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Showing posts from August, 2010

Conway's Soldiers (CheckerBoard Unreachable Line)

Source: Asked to me by Amol Sahasrabudhe (Morgan Stanley Quant Associate)

Problem:
An infinite checkerboard is divided by a horizontal line that extends indefinitely. Above the line are empty cells and below the line are an arbitrary number of game pieces, or "soldiers". A move consists of one soldier jumping over an adjacent soldier into an empty cell, vertically or horizontally (but not diagonally), and removing the soldier which was jumped over. The goal of the puzzle is to place a soldier as far above the horizontal line as possible.

Prove that there is no finite series of moves that will allow a soldier to advance more than four rows above the horizontal line.

I could get the correct direction in 5 min. Spent enough time on the problem but could not solve it. :( Give it a go! \m/

Update: Sep 07, 2010
Solution: Posted by Siddhant Agarwal (Senior Undergraduate, Elec IITB) in comments!!

Magic Money Machine

Source:CMU Puzzle Toad

Problem: The wizards at Wall Street are up to it again. The Silverbags investment bank has invented the following machine. The machine consists of 6 boxes numbered 1 to 6. When you first get the machine, it contains 6 tokens, one in each box. You have two buttons A, B on the machine and you can press them as many times as you like and in any order.


Button A Choose a number i from 1 to 5 and then take one token from box i and magically two tokens will be added to box i + 1.
Button B Choose a number i from 1 to 4 and then take one token from box i and then the contents of boxes i + 1 and i + 2 will be interchanged.

The machine sells for one trillion dollars. The contract says that you can take the machine back to the bank at any time and then the bank will give you one dollar for each token in the machine. Is the machine worth buying?

Update (Nov 10, 2010):
This problem is an IMO 2010 Problem. Solution available at artofproblemsolving link
Solution posted by Siddhan…

Number of Sons

Problem:

A man has two children. He says one of them is a son. What is the probability that the other one is also a son? (Hint: Answer is not 0.5 :P)

I have read this problem many times in many different forms. But still looks fresh :P

Solution:

Read the wikipedia article of a famous and related problem (Link)

Puzzle on Sorting

Source:Saurabh Joshi's Blog

Problem: (copy-pasting from the source)
Suppose we have 13 cards arranged in descending order: K Q J 10 9 8 7 6 5 4 3 2 1
At any move, you can take a substring of cards and move it elsewhere. For example, 10 9 8 could be move to the right by 2 steps to get K Q J 7 6 10 9 8 5 4 3 2 1. The goal is to sort the string in increasing order minimize the number of such moves.

It must be absolutely obvious that you can do it with 12 moves but the very fact that I’m asking you this question means that you can do better. How many moves do you need?

Now the obvious extension to this problem is to generalize this technique for any n.

Awesome Problem!!

Update (Sep 07, 2010):
Solution: At Saurabh Joshi's blog (Link)