A box is filled with 'N' slips of paper. On each slip of paper is written some positive integer (note that any positive integer may appear on the slips - not just the integers from 1 to 'N'). The integers do not necessarily appear in any sequence or pattern. Each of the slips has a different integer on it, so there is just one slip with the greatest integer.

A person who has no prior knowledge of which numbers appear on the slips - but who does know that there are 'N' slips - is to blindly pull slips from the box one by one. The person looks at each slip, then either agrees to accept that number (of Rupees) and quit or decides to go on and choose another slip.

Note that the person looks at each slip as he/she proceeds, and then decides whether to quit or to go on. That person can go forward, but cannot go back. If no choice is made by the time the 'N'th slip is reached, then the person must accept the number (of Rupees) on the 'N'th slip.

Does there " EXIST " a 'Best Strategy' for the person ? If " YES ", then what is that strategy ? (Here the term " Best Strategy" means that the person will get the greatest amount of Rupees).

A difference between this and the classic problem is that in the classic problem, the selector does not get to keep ANY money if he has not selected one with the highest value, and so must maximize the probability of selecting the highest one--second best just won't do at all.

In the present problem, the person accepting or rejecting gets to keep whatever amount is accepted, even if it is not the highest one. Thus if the idea is to maximize the expected value, the strategy can be different.

BTW, Cory's memory is indeed better than mine, and also e is indeed the correct denominator for the classical puzzle where one seeks to get the absolute highest value and nothing less.

Posted by Charlie on 2003-03-18 05:31:04