Reinventing Discovery_ The New Era of Networked Science - Michael Nielsen [37]
Most people don’t find this an easy puzzle. But it’s worth struggling with it for a few minutes before reading on. If you do, and you try laying out imaginary (or real) dominoes on a chessboar If you dou’ll discover that no matter how hard you try, you can’t quite do it. It’s as though there’s an unseen obstruction that is somehow preventing you from succeeding. In fact, there is no way of covering the board in the way requested. Here’s why. The key is to notice that if you put a domino down on the board, no matter where you put it, it will cover a total of one black square and one white square. So if you put two dominoes down, there will be a total of two black squares covered, and two white squares. Three dominoes means three black squares covered and three white squares covered. And so on. No matter how many dominoes you put down, the total amount of black and white covered will be the same. But notice that both the bottom left and top right squares on the chessboard are black. So to reach a situation where they are the only squares uncovered, you need to somehow cover 32 white squares and 30 black squares. That’s an unequal number, so there’s no way it is possible.
Figure 5.1. The puzzle starts with an empty eight-by-eight chessboard, as shown on the left. You’re asked if it’s possible to cover the chessboard with one-by-two dominoes, so that only the bottom left and top right squares remain uncovered. On the right, I’ve shown a failed attempt to do this, which leaves two extra squares in the top right corner uncovered.
Although most people find it hard to solve this puzzle, when the solution is explained they quickly say, “Aha, I see it!” It’s much easier to recognize the insight that solves the problem than it is to have that insight. Put slightly differently, there’s a gap between the difficulty of recognizing the insight and the difficulty of having the insight in the first place. A similar gap is present in examples such as the Polymath Project, Kasparov versus the World, and the MathWorks competition. Consider the MathWorks competition. It requires tremendous ingenuity to write programs that quickly pack CDs nearly full of songs. But, as we’ve seen, it’s easy to recognize when someone has written a good program: simply run the program on a few test inputs, and check that it runs fast, and leaves little space left over on the CD. It’s that gap between the difficulty of writing programs and the ease of evaluating them that fuels collective progress in the MathWorks competition. In chess, recognizing valuable insight isn’t quite as straightforward, but a competent chess player such as Krush can recognize and understand an exceptionally insightful analysis of a particular position, even if she couldn’t have come up with the analysis on her own. The best analyses may even stimulate the same feeling of “Aha, how clever!” as in the domino puzzle. Krush can’t play consistently at Kasparov’s level, but she’s good enough to recognize when other people are (momentarily, at least) playing at that level, and to understand their analyses. And in the Polymath Project, participants could recognize when others had mathematical insights that exceeded their own, and could incorporate those insights into their collective knowledge. Again, it’s that “Aha!” feeling stimulated by a clever insight. Each project thus has used this gap between our ability to have and to recognize useful insights, in order to convert individual insight into collective insight.
The problem in the Stasser-Titus experiments is that the small group discussions did not reliably convert individual insight into collective insight. Intellectually, many of the students participating the experiments would no doubt have agreed that the way to go was to systematically pool all their information, and then to make a decision based on the combined profiles so constructed. But in practice they didn’t do that. And, given the context, this is not