From d4c900aa98950a589e2676a0a64f4c3266740f08 Mon Sep 17 00:00:00 2001
From: Felix Martin <mail@felixm.de>
Date: Sat, 19 Sep 2026 17:03:07 -0400
Subject: [PATCH] Solve problem 323

---
 python/e323.py | 77 ++++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 77 insertions(+)
 create mode 100644 python/e323.py

diff --git a/python/e323.py b/python/e323.py
new file mode 100644
index 0000000..68b9ca2
--- /dev/null
+++ b/python/e323.py
@@ -0,0 +1,77 @@
+from random import randint
+from math import factorial
+from collections import defaultdict
+from fractions import Fraction
+
+
+def binomial(n: int, k: int) -> Fraction:
+    return Fraction(factorial(n), (factorial(k) * factorial(n - k)))
+
+
+def f(k: int, n: int) -> Fraction:
+    # Probability that k out of n trials succeed with rate p per trial.
+    p = Fraction(1, 2)
+    return binomial(n, k) * p**k * (1 - p) ** (n - k)
+
+
+def count(exp_n: Fraction, step: int, dist) -> Fraction:
+    step += 1
+
+    # For each number `n` of remaining "ones" calculate the probability that
+    # 0..`n` ones are remanining for that probability. We can then calculate a
+    # new probability mass by multiplying the current with the new probability
+    # for a given `n`.
+    ndist = defaultdict(Fraction)
+    for n, p in dist:
+        for i in range(n + 1):
+            np = f(i, n)
+            ndist[n - i] += p * np
+
+    # Probability that we reach `0` at this step counts to total expected
+    # number of steps.
+    exp_n += ndist[0] * step
+    del ndist[0]
+
+    # Drop probabilities small enough that they seem englible for the precision
+    # we need which is 10 digits after the decimal point.
+    dist = []
+    for n, p in ndist.items():
+        if p > Fraction(1, 10**15):
+            dist.append((n, p))
+    if dist:
+        return count(exp_n, step, dist)
+    return exp_n
+
+
+def stackoverflow():
+    # I found this expression when googling (geminiing) an expression for this
+    # type of problem. However, it did not yield the correct solution because I
+    # did not use Fractions. I then came up with `count` and was somehow clever
+    # enough to realize that I would have to use Fractions and after I came
+    # back I updated stackoverflow accordingly just to see if I had implemented
+    # the expression correctly.
+
+    # https://mathoverflow.net/questions/41604/what-is-the-expected-maximum-out-of-a-sample-size-n-from-a-geometric-distribut
+    t = Fraction()
+    p = Fraction(1, 2)
+    q = 1 - p
+    n = 32
+    for i in range(1, n + 1):
+        c = binomial(n, i)
+        c *= (-1) ** (i + 1)
+        c *= Fraction(1, 1 - q**i)
+        t += c
+    return float(t)
+
+
+def euler_323():
+    n_stackoverflow = stackoverflow()
+    n = count(Fraction(), 0, [(32, Fraction(1, 1))])
+    assert round(float(n_stackoverflow), 10) == round(float(n), 10)
+    return round(float(n), 10)
+
+
+if __name__ == "__main__":
+    solution = euler_323()
+    print(f"e323.py: {solution}")
+    assert solution == 6.3551758451