Will You Be A Ghostbuster Or A World Destroyer?

Riddler Classic

Here are four questions of increasing difficulty about finding sticks in the woods, breaking them and making shapes:

Problem 1

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?

Solution

from random import random

runs, s = 1000000, 0
for _ in range(runs):
    r1,r2 = random(), random()
    if r2 > r1:
        x1, x2, x3 = r1, r2 - r1, 1 - r2
    else:
        x1, x2, x3 = r2, r1 - r2, 1 - r1
    if (x1 + x2 > x3) and (x2 + x3 > x1) and (x1 + x3 > x2): s += 1
print(s/runs)

The probability is $0.25$.

Problem 2

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?

Solution

from random import random

runs, s = 1000000, 0
for _ in range(runs):
    x1,x2,x3 = random(), random(), random()
    if (x1 + x2 > x3) and (x2 + x3 > x1) and (x1 + x3 > x2): s += 1
print(s/runs)

The probability is $0.079256$.

Problem 3

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?

Solution

from random import random
runs, s = 1000000, 0

for _ in range(runs):
    r1,r2 = random(), random()
    if r2 > r1:
        x1, x2, x3 = r1, r2 - r1, 1 - r2
    else:
        x1, x2, x3 = r2, r1 - r2, 1 - r1
    if (x1 + x2 > x3) and (x2 + x3 > x1) and (x1 + x3 > x2): 
        cos_a = (x1**2 + x2**2 - x3**2)/2*x1*x2
        cos_b = (x1**2 + x3**2 - x2**2)/2*x1*x3
        cos_c = (x2**2 + x3**2 - x1**2)/2*x2*x3
        if  cos_a >= 0 and cos_b >= 0 and cos_c >=0:
            s += 1
print(s/runs)

The probability is $0.214148$.

Problem 4

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?

Solution

from random import random
runs, s = 1000000, 0

for _ in range(runs):
    x1, x2, x3 = random(), random(), random()
    if (x1 + x2 > x3) and (x2 + x3 > x1) and (x1 + x3 > x2): 
        cos_a = (x1**2 + x2**2 - x3**2)/2*x1*x2
        cos_b = (x1**2 + x3**2 - x2**2)/2*x1*x3
        cos_c = (x2**2 + x3**2 - x1**2)/2*x2*x3
        if  cos_a >= 0 and cos_b >= 0 and cos_c >=0:
            s += 1
print(s/runs)

The probability is $0.214148$.