<p>Just wanted to throw in the observation that E. T. Bell, in Men of Mathematics, wrote that Gauss was 10 when he came up with the summation method spontaneously (rather than 6 or 7, which you might expect for a first-grader). </p>
<p>Sigma Xi has a site with multiple tellings of this tale, over the years: <a href=“404”>Error;
<p>The excerpt from work by A. Galle in 1916 stated that Gauss was 9 at the time. Ludwig Haenselmann in 1878, asserted that Gauss was “in his ninth year,” which I take to mean 8 + some months. If I had to bet, I’d go with Haenselmann. Spotting the summation trick at 8 or even 10 is still plenty of an illustration of genius, especially in Gauss’s era.</p>
<p>Still, it is worth noting that while Gauss had not been shown this particular method, he was not completely untutored in mathematics beforehand. I have read that his uncle set mathematical puzzles for Gauss when Gauss was a young boy (of preschool age).</p>
<p>I am firmly of the belief that anyone can increase his/her mathematical problem-solving skill by working at it. If you try to approach your study by just learning algorithms, it will most likely give you a few extra points in the math contests, but not take you very far. If you approach your study by trying to really understand things, I think you will find that your creativity blossoms, too.</p>
<p>In a related vein, when I listened to seminars or read scientific papers or books back in college, I would generate a few questions, but they weren’t all that great–mostly pretty detail-oriented. Now, after many years, I am all about questions. Every scientific issue I encounter inspires large numbers of questions. </p>