Riddler Classic

This week’s Classic, from Spreck Rosekrans, continues our camping theme. Here are four questions of increasing difficulty about finding sticks in the woods, breaking them and making shapes:

If you break a stick in two places at random, forming three pieces, what is the probability of being able to form a triangle with the pieces?
If you select three sticks, each of random length (between 0 and 1), what is the probability of being able to form a triangle with them?
If you break a stick in two places at random, what is the probability of being able to form an acute triangle — where each angle is less than 90 degrees — with the pieces?
If you select three sticks, each of random length (between 0 and 1), what is the probability of being able to form an acute triangle with the sticks?



In [48]:

    
import random



In [54]:

    
N = 1000000

1



In [66]:

    
triangle_count = 0
for _ in range(N):
    a,b = sorted((random.random(), random.random()))
    x,y,z = (a,b-a,1-b)
    if x<0.5 and y<0.5 and z<0.5:
        triangle_count += 1
        
triangle_count / N









    Out[66]:





0.250935

2



In [67]:

    
triangle_count = 0
for _ in range(N):
    sticks = sorted((random.random(), random.random(), random.random()))
    if sticks[2] < sticks[0] + sticks[1]:
        triangle_count += 1
                    
triangle_count / N









    Out[67]:





0.499781

3



In [68]:

    
triangle_count = 0
for _ in range(N):
    a,b = sorted((random.random(), random.random()))
    x,y,z = (a,b-a,1-b)
    if (x**2 + y**2 >  z**2) and (x**2 + z**2 > y**2) and (z**2 + y**2 > x**2):
        triangle_count += 1
        
triangle_count / N









    Out[68]:





0.079496

4



In [69]:

    
triangle_count = 0
for _ in range(N):
    x,y,z = (random.random(), random.random(), random.random())
    if (x**2 + y**2 >  z**2) and (x**2 + z**2 > y**2) and (z**2 + y**2 > x**2):
        triangle_count += 1
                    
triangle_count / N









    Out[69]:





0.21426



In [ ]: