Interesting question

Homework by any chance? Generally prefacing your question by the phrases

1. this is interesting...
2. hey, can anyone show...
3. now I fouund this neat thing, can some one explain it...

increase the chances "exponentially"

 

It is intuitively obvious that . . .

 

Here's a cute, well known problem (it's referred to as the "missing area problem").

Take a circle, and mark n points around the circumference of the circle NOT evenly spaced. Draw all secants connecting the points. How many areas does that divide the circle into? (If n were even and you spaced all points equally, all "diameters" would cross at the center. Moving some of the points slightly will move some secants off the diameter so you will have line crossing near the center, increasing the number of areas. The point of "NOT evenly spaced" is to get the maximum number of areas.)

For example if there is only one point, there are no secants and the circle remains 1 area.
If there are two points, there is one secant and the circle is divided into two area.
If there are three points, there are 3 secants, forming a triangle. There is the area inside the triangle and 3 areas between the there sides and the circle: total of 4 areas.
If there are 4 points, there are 6 secants, four forming a quadrilateral and 2 diagonals. The two diagonals divide the quadrilateral into 4 areas and we have the four areas between the sides and the circle: total of 8 areas.
If there are 5 points, it's harder to count but there are 16 areas.

There is, believe it or not a simple formula for the number of area given n points!

How many areas if there are 6 points? You can draw a picture and count the areas or use the formula: there are 31!

(Hey, I said it was a "missing area" problem.)

In case you were wondering with 7 points, you get 57 areas.

 

Ok I like that one HallsofIvy. So next time someone gives me an annoying "1, 2, 4, 8, 16 ... what comes next question, then I can say 31. Nice one. :)

 

I once had a teacher who gave us the sequence "35, 34, 33, 32, 31, 30, 29,..." and asked "what comes next". The correct answer? "61", of course!

(These were the numbers on the subway stops on his way in to work. Between stop "29" and "28", the train switch on to a different line.)

 

I just did a search on google on 'missing area' and came up with this page:

http://www.reed.edu/~mcphailb/puzzles/triangle.html [Broken]

It puzzles me. I don't see how a triangle 2 units high and 5 units wide can have the same gradient as a triangle 3 units high and 8 units wide. Did they cheat?

 

they aren't similar triangles. if they were you'd have a serious problem, they just appear close enugh for you to believe it when you look at it

 

Yes, they only appear to have the same gradient (as matt said). Try drawing the same problem, with great accuracy, on a piece of paper.