[Math Lair] Insight

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Perhaps the best known story about the ancient Greek mathematician Archimedes is that, when the solution to a problem occurred to him in a bath, he ran home, naked, through the streets of Syracuse, shouting "Eureka!" ("I have found it") as he ran. While you've probably never done something like that, you probably have had the experience of having a sudden flash of insight, out of the blue, that allows you to solve a problem that you're working on.

These flashes of insight often arise all of a sudden, often when you're not consciously working on the problem. To take another example, in 1843 William Rowan Hamilton had been unsuccessfully trying to multiply triplets. The key insight, which allowed him to discover quaternions, came to him on a walk with his wife along a canal in Dublin. Staggered by this insight, he carved the key to the problem, i² = j² = k² = ijk = −1 on Brougham Bridge. Many other examples of these sudden insights occur in mathematics and also in many other fields of study.

While you may be no Archimedes or William Rowan Hamilton, it's likely that you'll encounter many problems, both in mathematics and in other areas, that, while ostensibly difficult, are quite simple if you look at them in the correct manner. So, how can you come up with these creative insights? Well, it isn't entirely clear how to do that. It isn't clear how these creative hunches arise or what occurs in a creative person's head to cause these insights to arise. However, you can increase the likelihood of these insights occurring. Here are some useful suggestions:

  1. Gain an understanding of both the subject area in question and of mathematics in general can help.
  2. Learn how how to solve problems, including knowing various common heuristics.
  3. Practice thinking in a creative, offbeat, non-linear manner. Learning about lateral thinking can be helpful.

There are several pages of puzzles and problems on this site that have surprising answers, but are straightforward if you look at them in the correct way. For example, see:

Sources used (see bibliography page for titles corresponding to numbers): 41.