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exponential increase [try it]


Take a chess board and some rice. Put a grain of rice on the first square, Rook’s a1. Put two grains on the next one, Knight’s b1. Keep going down the row, doubling the grains at every square (4 on Bishop’s c1, 8 on King’s d1, etc), and continue the same way for row 2, and so on, for as long as you can.

One simple trick to double the grains without counting them out each time is to do this: sweep up the grains from all the filled up squares into the next one, and add one grain. (And of course, you would have to fill in all the previous squares again before you move on to the next one.) That is, 2N = 1 + Sumk=0k=N-1 2k , which is quite straightforward to show if you write it out in binary notation.

At which square do you need a kilogram of rice? At what stage do you run out of all the rice in your house? When do you reach the yearly capacity of the country?