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).
In my interpretation of the problem, the posted solution is not quite correct, as it does not address two issues:
1: the highest number viewed in the first k slips being much less than n. In this case, because we know that there are no two slips the same, and that all slips contain positive integers, we know that there must be at least one slip that is n rupees (if no integer is missed, and the smallest value is 1). So in this case, the user can simply look for the first slip of n or greater.
2: The situation changes as the user approaches the end of the game, drastically so when looking at the penultimate (never used that word before!) slip of paper. For example, with 2 thousand slips to look at, in the first k slips the highest value seen is 10000. Systematically viewing and rejecting slips all along, the player finds himself looking at the second last slip, which contains the value 9995. Also, having seen each of the other slips, has noted that all the values from 1-1997 have been shown except for 104. This makes it very likely that the last slip is in fact 104, and the player will maximize his return by breaking the strategy as posted. Of course this example shows only the general condition of the exception to illustrate my point.
The first situation should be included in the strategy for sure, and the second only if the player receives money no matter what slip they choose (as opposed to only receiving the money if they manage to choose the largest, in which case reject the second exception).
regarding posted solution
