Strictly speaking this algorithm has an O(n^2) part in it, but it runs under a second so...

51/main.py

@@ -8,6 +8,7 @@ Find the smallest prime which, by replacing part of the number (not necessarily
 
 import numpy as np
 import math
+import time
 
 def sieve(n):
     assert n > 1
@@ -24,13 +25,52 @@ def sieve(n):
 
     return [i for i,val in enumerate(ns) if val][2:]
 
+# Does generate all families that can be made for that number, so that consists of multiple families together
+def families(num):
+    result = []
+    variedDigits = set()
+    digits = [int(d) for d in str(num)]
+
+    for i in range(len(digits)):
+        if not i in variedDigits:
+            buddies = []
+
+            for j in range(len(digits)):
+                if digits[i] == digits[j]:
+                    buddies.append(j)
+                    variedDigits.add(j)
+
+            r = digits[:]
+
+            for n in range(10):
+                for bud in buddies:
+                    r[bud] = n
+                maybeResult = int("".join([str(_r) for _r  in r]))
+                result.append(maybeResult)
+    
+    return result
+
+def findPrimeFamily(length):
+    primes = sieve(1000000)
+    primeSet = set(primes)
+
+    for p in primes:
+        primeFamilies = families(p)
+
+        for i in range(len(primeFamilies) // 10):
+            counter = 0
+            for pf in primeFamilies[10*i:10*i+10]:
+                if pf in primeSet and len(str(pf)) == len(str(p)):
+                    counter += 1
+            if counter >= length:
+                return [x for x in primeFamilies[10*i:10*i+10] if x in primeSet]
+
 def main():
     print("Hello this is Patrick")
 
-    primes = sieve(1000000)
+    t0 = time.time()
 
-    # So instead of going past all number families and see if they are prime I think it's better to look for families in the primes
+    print(findPrimeFamily(8), time.time() - t0)
-    # Quite the combinatorial problem indeed, lots of permutations and I don't immediately see an easy way to fix it
 
 if __name__ == "__main__":
     main()