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Showing posts from September, 2010

Prime Number Strategy Game

Disclaimer: Difficult Problem

Source: Amol Sahasrabudhe (Associate, Morgan Stanley) - Got his permission to put it on blog \m/ \m/

Problem: Lets consider a two player game in which a number is given. The first person gets a chance to choose a number of the form (prime-1) and subtract it from the given number. Then the second person gets a chance to do the same and so on. The person who makes the number zero wins.

So, if the chosen number is of the form (prime-1), the first person would win in the first chance. There are infinite numbers for which first person has a winning strategy. Prove that second person also has a winning strategy for infinite numbers. :)

Bonus: Given a chance to choose whether you want to be first person or second person, what would you choose?

Update (25-09-10)
Solution: Posted by Siddhant Agarwal (4th year, EE, IITB) in comments!

Number Games

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)

Equal numbers in a circle

Source: Stanford Math Circle Sunday May 30, 2010

Problem: A circle is divided into 6 sectors. The numbers 1, 0, 1, 0, 0, 0 are written into the sectors in the counter clock-wise direction. You may increase any two neighboring numbers by 1. Is it possible to make all of the numbers equal?

Disclaimer: Think simple. Very simple problem. Solved it in less than 30 seconds. :)

Solution: Posted by Aman in comments!! He claims he took less than 10 seconds \m/ \m/. This is one thing where it feels good to win, but feels awesome to be defeated \m/ \m/