Question

Well it is not a program problem. Is there any hint for such quiz?
I am thinking about focusing on two random R1, R2, both of which is in range (0, 1). and supposing R2 > R1
and then fulfill two equation:

R1 + (1 - R2) > R2 - R1 // two sticks sum longer then the rest one
|R1 - (1 - R2)| < R2 - R1 // the difference of these two should be shorter the rest one

but I cannot move further…

Answer 1

The answer is 1/4.
Here is the explanation.

Let x is the length of the leftmost stick and y is the length of the rightmost stick.
Then the middle stick has length n-x-y, if the original stick’s length was n.

The possible values for x,y are those for which:

1. x > 0
2. y > 0
3. x + y < n

In the plane Oxy this is equivalent to say that the point (x,y) lies within the triangle with vertices (0, 0), (n, 0), (0, n).

Now these three numbers (x, y, n-x-y) form a triangle if all of the three are satisfied:

1. x + y > n – x – y <=> x + y < n/2
2. x + (n – x – y) > y <=> y < n/2
3. y + (n – x – y) > x <=> x < n/2

Again in the Oxy plane these are satisfied when the point (x,y) lies within the triangle with vertices (0, n/2), (n/2, n/2), (n/2, 0).

The area of this triangle is a quarter of the area of the (0, 0), (n, 0), (0, n) triangle, since it’s the ‘middle’ triangle (whose vertices are the midpoints) of the bigger one.

Here is a simple C# program to verify the answer:

Random r = new Random();
int count = 0, total = 0, tries = 1000000;
double x, y;

for (int i = 0; i < tries; i++)
{
    x = r.NextDouble();
    y = r.NextDouble();
    if (x + y > 0.5 && x < 0.5 && y < 0.5) ++count;
    if (x + y < 1.0) ++total;
}

Console.WriteLine((double)count / total);