import random
"""
Generate k random positive integers that sum to n.
Algorithm (stick-breaking / random composition):
------------------------------------------------
1. We choose (k - 1) random "cut points" in the range [1, n - 1].
These represent where we "break" the integer n into k parts.
2. Sort the cut points.
3. Compute the differences between consecutive points
(including 0 and n as boundaries). These differences
are the k positive integers that sum to n.
This produces a uniformly random composition of n.
"""
"""
Paper Run (Dry Run) of the Random-Sum Program
---------------------------------------------
Input:
n = 50
k = 7
We need (k - 1) = 6 random cut points in the range [1, 49].
Simulated rand() outputs (rand() % 49):
12, 3, 40, 25, 7, 33
After adding +1 (because the code uses 1 + rand() % (n - 1)):
cuts = {13, 4, 41, 26, 8, 34}
Step 1: Sort the cut points
cuts → {4, 8, 13, 26, 34, 41}
Step 2: Convert cut points into segment lengths
prev = 0
out[0] = 4 - 0 = 4
prev = 4
out[1] = 8 - 4 = 4
prev = 8
out[2] = 13 - 8 = 5
prev = 13
out[3] = 26 - 13 = 13
prev = 26
out[4] = 34 - 26 = 8
prev = 34
out[5] = 41 - 34 = 7
prev = 41
out[6] = 50 - 41 = 9 (final segment)
Final result:
parts = {4, 4, 5, 13, 8, 7, 9}
Verification:
4 + 4 + 5 + 13 + 8 + 7 + 9 = 50
Output:
Random parts that sum to 50:
4 4 5 13 8 7 9
Sum = 50
"""
def generate_random_sum(n: int, k: int) -> list:
if k <= 0 or n < k:
# k positive integers must sum to n → minimum sum is k
raise ValueError("Invalid n or k")
# Generate (k - 1) random cut points in [1, n - 1]
cuts = [random.randint(1, n - 1) for _ in range(k - 1)]
# Sort the cut points so we can compute segment lengths
cuts = sorted(cuts)
result = []
prev = 0
# Convert cut points into segment lengths
for c in cuts:
result.append(c - prev)
prev = c
# Last segment: from last cut to n
result.append(n - prev)
return result
def main():
n = 50 # total sum
k = 7 # number of random parts
parts = generate_random_sum(n, k)
print(f"Random parts that sum to {n}:")
print(*parts)
# Verify sum
print("Sum =", sum(parts))
if __name__ == "__main__":
main()
"""
run:
Random parts that sum to 50:
16 3 7 4 10 8 2
Sum = 50
"""
