The legend of Sisa
The legend of the origin of chess
Supplement to the encyclopedic historical dictionary
Written by Vicenç Joaquín Bastús i Carrera (Barcelona – 1833)
At the beginning of the fifth century of the Christian era, there was in the Indies a most powerful prince, whose dominions were located on the banks of the Ganges; which had taken the famous title of King of the Indies. His father had forced a large number of sovereigns to pay him tribute and submit to his rule.
The young monarch forgot very early that kings must be the fathers of their people: that the love of the vassals to their kings is the only solid support of the throne, that only that love can truly unite the peoples with the prince who governs them , and from whom do all the strength and power: that a king without vassals will have no more than a vain title, nor gain any advantage over other men.
The brahmins and rajales, that is, certain philosophers, and the great ones represented all these things to the king of the Indians; but intoxicated with the idea of his greatness, which he contemplated eternally, he despised his wise representations. Having continued these and the complaints he took offense, and to avenge his authority, which he believed despised by those who dared to disapprove his conduct, made them perish among torments. This example frightened everyone else, and sealed their lips.
The prince, abandoned to himself, and what was even more dangerous to him and more terrible to his people, devoted to the pernicious advice of the flatterers, let himself be carried away to the last excesses. The peoples, overwhelmed by the weight of an unbearable tyranny, proved to the extreme how odious they had become an authority that only used them to make them unhappy. The tributary princes, persuaded that, having lost the King of the Indies the love of their peoples, had lost all their strength, prepared to shake off the yoke and bring the war to their states.
Then a brahma named Sisa, son of Daher, penetrated the misfortunes of his country, tried to make the prince open his eyes to the fatal effects that his behavior would produce; but taught by the example of those who had preceded him, he knew that his lesson would not be useful but taking the prince by himself without noticing that he received it from another.
With this object he invented the game of chess, in which the king, although he is the main one of the pieces, can not attack or even defend himself against his enemies without the help of his vassals and his soldiers. The new game became famous very soon; the King heard about him and wanted to learn it. Brahman Sisa was chosen to teach him, and with the pretext of explaining the rules and showing him with what art the other pieces needed to be used in defense of the king, he made him see and like the truths he had refused to hear until then.
The prince, born with a spirit and virtuous feelings that the maxims of the courtiers had not been able to quell entirely, applied these lessons of the philosopher, and realizing that the love of the peoples to their king makes all his strength, changed his conduct, and thus he foresaw the misfortunes that threatened him;
and sensible and recognized he left the brahman the choice of reward: he asked to be given the grains of wheat that summed the number of houses on the board in this way: one for the first, two for the second, four for the third, doubling thus by the others until the 64. Admirado the king of the apparent shortness of the request it granted it to the instant and without examination; but having been calculated by his treasurers, they found that he had compelled a thing, for the satisfaction of which all his treasures and vast states would not suffice.
In effect, they saw that the sum of the grains of wheat had to be evaluated in 16,284 cities, of which each one had 1024 granaries, that in each of them there were 174,762 measures, and in each of these 32,768 grains. The philosopher then used the occasion to make the prince understand how much it matters to kings to be careful against those around them, and how much they should fear that their best intentions are abused.
Problem of wheat and chessboard
El denominado problema del trigo y del tablero de ajedrez (a veces puede aparecer expresado en términos de granos de arroz), es un problema matemático cuyo enunciado es el siguiente, palabras más, palabra menos:
“Si se colocase sobre un tablero de ajedrez (lo suficientemente grande) un grano de trigo en el primer casillero, dos en el segundo, cuatro en el tercero y así sucesivamente, doblando la cantidad de granos en cada casilla, ¿cuántos granos de trigo habría en el tablero al final?”
Solution by brute force
La solución de fuerza bruta consiste en duplicar manualmente cada potencia de dos e ir acumulando el sumatorio correspondiente a esa serie geométrica.
donde corresponde al número total de granos.
La serie puede ser expresada como exponentes:
y representarse en notación de sumatoria (sigma mayúscula) como:
También puede resolverse de forma mucho más fácil por medio de:
Una prueba de lo cual es:
Multiplicar cada lado por 2:
Restar o sustraer la serie original de cada lado:
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