26-Apr-2012, 7:55 AM
One of my favorite explanations of why the next number can always have more than one answer.
Take a circle. Put a point on it. Now draw all the lines between all the points. (In this case, 0 lines.) Now count how many regions the circle is broken up into (1). Now add a second point, draw all the lines (1) and count the sections (2). Now add a third point, draw all lines between all points that don't already exist (for a total of three lines) and count the sections (4). So you have 1,2,4....
If you repeat this process up to 6, and MAXIMIZE the number of sections (no intersections are of more than two lines), the sequence is 1,2,4,8,16,31,.... I don't know who discovered this, but it's mostly used just to prove this point.
[in this discussion, but "line" I mean "chord or diameter" line segment.]
Also 3,1,4,2,8 are the first five digits of 22/7, a common approximation for pi (derived, I believe from the continued fraction.)
This is where my math-geekiness kills a thread.
A, B, C, D, A -- All Big Cats Dine on Antelope.
Take a circle. Put a point on it. Now draw all the lines between all the points. (In this case, 0 lines.) Now count how many regions the circle is broken up into (1). Now add a second point, draw all the lines (1) and count the sections (2). Now add a third point, draw all lines between all points that don't already exist (for a total of three lines) and count the sections (4). So you have 1,2,4....
If you repeat this process up to 6, and MAXIMIZE the number of sections (no intersections are of more than two lines), the sequence is 1,2,4,8,16,31,.... I don't know who discovered this, but it's mostly used just to prove this point.
[in this discussion, but "line" I mean "chord or diameter" line segment.]
Also 3,1,4,2,8 are the first five digits of 22/7, a common approximation for pi (derived, I believe from the continued fraction.)
This is where my math-geekiness kills a thread.
A, B, C, D, A -- All Big Cats Dine on Antelope.
"Bad news, bad news came to me where I sleep / Turn turn turn again" - Bob Dylan