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In the top row of an 8 \times 8 chessboard, Tom writes the values 1, 2, 4, 8, 16, 32, 64, 128. In the leftmost column, Tom writes the values 1, 3, 9, 27, 81, 243, 729, 2187. In every other square that doesn't have a number yet, Tom writes the product of the leftmost number in that square's row and the topmost number in that square's column. What is the sum of all the numbers on the chessboard?

Guest Jan 25, 2015

Best Answer 

 #1
avatar+80941 
+10

So we have something like this...

1  2  4  8  16  32   64   128  

3

9

27

81

243

729

2187

Let's see if we can discover a pattern.....!!!

For the first row the sum is just 1(1-2^8)/(1-2)  = 255

For the second row, the sum is just 3(1-2^8)/(1-2) = 765

That leads us to believe that the sum of all numbers on the chessboard is given by the sum of the numbers in the first column times (1-2^8)/(1-2)

And the sum of the numbers in the first column is just 1(1-3^8)/(1-3) = 3280

So the total sum of numbers on the chess board is given by 3280(1-2^8)/(1-2) = 836,400...!!!

Another way to look at this is to notice that every successive row sum is triple the previous one

So we would have  255(1-3^8)/(1-3) = 836,400...just as we expected..!!!

 

CPhill  Jan 25, 2015
Sort: 

2+0 Answers

 #1
avatar+80941 
+10
Best Answer

So we have something like this...

1  2  4  8  16  32   64   128  

3

9

27

81

243

729

2187

Let's see if we can discover a pattern.....!!!

For the first row the sum is just 1(1-2^8)/(1-2)  = 255

For the second row, the sum is just 3(1-2^8)/(1-2) = 765

That leads us to believe that the sum of all numbers on the chessboard is given by the sum of the numbers in the first column times (1-2^8)/(1-2)

And the sum of the numbers in the first column is just 1(1-3^8)/(1-3) = 3280

So the total sum of numbers on the chess board is given by 3280(1-2^8)/(1-2) = 836,400...!!!

Another way to look at this is to notice that every successive row sum is triple the previous one

So we would have  255(1-3^8)/(1-3) = 836,400...just as we expected..!!!

 

CPhill  Jan 25, 2015
 #2
avatar+91435 
+5

Chris drew my attention to this question.  He really liked it and it does look very interesting. 

I have given it a puzzle icon and added it to our puzzle thread.   Thanks Chris    

Melody  Jan 26, 2015

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