Today I will tell the famous inventor of Ajedrez.Dice legend and history: the King of Persia dead boring at times, suddenly became fascinated by the game of chess, which presented a witty and clever inventor. It is said that he was so grateful that Eastern mathematician king offered what he wanted. The inventor replied
- I'll settle for 1 grain of wheat for the first square on the board, 2 for second, 4 for third, 8 for fourth and so on until the box 64 of the board. (Ie the sum of the 64 first terms of a ratio PG 2 and whose first term is 1).
- I'll settle for 1 grain of wheat for the first square on the board, 2 for second, 4 for third, 8 for fourth and so on until the box 64 of the board. (Ie the sum of the 64 first terms of a ratio PG 2 and whose first term is 1).
King taunted the minutiae thinking I was asking and asking his vizier to prepare the award requested, did the calculations and gave that it was impossible to enforce the order, as the sum of the grains of the 64 squares was nothing less than the amount of 18,446,744,073,709,551,616 grains (each kilogram of wheat fit 28220 about a grain, so that the result would be about 653,676,260,585 tons, which would occupy a cube-shaped deposit of more than 11.5 km squares. To produce such a quantity of wheat would need to be cultivating the Earth, including oceans, for eight years).
A second part of the story, which is the next king because of the embarrassment of having to accept that there was enough grain to pay, talk to other intelligent and witty man of his court for him to pull out of trouble and this he proposed the following:
- the inventor to see how generous you are, offer him not only the sum of the 64 first terms, but the sum of an infinite chess board.
To which the king said:
- You're crazy! If I have to pay what a normal chess board, as I do to extend the sum to infinity would be infinite grains ...
ingenious
But the assistant said
"Take me to the inventor of chess and trust me, everything will be alright !!!!:
Once assembled, the witty assistant will proposed to the inventor, that the king was so pleased and happy with the game of chess and was so generous, not only offered to give the sum of 64 squares, but the sum of the squares of a chess board infinity. To which, the inventor shrug accepted. And the king's aide went on to explain: Let's call
S = 1 +2 +4 +8 +16 +..... (The sum of the infinite grains of wheat board infinity). Now multiply by 2, so we will have 2 • S = 2 +4 +8 +16 +32 +.... Then subtract 2 • S - S = (2 +4 +8 +16 +8 +4 +2 +....)-( a +....) +16
thus, removing the parentheses , the summands of the second brackets change every sign and we are canceling the first parénteis that are positive and thus
we have that S = -1.
words: not only no longer had to pay the inventor, but he owed over a grain. Amazing is not it ?!!!!!!
Where would you say is the explanation for this problem? Well ..... I await your answers and I answer in a few days. And have no reward THINK!!
0 comments:
Post a Comment