Legend has it that the game of chess was invented by a mathematician who worked for a king. The king was very pleased. He said, “I want to reward you.” The mathematician said “My needs are modest. Please take my new chess board and on the first square, place one grain of wheat. On the next square, double the one to make two. On the next square, double the two to make four. Just keep doubling till you’ve doubled for every square, that will be an adequate payment.” We can guess the king thought, “This foolish man. I was ready to give him a real reward; all he asked for was just a few grains of wheat.”
But let’s see what is involved in this. We know there are eight grains on the fourth square. I can get this number, eight, by multiplying three twos together. It’s 2x2x2, it’s one 2 less than the number of the square. Now that continues in each case. So on the last square, I’d find the number of grains by multiplying 63 twos together.
Now let’s look at the way the totals build up. When we add one grain on the first square, the total on the board is one. We add two grains, that makes a total of three. We put on four grains, now the total is seven. Seven is a grain less than eight, it’s a grain less than three twos multiplied together. Fifteen is a grain less than four twos multiplied together. That continues in each case, so when we’re done, the total number of grains will be one grain less than the number I get multiplying 64 twos together. My question is, how much wheat is that?
You know, would that be a nice pile here in the room? Would it fill the building? Would it cover the county to a depth of two meters? How much wheat are we talking about?
The answer is, it’s roughly 400 times the 1990 worldwide harvest of wheat. That could be more wheat than humans have harvested in the entire history of the earth. You say, “How did you get such a big number?” and the answer is, it was simple. We just started with one grain, but we let the number grow steadily till it had doubled a mere 63 times.
Now there’s something else that’s very important: the growth in any doubling time is greater than the total of all the preceding growth. For example, when I put eight grains on the 4th square, the eight is larger than the total of seven that were already there. I put 32 grains on the 6th square. The 32 is larger than the total of 31 that were already there. Every time the growing quantity doubles, it takes more than all you’d used in all the proceeding growth.Each doubling produces twice the amount previously produced
At another point in the lecture, Bartlett gives the example of bacteria filling a bottle. The doubling rate is every minute, so Bartlett poses this challenge to his students:
If you were an average bacterium in that bottle, at what time would you first realize you were running of space? Well, let’s just look at the last minutes in the bottle. At 12:00 noon, it’s full; one minute before, it’s half full; 2 minutes before, it’s a quarter full; then an 1/8th; then a 1/16th. Let me ask you, at 5 minutes before 12:00, when the bottle is only 3% full and is 97% open space just yearning for development, how many of you would realize there’s a problem?
- Cheap (without reference to social, environmental, disposal or clean-up costs);
- Durable (even indestructible by natural decay processes);
- Lightweight (even buoyant) and compact.