## 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
= 1000000, 0
runs, s for _ in range(runs):
= random(), random()
r1,r2 if r2 > r1:
= r1, r2 - r1, 1 - r2
x1, x2, x3 else:
= r2, r1 - r2, 1 - r1
x1, x2, x3 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
= 1000000, 0
runs, s for _ in range(runs):
= random(), random(), random()
x1,x2,x3 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
= 1000000, 0
runs, s
for _ in range(runs):
= random(), random()
r1,r2 if r2 > r1:
= r1, r2 - r1, 1 - r2
x1, x2, x3 else:
= r2, r1 - r2, 1 - r1
x1, x2, x3 if (x1 + x2 > x3) and (x2 + x3 > x1) and (x1 + x3 > x2):
= (x1**2 + x2**2 - x3**2)/2*x1*x2
cos_a = (x1**2 + x3**2 - x2**2)/2*x1*x3
cos_b = (x2**2 + x3**2 - x1**2)/2*x2*x3
cos_c if cos_a >= 0 and cos_b >= 0 and cos_c >=0:
+= 1
s 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
= 1000000, 0
runs, s
for _ in range(runs):
= random(), random(), random()
x1, x2, x3 if (x1 + x2 > x3) and (x2 + x3 > x1) and (x1 + x3 > x2):
= (x1**2 + x2**2 - x3**2)/2*x1*x2
cos_a = (x1**2 + x3**2 - x2**2)/2*x1*x3
cos_b = (x2**2 + x3**2 - x1**2)/2*x2*x3
cos_c if cos_a >= 0 and cos_b >= 0 and cos_c >=0:
+= 1
s print(s/runs)
```

The probability is \(0.214148\).