From f0d81aba242f49791c7fd9a4c5121df2a83649a4 Mon Sep 17 00:00:00 2001
From: Felix Martin <mail@felixm.de>
Date: Sat, 27 Jun 2026 08:16:43 -0400
Subject: [PATCH] Solve problem 820

---
 python/e820.py | 81 ++++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 81 insertions(+)
 create mode 100644 python/e820.py

diff --git a/python/e820.py b/python/e820.py
new file mode 100644
index 0000000..6a7343a
--- /dev/null
+++ b/python/e820.py
@@ -0,0 +1,81 @@
+from math import floor
+from time import time_ns
+
+
+def d(_num: int, den: int, n: int) -> int:
+    """ (10^n // d) % 10 -> (10^n % 10*d) // d % 10 """
+    return pow(10, n, 10 * den) // den % 10
+
+
+def d_naiv(num: int, den: int, n: int, debug: bool = False) -> int:
+    """Return the nth digit of the factional part of num / den."""
+    if num == den:
+        if debug:
+            print("1.0")
+        return 0
+    assert num < den
+    if debug:
+        print("0.", end="")
+
+    x = None
+    r = 0
+    i = 0
+    num *= 10  # 0.
+    nums = {}
+    while i < n:
+        if num == 0:
+            r = 0
+            break
+        elif num < den:
+            r = 0
+            if debug:
+                print(0, end="")
+        else:
+            r = num // den
+            num %= den
+            if debug:
+                print(r, end="")
+        i += 1
+        if x is None:
+            if num in nums:
+                j = nums[num]
+                delta = i - j
+                x = floor((n - i) / delta)
+                if debug:
+                    print("---")
+                    print(f"{den=} {j} -> {i} (d={delta} mult={x})")
+                    print(f"\n{j} -{delta} {x}> {i} ")
+                i += x * delta
+            else:
+                nums[num] = i
+        num *= 10
+    if debug:
+        print()
+    # dr = (10**n % (10*den)) // den
+    dr = pow(10, n, 10 * den) // den
+    assert r == dr
+    # print(f"k = {den} n= {n}")
+    return r
+
+
+def s(n: int) -> int:
+    return sum(d(1, k, n) for k in range(1, n + 1))
+
+
+def euler_820():
+    d(1, 7, 10)
+    assert d(1, 2, 7) == 0
+    assert d(1, 3, 7) == 3
+    assert d(1, 5, 7) == 0
+    assert d(1, 6, 7) == 6
+    assert d(1, 7, 7) == 1
+    assert s(7) == 10
+    assert s(100) == 418
+    assert s(10000) == 43742
+    return s(10**7)
+
+
+if __name__ == "__main__":
+    solution = euler_820()
+    print(f"e820.py: {solution}")
+    assert solution == 44967734