6+(+Mathematics+)

In math class, we had one math problem.

Once, there was a king in the kingdom. But enemy attacked his kingdom, so his kingdom was destroyed by enemy. One day, one man appeared and he solved this situation. King was so happy because one man saved his kingdom. So king said "You save my kingdom, I want to give you something. You can choose. First one is you can get 1.000.000 dollars, or Second one is when you pass each square in chess board you can get cents like this. If you go 1st square you can get 1 cent, 2nd square you can get 2cents, 3rd square you can get 4 cents, and 4rd square you can get 8 cents. And you can have all the money that you passed square. So if you passed 4 square you can have 15 cents. You can choose,What do you want to get, first one or second one?" If you are that person, What do you choose?

 Question. On a chess board you got to put 1 cent on the first square, 2 on the next, 4 on the next, 8 on the next, 16.....all the way to 64. You get to keep all the money!
 *  Extension **

 For example : after the 3rd square you get to keep 1 plus 2 plus, 4 = 7cents.

 How much would you score in total 'loot'?

 Is it possible to find out a formule to solve this?


 *  Answer **

 A chess board have 64 squares. you can get cent from Each square. Formula, y=2^x-1(x=square number) For example you are in 6th square y=2^6-1=2^5=32. You get 32cents. But you can get all the money that you passed square. y=(2^x)-1(y=all the money, x=square number) For example you are in 6th square y=(2^6)-1=(64)-1=63. You can get all the money that 63 cents!

 A chess board have 64 squares. So you have to like this. 2^1=2 2^2=4 2^3=8 2^4=16 . . . . 2^64=18,446,744,073,709,551,616

 y=(2^64)-1=18,446,744,073,709,551,616-1=18,446,744,073,709,551,615 So you can get 18,446,744,073,709,551,615 cents,184,467,440,737,095,515.15 dollars.

 This is much bigger than 1.000.000 dollars.  help on how to format text
 * If I was that man, I choose __second__ one.**