Ciciora’s Corner: The Second Half of the Chess Board
Exponential growth might be problematic with grains of rice, but it's marvelous when talking about bits and bytes
Futurist Ray Kurzweil is reported to have coined the phrase “the second half of the chessboard” to illustrate the impact of exponential growth. The cable industry and the semiconductor industry have both experienced exponential growth. Let’s use this interesting idea to take a closer look at exponential growth.
The chessboard is the basis of a story often used to illustrate the power of an exponential increase. In this story, a man does something that pleases the powers that be and is told to name his reward. The hero cleverly says that he would like one grain of rice on the first square of the chessboard, and double that on the second, followed by double that again on the next square and so on until all eight rows and eight columns are filled. The King (or Emperor) can’t believe the request is so small. The reward is ordered to be measured out.
Soon it becomes clear that the request exceeds all the grain in the land. The hero is now a villain and is executed for embarrassing the King.
If you have a spreadsheet program on your computer, try it yourself. Put one in the first cell, double that in the next, and double again in the next cell until your reach sixty four. Total up the number of grains that would go on the chessboard. My calculation comes to:
9,223,372,036,854,780,000 grains on the sixty fourth square and
18,446,744,073,709,600,000 grains on the whole board
Now, a million is a thousand thousands, and a billion is a thousand millions. The progression is: thousands, millions, billions, trillions, quadrillions, and quintillions. So there are 18 quintillion grains on the whole board.
You can use your spreadsheet to graph this function: Now, that’s a hockey stick! Exponential growth is a serious thing.
Note that the number of grains on any square exceeds the total of all of the grains on all of the previous squares.
Since there are 64 squares on the chessboard, the first half has 32 squares and, from the spreadsheet, that square will have 2,146,483,648 grains and the total number of grains on the first half is 4,294,967,295. The president’s proposed U.S. budget is just under $4 trillion, in the same magnitude as the number of grains on the first half of our chessboard’s load.
If a grain of rice weighs 25 mg, I’ll let you verify that the first half of the chessboard holds 118,111 tons of rice. If you want to be astonished, figure out the weight held by the second half of the chessboard.
The purpose of all of this is to get an appreciation for the power of exponential growth. Carl Sagan, who apparently did not say “billions and billions,” did say “Exponentials can’t go on forever, because they will gobble up everything.” There are things you and I own in such huge numbers. For example, consider transistors. The transistor was invented in 1947. Moore’s Law says the number of transistors on a chip for a given price doubles about every 18 to 24 months. Let’s be aggressive with Moore’s Law. There have been 44 18-month periods in the last 66 years.
Look at the spread sheet you made.
The 44th square has more than 17.5 quadrillion grains. In our example, those grains represent bits. But we usually measure memory in terms of bytes. So divide by eight and get 2,199 gigabytes, or 2 terabytes. You likely have a couple of terabytes of memory in all of the old computers, laptops, cellphones, iPods, etc. Even more likely, you have a couple of terabytes of old hard drives lying around.
Now let’s try that being a bit more conservative, using 24 months for a doubling, or 33 doublings of transistor growth since 1947. Go back to the spreadsheet, and note that this is the first square on the second half of the chessboard. That square has 8,589,934,591 grains representing 8.5 gigabits – or 1 gigabyte. We certainly have gigabyte memory chips in our digital cameras! So we can conclude that Moore’s Law applies to digital transistors part way between the conservative and the aggressive forms.
More big numbers: A couple of years ago I was involved in a case concerning the amount of usage of a cable system for video versus telephony. The telephony equipment totaled the number of bits used for telephony each month. I decided to calculate the number of bits carried on the system in a month for video and form a ratio. That quickly got to huge numbers. The end result was a ratio of more than seven thousand video bits for every telephony bit. I was surprised and pleased with that result.