Una antiquísima leyenda cuenta que Sheram, príncipe de la india, quedó tan maravillado cuando conoció el juego del ajedrez, que quiso recompensar generosamente a Sessa, el inventor de aquel entretenimiento. Le dijo: "Pídeme lo que quieras". Sessa le respondió: "Soberano, manda que me entreguen un grano de trigo por la primera casilla del tablero, dos por la segunda, cuatro por la tercera, ocho por la cuarta, y así sucesivamente hasta la casilla 64".
El príncipe no pudo complacerle, porque el resultado de esa operación S = 1 + 2 + 4 + ... + 263 es aproximadamente 18 trillones de granos. Para obtenerlos habría que sembrar la Tierra entera 65 veces.
Pulula por los círculos matemáticos un sorprendente final de la historia. Sheram, preocupado al haber empeñado su palabra, mandó llamar al matemático del reino, un tal Pepe Martínez Aroza, el cual razonó de la siguiente manera:
"Alteza, puesto que no tenéis trigo suficiente para pagar la deuda contraida con Sessa, igual os daría deberle aún más. Sed, pues, magnánimo y aumentad vuestra recompensa a la cantidad S = 1 + 2 + 4 + 8 + ... hasta el infinito. Observad que, a partir de la segunda casilla, todas las cantidades a sumar son pares, lo cual nos permite escribir S = 1 + 2 × ( 1 + 2 + 4 + 8 + ... ), o lo que es lo mismo, S = 1 + 2 × S. Ahora, vos mismo podéis resolver esta sencilla ecuación de primer grado y, veréis que la única solución es S = -1. Podéis decir a Sessa que no solamente puede considerarse pagado con creces, ya que habéis aumentado enormemente vuestra recompensa, sino que actualmente os adeuda un grano de trigo."
English Version
An ancient legend says that Sheram, prince of India, was so amazed when he learned the game of chess, who wanted to generously reward Sessa, the inventor of that entertainment. He said: "Ask what you want. " Sessa said, "Sovereign, send me a grain of wheat delivered by the first square on the board, two for second, four for third, eight for the fourth, and so on until the box 64 ".
The prince could not please him, because the result of the operation S = 1 2 4 ... 263 is about 18 trillion grains. To get them requires growing the whole Earth 65 times.
Mathematical circles swarming around a surprise ending of the story. Sheram, worried about having pledged his word, he sent for the mathematical realm, a certain Aroza Pepe Martinez, who reasoned as follows:
"Highness, since you do not have enough wheat to pay the debt owed to Sessa, like you would owe more. You, therefore, magnanimous and increase your reward to the quantity S = 1 2 4 8 ... ad infinitum. Observe that from the second box to add all amounts are pairs, which allows us to write S = 1 2 × (1 2 4 8 ...) or what is, S = 1 2 × S . Now, you yourself can solve this simple equation of first grade and you will see that the only solution is S = -1. Sessa You can tell that not only can be considered more than repaid, and you have greatly increased your reward, but now I owe a grain of wheat. "
The prince could not please him, because the result of the operation S = 1 2 4 ... 263 is about 18 trillion grains. To get them requires growing the whole Earth 65 times.
Mathematical circles swarming around a surprise ending of the story. Sheram, worried about having pledged his word, he sent for the mathematical realm, a certain Aroza Pepe Martinez, who reasoned as follows:
"Highness, since you do not have enough wheat to pay the debt owed to Sessa, like you would owe more. You, therefore, magnanimous and increase your reward to the quantity S = 1 2 4 8 ... ad infinitum. Observe that from the second box to add all amounts are pairs, which allows us to write S = 1 2 × (1 2 4 8 ...) or what is, S = 1 2 × S . Now, you yourself can solve this simple equation of first grade and you will see that the only solution is S = -1. Sessa You can tell that not only can be considered more than repaid, and you have greatly increased your reward, but now I owe a grain of wheat. "
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