Insight
What people look for in mathematics is insight. Given the same problem that makes most people struggle, some people just see through it and reach a conclusion in a flash of light. When their answers are presented, people often marvel "What a genius." or "Ah-ha, how come I didn't see such a simple solution?"
Leonhard Euler was a man with insight. His solution to the zeta function, x(2n), using the infinite series for (sin Öx)/Öx left people pondering, "How did he come up with that crazy idea?" His solution to the Konigsberg bridge problem was just as amazing and led to the branch of mathematics called graph theory. It is people like Euler who push mathematics forward.
Of course, Euler accomplished much much more than what I just mentioned. In almost every branch of mathematics, you will hear his name. I think it is too complicated to go into the zeta function or the Konigsberg bridge problem to illustrate what insight is. Rather, I will give a couple simple examples here. You are welcome to take the challenge. Some of you might have seen the problem somewhere.
(1) In the U.S. Tennis Open, there are 128 single players. Two player play a match and the winner moves onto the next round. How many matches should be played in order to reach a champion?
Most of us would say, there are 64 matches in the first round, 32 the next, 16 following that, etc. Therefore the number of matches needed is 64 + 32 + 16 + 8 + 4 + 2 + 1 = 127.
However, if you think about it in a different manner, you will reach 127 much faster. Each match produces one loser. To reach a champion, there will be 127 losers. Therefore 127 matches are needed.
(2) Given a 3´3´3 wooden block, you are asked to cut it into 1´1´1 cubes. After each cut, you can arrange the pieces any way you want. What is the minimal number of cuts needed to accomplish this task?
You can try to figure out all the possible combinations and find the least number of cuts. However, since the central 1´1´1 cube needs 6 cuts (one for each face), 6 cuts are needed.
(3) Two buses approach each other at a distance of 10 kilometers. One at a speed of 20 km/hr another at 30 km/hr. In between is a bee that flies at 5 km/hr. The bee starts an the front of one bus and heads toward the other. As soon as it hits the other bus, it turns back and so forth. Assuming the bee doesn't take time to turn around, how long does the bee fly when the two buses meet?
It was said (probably never happened) that John von Neumann (one of the founders of game theory) was given this problem. He had tried to find the distance of the bee's flight by computing the length of its successive flights between the two trains, and then summing the resulting convergent series.
Well, that is the long way. Most of us probably heard about the solution. It takes 10/(20+30) = 1/5 hr for the buses to meet and the bee flies for a total of 1/5 hr. Therefore the distance is 5 ´ 1/5 = 1 km.
I hope the above three examples achieve my purpose. Insight is not something that just pops out from nowhere. It often takes years of hard working, constant thinking to possess this ability. But it is satisfying when you see the extraordinary out of the ordinary. And always try to look at something from a different perspective. You may discover a whole new world.