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June 20th, 2010, 12:25 AM  #1 
Member Joined: May 2010 Posts: 47 Thanks: 0  Drawing black tiles! Tough one
Hello everyone. Consider that there is a chess like board with 8 * 8 tiles. All the tiles are white!!! There is a magic dice which rolls 95% of the time white and 5% of the time black. I use this dice only once for every of the 8*8=64 tiles. After passing from all the tiles there might be a chance that some tiles changed their color from white to black. I would like to find what is the chance after having used this dice after each one to be at least one black square created! This is the hard part. I Would like to see one black square somewhere. So this is not I think and i.i.d problem. A black square might be created in A1,A2,B1,B2 or in A2,A3,B2,B3 and so on. I do not know how to try to think for these type of problem. I would like to have this calculated fro my thesis but as I am not a mathematician student there is almost no support from my department. I would like to thank you in advance for your help. Best Regards Alex. 
June 20th, 2010, 02:25 AM  #2 
Math Team Joined: Apr 2010 Posts: 2,780 Thanks: 361  Re: Drawing black tiles! Tough one
Hello Alex, For the square, at least 4 tiles must be turned black. How many options do you have to find such a square? This helps, but I´m not sure about the answer. Hoempa 
June 20th, 2010, 03:00 PM  #3 
Member Joined: May 2010 Posts: 47 Thanks: 0  Re: Drawing black tiles! Tough one
Exactly this is the hard part of the problem.. What is the chance of having one square created? I do not know this answer. I must find a way (that is the reason of making the post here) to calculate it. Best Regards Alex 
June 25th, 2010, 04:19 PM  #4 
Senior Member Joined: Feb 2009 From: Adelaide, Australia Posts: 1,519 Thanks: 3  Re: Drawing black tiles! Tough one
By "black square" do you mean a 2x2 block of black tiles? If so, the chance (given that one tile in twenty is black) is about 0.03 %.

June 30th, 2010, 01:53 AM  #5 
Member Joined: May 2010 Posts: 47 Thanks: 0  Re: Drawing black tiles! Tough one
Thanks for your answer. Yes a square is defined as a 2*2 placement. The problem is that 4 squares aligned in a straight line do not count as a square. Also I would like to calculate the chance at least one such box to be created. If we finally manage to solve this ... then I ll provide an extension to this Best Regards Alex. 
July 1st, 2010, 03:17 AM  #6 
Senior Member Joined: Feb 2009 From: Adelaide, Australia Posts: 1,519 Thanks: 3  Re: Drawing black tiles! Tough one
I used a recurrence and found that of the 2^64 possible patterns, exactly 16477833186525760257 contain at least one 2x2 block. That's about 89.3265 % of them. But if a cell is coloured in black only 5 % of the time, the chance of a 2x2 block somewhere on the board falls to 0.03 %.

July 2nd, 2010, 07:02 AM  #7 
Newbie Joined: May 2009 Posts: 25 Thanks: 0  Re: Drawing black tiles! Tough one
The change that any given 4 square block is coloured black is 0.05^4 There are 49 such blocks on an 8 x 8 board. Therefore the chance at least 1 is black is black = (1  0.05^4)^49 = 0.000306 or 0.0306% 
July 2nd, 2010, 07:17 AM  #8  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Drawing black tiles! Tough one Quote:
 

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