Source: Puzzle Toad, CMU
Problem: Its raining outside and Alfonso and Bernadette are bored.
Alfonso suggests the following games:
(a) Two players alternatively erase some 9 numbers from the sequence 1,2,...,101 until only two remain. The player that starts wins x−54 dollars from the player that plays second. Here x is the difference between the remaining two numbers. Would you rather be the first or the second player?
(b) Two players alternatively erase one number from the sequence 1,2,...,27 until only two numbers remain. The first player wins if the sum of these numbers is divisible by 5; otherwise the second player wins. Who has a winning strategy?
Update (Oct 10, 2010):
Solution posted by Siddhant Agarwal (Senior Undegraduate, EE, IITB) in comments!! Also at Puzzle Toad page (http://www.cs.cmu.edu/puzzle/solution31.pdf)