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There is a fable that tells of an ancient Indian mathematician who invents the game of chess. The emperor of India is so impressed that he offers the mathematician anything he wishes. According to the fable, he "simply" asks for a single grain of rice to be placed on the first square of the chessboard, then doubled on each successive square.
A simple request? In reality, he knows that he is asking for more rice than can possibly be provided for him. (Apparently, mathematicians have a long history of being tricksters and con artists.) I can create superscripted exponents on my computer but I don't know if they will show up correctly on yours, so let's agree to use the "^" symbol to indicate raising something to a power.
If you think about the grains of rice placed on just the first row of a chessboard, you may recognize that they are all powers of 2...
2^0 = 1
2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 16
2^5 = 32
2^6 = 64
2^7 = 128 and so on.
The total amount of rice in the first row, then, would be
1+2+4+8+16+32+64+128 = 255 grains of rice
Is there a formula that might help us find this more quickly? Yes. Take 2 and raise it to the number of squares, then subtract 1. Try it...
2^8 - 1 = 255 grains
So the total amount of rice for the first two rows would be
2^16 - 1 = 65,535 grains
You can either trust me on this or add rice on the 16 squares yourself. The formula does work, though!
This means that the total amount of rice on the first half of the board would be
2^32 - 1 = 4,294,967,295 grains
Over 4 billion grains of rice -- around 100,000 kilograms of the stuff -- and we're only on the first half of the board. Because we're increasing exponentially, it turns out that there is more than one BILLION times more rice on the second half of the board than there was on the first!
Again using our formula, the entire board would contain
2^64 - 1 = 18,446,744,073,709,551,615 grains of rice
This is over 18 QUINTRILLION grains of rice, which would weight over 46 million metric tons! To give this a bit of perspective, this would be more than the weight of six million ships the size of the Queen Elizabeth II.
No emperor could possibly provide that much rice!
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