006. Sessa, the Indian inventor of chess who made a great king bewildered

Page created: Year 2014

Sessa, the Indian inventor of chess who made a great king bewildered

Source: from a Russian book, Mathematics can be fun, Mir Publishers, Moscow.
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The Legend about a Checkered Board. Chess is one of the oldest games in the world. It was invented many, many centuries ago and it is not surprising, therefore, that there are so many legends about it, legends that it is, of course, impossible to verify. I should like to relate one of them. It is not necessary to know how to play chess to understand the legend: it is enough to know that it is played on a checkered board with 64 squares.
Chess, legend has it, comes from India.
King Sheram was thrilled by the huge number of clever moves one could make in the game. Learning that its author was one of his subjects, he commanded the man to be brought before him in order to reward him personally for his marvellous invention. The inventor, a man called Sessa, appeared before the king, a simply clad scholar who made his living by teaching.
"I wish to reward thee well for thy wonderful invention," the king greeted Sessa.
The sage bowed.
"I am rich enough," the king continued, "to satisfy thy most cherished wish. Just name what thou wouldst have and thou shalt have it."
Sessa was silent.
"Don't be shy," the king encouraged him. "Say what thou wouldst like to have. I shall spare nothing to satisfy thy wish."
"Thy kindness knows no bounds, ? Sire," the scholar replied. "But give me time to consider my reply. Tomorrow, after I have well thought about it, I shall tell thee my request."
The next day Sessa surprised the king by his extremely modest request.
"Sire," he said, "I should like to have one grain of wheat for the first square on the chessboard."
"A grain of ordinary wheat?" The king could hardly believe his ears.
"Yes, Sire. Two for the second, four for the third, eight for the fourth, 16 for the fifth, 32 for the sixth"....
"Enough," the king was irritated. "Thou shalt get thy grains for each of the 64 squares of the chessboard as thou wishest: for each square double the amount of the preceding square. But know thou that thy request is not worthy of my generosity. By asking for such a trite reward thou hast shown disrespect for me. Truly as a teacher, thou couldst have shown a better example of respect for thy king's kindness. Go! My servants shall bring thee thy sack of grain."
Sessa smiled and went out, and then waited at the gate for his reward.
At dinner the king remembered Sessa and inquired whether the "foolhardy" inventor had been given his miserable reward.
"Sire," he was told, "thy command is being carried out.
Thy sages are calculating the number of grains he is to receive."
The king frowned. He was not accustomed to seeing his commands fulfilled so slowly.
In the evening, before going to bed, the king again asked whether Sessa had been given his bag of grain.
"Sire," was the reply, "thy mathematicians are working incessantly and hope to compute the sum ere dawn breaks."
"Why are they so slow?" the king demanded angrily.
"Before I awake Sessa must be paid in full, to the last grain. I do not command twice!"
In the morning, the king was told that the chief court mathematician had asked for an audience.
The king ordered him to be admitted.
"Before thou tellest me what thou hast come for," King Sheram began, "I want to know whether Sessa has been given the niggardly reward he asked for."
"It is because of this that I have dared come before thy eyes so early in the morning," the old sage replied. "We have worked conscientiously to calculate the number of grains Sessa wants. It is tremendous, indeed "
"However tremendous," the king interrupted him impatiently, "my granaries can easily stand it. The reward has been promised and must be paid!"
"It is not within thy power, ? Sire, to satisfy Sessa's wish. Thy granaries do not hold the amount of grain Sessa has asked for. There is not that much grain in the whole of thy kingdom; in fact, in the whole world. And if you wouldst keep thy word, thou must order all the land in the world be turned into wheat fields, all the seas and oceans drained, all the ice and snow in the distant northern deserts melted. And if all this land is sown to wheat, then perhaps there will be enough grain to give Sessa." The king listened awe-struck to the wise man.
"Name this giant number," he said thoughtfully. "It is 18 446 744 073 709 551615. ? Sire!" the sage replied.
So goes the legend. We do not know whether it was really so, but that the reward would run into such a number is not difficult to see: with a little patience we can calculate it ourselves.
Starting with one we must add up the numbers: 1, 2, 4, 8, etc. The result of the 63rd power of 2 will show us how much the inventor was to receive for the 64th square. Following the pattern shown on page 65 we shall easily find the number of grains if we find the value of 264 and subtract 1.
In other words, we must multiply 64 twos:
2x2x2x2x2x2, etc. 64 times.
To facilitate calculation we shall divide these 64 factors into 6 groups of 10 twos, the last group to contain 4 twos. The product of ten twos is 1 024, and of four twos is 16. Hence, the value we seek is
1 024 x 1 024 x 1 024 x 1 024 x 1 024 x 1 024 x 16.
Multiplying 1 024 by 1 024 we get 1 048 576.
What we have to find now is
1 048 576 x 1 048 576 x 1 048 576 x 16
and subtract 1 from the result, and then we shall know the
number of grains:
18 446 744 073 709 551 615.
If you want to have a clear picture of what this giant number is really like, just imagine the size of the granary that will be required to store all this grain. It is well known that a cubic metre of wheat contains 15 000 000 grains. Hence, the reward asked by the inventor of chess would require a granary of approximately 12 000 000 000 000 cubic metres or 12 000 cubic kilometres. If we take a granary 4 metres in height and 10 metres in width, its length must be 300 000 000 kilometres, i. e. twice the distance from the earth to the sun.

Simple direct calculation
1 = 1

1 x 2 = 2

2 x 2 = 4

4 x 2 = 8

8 x 2 = 16   

16 x 2 = 32

32 x 2 = 64          grains for the 7^th. The king must have calculated in his mind till here only

64 x 2 = 128

128 x 2 = 256

256 x 2 = 512

512 x 2 = 1024

1024 x 2 = 2048

2048 x 2 = 4096     grains for 13^th square

4096 x 2 = 8192

8192 x 2 = 16384

16384 x 2 = 32768

32768 x 2 = 65536

65536 x 2 = 131072

131072 x 2 = 262144

262144 x 2 = 524288      grains for 20^th square

524288 x 2 = 1048576

1048576 x 2 = 2097152
2097152 x 2 = 4194304
4194304 x 2 = 8388608
8388608 x 2 = 16777216
16777216 x 2 =  33554432        grains for 26th square
33554423 x 2 = 67108864
67108864 x 2 = 134217728
134217728 x 2 = 268435456
268435456 x 2 = 536870912      536 million+ grains or 53 crores 68 lakhs+ grains
536870912 x 2 = 1073741824
1073741824 x 2 = 2147483648
2147483648 x 2 = 4294967296      grains for 33rd square
4294967296 x 2 = 8589934592
8589934592 x 2 =  17179869184     
                               17 billion 179 million+ grains or 1717 crores 98 lakhs+ grains
17179869184 x 2 = 34359738368
34359738368 x 2 =  68719476736       grains for 37th square

How the king could have handled the situation

The king was unable to satisfy Sessa's request. But had he been clever in mathematics, he would have easily avoided promising such a huge reward, all he should have done was to offer Sessa to count the grains himself, one by one and take them to home. Indeed, if Sessa had counted the grain day and night, without stopping, taking a second for each grain, he would have counted 86 400 grains on the first day. One million grains would have taken him no fewer than 10 days to count. It would have taken him about six months to count the grains in one cubic metre of wheat - that would have given him 27 bushels. Counting without interruption for 10 years, he would have counted off about 550 bushels. You will see that even if Sessa had devoted all the remaining years of his life to counting the grain, he would have got only an insignificant part of the reward.