By William M. Esposo (The Philippine Star)
The Catholic Church in our country is so messed up that it cannot...
A Tale of the Two CEOs
One day, two bold CEOs decided to play a game of chess where the winner gets to ask anything he wants from the loser. After the game, the winning CEO asked the losing CEO to choose between two payments. The first payment involves the losing CEO giving half of his company’s assets to the winning CEO. The second payment involves placing 1¢ in the first square of the chessboard, 2¢ in the second square, 4¢ in the third, 8¢ in the fourth and so on until all the 64 squares of the chessboard are filled. Thinking that it will allow him to get off easy, the losing CEO agreed to the pay the winning CEO the second reward. But the losing CEO made a very serious mistake. In the process of trying to pay the winning CEO the reward, the losing CEO ended up going bankrupt and buried in debt. In fact, the losing CEO may never be able to give the reward money even if he spends his whole life working for it.
Double, Double, Double….Jeopardy!
Human intuition evolved to understand linear progressions and patterns only. For many everyday purposes, this intuition is a quick and effective tool in assessing odds and projecting future values. The losing CEO’s big mistake is that he used the said intuition on an example where it is not applicable, an example that involved not a linear progression but a geometrical one.
When you add up the terms of an increasing geometric progression, what you get isexponential growth. As with geometric progressions, the human brain is notoriously ill equipped in understanding the power of exponential growth. This is shown by the fact that, without the aid of mathematics, almost all of us find it difficult to understand why the losing CEO made such a grave error. In order to comprehend the gravity of the losing CEO’s mistake in choosing the second payment option, let us get rid of our intuition for the moment and let us turn to mathematics.
Imagine starting with x of something. If you double that number, it becomes twice the original, 2x. If you double the previous result, you get four times the original, 4x. If you keep on doubling the most recent result, you’d successively get 8x, 16x, 32x, 64x and so on. Notice that doubling once gives you 2x or 21x while doubling twice gives you 4x or 22x. Meanwhile, doubling thrice gives you 8x or 23x and doubling four times gives you 16x or 24x. Following this pattern, we can see that doubling x an n number of times gives you 2nx.
Recall that the losing CEO started with a mere 1¢ (that is, x = 1¢). By the 8th square (the last square in the first row) he is required to double the original 1¢ seven times. This means that he must place 27 times 1¢ on the 8th square. Using a simple calculator, one can easily confirm that 27 = 128. This means that the 8th square must contain 128¢ or $1.28. So far, the losing CEO still feels he’s having it easy. However, when he reaches halfway through the chessboard (the 32ndsquare), he would have doubled the original value 31 times. This means that the 32nd square must contain 231 times 1¢. Using a calculator, one can compute that this amounts to 2 147 483 648¢ or around 21.5 million dollars! But the tragedy of the losing CEO is only beginning; even though at this point he is halfway through the chessboard, the losing CEO is still very far from paying half his due. By the time he reaches the last chess square, he is going to need a whopping 92 million billion dollars! But wait, there’s more. The said 92 million billion dollars is for the last square only. Adding up the amount of money he must place on all 64 squares of the chessboard, the total amount of money the losing CEO owes the winner is approximately 184 million billion dollars!