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I just took a quiz and am dumbfounded by my lack of insight. Consider what kind of idiot I'd have to be to do the following:

Point A = (8,-15) and point B = (-8,15). P is the locus of points (x,y) such that AP • BP = 0. Describe the elements in P.

Immediately I graphed A and B and then observed that their dot product was 0, which meant that vectors AP and BP are perpendicular. Now, out of some stupidity, I immediately drew the vertical and horizontal lines through A and B and concluded that the two points of intersection are the only elements of the locus. Clearly, though, the locus is a circle, because if we inscribe a right angle into a circle the hypotenuse is the diameter. Since the intersection of A and B is a right angle, the diameter is AB. The radius is the distance between A and B divided by 2, which is 17. The locus is thus the circle x^2 + y^2 = 289. A simple, easy problem with an intuitive solution.

Why did I do this incorrectly? How can I think better in test situations and at home (similarly, I find myself spending 2-3 hours on math homework daily while the A+ student in my class spends 45 minutes maximum)? Indeed, in regular math I got an 100 on every single assessment without exception. But this is plug and chug pseudo-math. Honors math is the real thing - learning concepts and using them to see connections and draw new things based on old knowledge. How effectively one does this determines the grade in honors math.

The bottom line is - how do I get amazing at real math? How do I think and realize things during a quiz or test? How do I come up with ideas about new problems related to old ideas? Really - how do I become mathematically intelligent?

Nursultan Tulyakbay
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    Everybody makes such mistakes (even chess grandmasters sometimes make obvious mistakes). You should work harder than you do now. And don't stop, just practice, practice and practice. – pointer Oct 03 '14 at 17:48
  • If your intuition often lets you down, try not to use it as much by identifying problems that does not require it. This way you might be able to develop better skills of solving the problems simply. For example, in this case, you do not need to draw it and think about it - it is sufficient to express the dot product as (|8-x|,|-15-y|)*(|-8-x|,|15-y|) and solve the equation. Also, you should train your skills on lots of small, easy problems to get it under your skin – Jindra Helcl Oct 03 '14 at 18:05
  • Look at it from a probabilistic perspective, there are usually countless ways to screw up and only a finite number of ways of getting right. So, be happy when you get things right, because the odds are against you :-). – copper.hat Oct 03 '14 at 18:05
  • And I have made countless mistakes. Once I accidentally sent a confidential memo. to an essentially worldwide audience because I transposed the subject and to line in an email. I had, unfortunately, set up an email alias that matched one of the words in the subject and happily launched the memo. Over 15 years ago and it still hurts :-). – copper.hat Oct 03 '14 at 18:10

2 Answers2

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If you are interested in mathematics, as you seem to be, I suggest you start solving problems in your free time as a hobby. You will get better and learn many fundamental strategies that do not come from studying. A good book on problem solving is "arts and crafts of problem solving" by Paul Zeitz.

A problem you can work on below:
n points are placed on a circle, what is the maximum number of regions you can divide the circle into by connecting these points.

(hint : not $2^{n-1}$)

qwerty314
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  • I don't have a pen or paper right now, but to start on this problem I would first try to work with what I know about intersections and combinatorics: – Nursultan Tulyakbay Oct 03 '14 at 18:49
  • To maximize the number of regions, we'd have to have as many intersections as possible. I would first try thinking about polygons. 1 point, 1 region. 2 points, 2 regions. 3 points, 6 regions. 4 points, 8 regions. 5 points, 5 + the internal intersections of the diagonals. This now shows us the generalization - the maximum number of intersections is n + q, where q is the maximum number of internal regions. – Nursultan Tulyakbay Oct 03 '14 at 19:06
  • To find q, we must break it up into its components. First, let's find the number of distinct diagonals a polygon can have. If there are n sides, there are (n(n-3))/2 diagonals because if we consider that each vertex can form diagonals with all the other vertices besides the two adjacent ones, then we have (n(n-3))/2 distinct diagonals, because for each diagonal made from one vertex there is a duplicate one made from the opposite one. – Nursultan Tulyakbay Oct 03 '14 at 19:16
  • Now we consider what drawing a diagonal actually does to the inside. Without intersections, drawing a line between two points on the circle multiplies the number of regions by 2. For each additional diagonal, we have another multiplication by 2, plus 2(the number of intersections with other diagonals). – Nursultan Tulyakbay Oct 03 '14 at 19:25
  • Should that be divide the disc, by any chance? It took me a second to see why the obvious answer wasn't $n$ ;) – Jessica B Oct 03 '14 at 21:11
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Enjoy whatever you do in maths taking your own sweet time. Attempt to be correct and don't worry about other guy's speed.With constant practice on a good work book comes the skill and recognising connections will automatically pick up. Good luck!

Narasimham
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