## Tuesday, December 22, 2009

### Fox 217

Have beautiful days with your families
and the loved ones!
See you guys next year!
Yourself should show mercy on yourself.
Do something out of the box.
While enjoying the time...
NEVER QUIT THINKIN' !

1. Come on... No ideas here??
I think we can at least set a bound for the answer. Any thoughts?

2. Two lines cannot enclose a region.

Here, I measure angles as a fraction of a full circle e.g 90 degrees is typed as 1/4.

In 3 steps, if the second leaves an angle a between the 2 lines and the third leaves an angle of b, a region will be enclosed if 2a+b<=1/2. Given an angle a, the probability that b<=1/2-2a is 1/2-2a, or 0 if a<1/4. Integrating for a between 0 and 1/2 (a can be clockwise or anticlockwise, but b,c etc are measured all in the same direction as a) gives a probability of 1/4.

3. Apparently, the average number of steps taken is about nine. The process can easily be simulated programmatically, and the results of such a simulation show an average of 8.877 steps. Nearly half of the processes ended in seven or fewer steps. The greatest number of steps taken was 76, though the process could continue indefinitely.

Contrary to the previous post, the probability of closing an area in three steps is approximately, if not exactly, 1/12.

4. The triangle version indeed gives an area of 1/12. I posted some details on another forum here. It's also another one of the Fox questions somewhere or other.

5. 1/12 was the answer for the triangular version. It must be somewhere here in this blog, but couldn't find it. There was even a fancy "text-based" solution.

6. I guess you mean the last comment at: