Mathematics+Activity

Mathematics activity

In math class, we had one math problem

Once, there was a king in the kingdom. One day, the enemy attacked his kingdom. So his kingdom was destroyed by enemy. The king want a person who can save the kingdom. The kingdom as one pleases, one man appeared and he solved this situation. King was so happy. So king said "You save my kingdom, I want to give you something. For examples, you can get 1.000.000 dollars. He said " I want that, if I pass each square in chess board I can get cents. If I 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. Then I can have all the money that you passed square. Can you allows that my opinion?" The king said "Of,course."


 *  Extension **

You have won an amazing competition!  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!

 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!

<span style="font-family: Tahoma,Geneva,sans-serif;"> A chess board have 64 squares. So you have to like this. <span style="font-family: Tahoma,Geneva,sans-serif;"> 2^1=2 <span style="font-family: Tahoma,Geneva,sans-serif;"> 2^2=4 <span style="font-family: Tahoma,Geneva,sans-serif;"> 2^3=8 <span style="font-family: Tahoma,Geneva,sans-serif;"> 2^4=16 <span style="font-family: Tahoma,Geneva,sans-serif;">. <span style="font-family: Tahoma,Geneva,sans-serif;">. <span style="font-family: Tahoma,Geneva,sans-serif;">. <span style="font-family: Tahoma,Geneva,sans-serif;">. <span style="font-family: Tahoma,Geneva,sans-serif;"> 2^64=18,446,744,073,709,551,616

<span style="font-family: Tahoma,Geneva,sans-serif;"> y=(2^64)-1=18,446,744,073,709,551,616-1=18,446,744,073,709,551,615 <span style="font-family: Tahoma,Geneva,sans-serif;"> So you can get 18,446,744,073,709,551,615 cents,184,467,440,737,095,515.15 dollars.

<span style="font-family: Tahoma,Geneva,sans-serif;"> This is much bigger than 1.000.000 dollars.

<span style="font-family: Tahoma,Geneva,sans-serif;"> **I think he is clever**