# Ancient Indian Legend about the invention of chess

There is an ancient Indian legend about the invention of chess. Ruler of India was so pleased with one of his wise men, who had invented the game, that he offered this wise man a reward of his choosing. The wise man said:

"Give me one grain of wheat for the first square of the chessboard, two grains for the next square, four for the next and so on for all squares, with each square having double the number of grains as the square before."

The ruler agreed, surprised that the man had asked for such a small reward. After a couple of days, his treasurer came back and informed him that the reward would add up to an astronomical sum, far greater than all the wheat that could conceivably be produced in many years!

Can you count how many grains of wheat would be on the chessboard at the finish?

The correct answer is below.

Simple solution is to manually double and add each step of the series:

TOTAL = 1 + 2 + 4 + 8 + 16 + ... + 9 223 372 036 864 775 808 = 18 446 744 073 709 551 615

The series may be expressed using exponents:

TOTAL = 2^{0} + 2^{1} + 2^{2} + 2^{3} + 2^{4} + ... + 2^{63} = 2^{64} - 1

because

a = 2^{0} + 2^{1} + 2^{2} + 2^{3} + 2^{4} + ... + 2^{63} (multiply each side by 2)

2a = 2^{1} + 2^{2} + 2^{3} + ... + 2^{64}

2a = a - 2^{0} + 2^{64 }

2a - a = 2^{64} - 2^{0} (remember that: 2^{0} = 1)

a = 2^{64} - 1

to sum up**TOTAL = 2 ^{64} - 1 = 18 446 744 073 709 551 615**

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