The classic chessboard problem

There is a story about the invention of chess which illustrates some interesting facts about exponential growth and large numbers. The story is that the mathematician who invented chess showed it to the king of India who was pleased and said the inventor could ask for any prize he wished. What he asked for seemed small as he asked for just one grain for the first square, two for the second, four for the third and so on. This can be seen to follow a exponential increase where each square has 2(n-1) grains of rice. 
Two is raised to the (n-1) because the first square has 1 or 20=2(1-1)=1 grains. At first this seems like quite a small request when he could have asked for chests of gold and diamonds or a large piece of land to live on. However the amount of rice rises exponentially and even just looking at the final square shows the total is going to be a huge amount of rice.
The final square or the 64th square as that is how many are on a chessboard will have 2(64-1)=263=9.22*1018 grains of rice. The total number of grains of rice is going to be 264-1=18.44*1018 which can be found from the exponential sum (where r=2 and N=64 using the formula in this link) or graphically from this link (where the 1/2 box is equal to the 64th square and therefore the total is double [minus one because there is no 1/2, 1/4, 1/8... grains])
Now its hard to imagine how much rice this is as 1018 can't be easily imagined so i'm going to relate it to other things. 
1) By Weight
A grain of rice weighs about 25 milligrams or .025 grams. So 18.44*1018 grains of rice weigh (18.44*1018)*(.025 grams)=4.612*1017 grams. This is also a huge number which is hard to visualize. It is equal to 461 billion metric tons, which is over five times the weight of every living plant and animal on earth. This is also equal to the amount of rice produced worldwide in a year multiplied by 625 (it would take 625 years of the whole world producing rice at our current rate to collect that much)
2) By Volume
A grain of rice is about 5 millimeters long by 2.5 millimeters wide. If we arranged this many grains of rice in a line they would be 264*(.005m)=9.22*1016m which is 9.75 light years or more than enough rice to stretch from earth to proxima centauri (the closest star) and back. If we were to compare the distance to something a little closer like the moon, which is 3.939*108m away we would find we have enough rice to stretch to the moon (.005)264/(3.939*108)=2.34*108 times. If we had a square shaped tower of rice to the moon each side would be 15,302=sqrt(2.34*108) grains of rice wide. 15,302 grains of rice is 38.25 meters wide (15302*.0025) or 125 ft.
Using just the the rice from a chessboard of exponential growth we end up with
461 billion metric tons
625 years of worldwide rice production
9.75 light years of rice laid end to end
A line of rice from here to the nearest star and back
A tower of rice 125' feet on each side rising all the way to the moon
Obviously this is a lot of rice, and maybe the power of exponential growth is a little clearer.


Jared G.

Math and Physics from Elementary to University

800+ hours
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