The Immense Power of Exponential Growth
My father taught me to play Chess. And I am still a terrible player. However, I do remember a wonderful little story about exponential growth he wove around the origins of the game.
Legend has it that an Indian King was presented with a beautifully hand-crafted chess board by a chess-master mathematician. Delighted with the magnificent piece the king asked what the mathematician might like in return. Humbly, the mathematician requested that a grain of rice be put on the 1^{st} square, 2 on the second square, 4 on the third….. doubling at each successive square. The King quickly agreed to his humble request.
Things were going really well (at first) and are summarised in the following table.
Square Number |
Grains on that Square | Total Rice on Board |
1 |
1 | 1 |
2 |
2 |
3 |
3 |
4 | 7 |
4 |
8 |
15 |
5 |
16 |
31 |
6 |
32 |
63 |
7 | 64 |
127 |
8 | 128 |
255 |
After the first row, the total amounts to 255 grains of rice – barely half a cup full. And some interesting relationships emerge. The total rice on the board is given by
2^{(Square Number)} -1.
For example:
Square Number |
2^{(Square Number) }– 1 |
Total Rice on Board |
1 |
2^{1 }– 1 |
1 |
4 |
2^{4} – 1 |
15 |
8 |
2^{8} – 1 |
255 |
Then, square 12 = |
2^{12} – 1 |
4095 |
Based on this how much rice might the King owe at the end of the second row of the Chess Board?
And at the end of the first half of the board? I.E. on the 32^{nd} square?
16 |
2^{16} – 1 |
65,535 |
32 |
2^{32} – 1 |
4,294,967,295 |
So at the half way point (i.e. the 32^{nd} square on the Chess Board), the King owes the mathematician over 4 Billion grains of rice.
Clearly the mathematician had a plan…