(1+1000000√(1+1000001√(1+1000002√(1+1000003√(1+1000004√(1+…)) ) ) ) )
This is a famous solution of Ramanujan. The key observation is that:
√(N+1)2 =√(1+N 2 +2N)=√(1+N√(N+2) 2)
From recursion, it becomes clear that the nested radical expression above is equal to 1,000,001.
Solutions were submitted by Shyam Bihari Agarwal, Dave Andrist, Bob Conger, John Berglund, Kenneth Klinger, Luba Pesis, Misha Rajcoomar, Dave Schofield and William Volterman.
