Add files via upload

ipynb/Stubborn.ipynb

@@ -298,10 +298,10 @@
     {
      "data": {
       "text/plain": [
-       "['000000000000000000000000000000000000000000000000000000000000',\n",
+       "['0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000',\n",
-       " '000000000000000000000000000000000000000000000000000000000001',\n",
+       " '0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001',\n",
-       " '863811000557423423230896109004106619977392256259918212890625',\n",
+       " '3953007319108169802938509890062166509580863811000557423423230896109004106619977392256259918212890625',\n",
-       " '136188999442576576769103890995893380022607743740081787109376']"
+       " '6046992680891830197061490109937833490419136188999442576576769103890995893380022607743740081787109376']"
       ]
      },
      "execution_count": 8,
@@ -310,31 +310,7 @@
     }
    ],
    "source": [
-    "stubborn_endings(60)"
+    "stubborn_endings(100)"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 9,
-   "id": "73491ea5-a4c8-4221-8fc7-d6ce44d17bb4",
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "text/plain": [
-       "['000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000',\n",
-       " '000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001',\n",
-       " '055462996814764263903953007319108169802938509890062166509580863811000557423423230896109004106619977392256259918212890625',\n",
-       " '944537003185235736096046992680891830197061490109937833490419136188999442576576769103890995893380022607743740081787109376']"
-      ]
-     },
-     "execution_count": 9,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "stubborn_endings(120)"
    ]
   },
   {
@@ -344,33 +320,24 @@
    "source": [
     "This leads to a few new questions.\n",
     "\n",
-    "## Can each stubborn ending be extended exactly one way?\n",
+    "## Can each stubborn ending always be extended exactly one way?\n",
     "\n",
-    "We know that each stubborn ending of length *d* has to build on the endings of length *d* - 1, and so far, there has always been exactly one way (out of the ten possible digits) to extend each of the length *d* - 1 endings. Will that always be true? Might there be a case where two digits work with one of the endings, or no digits?\n",
+    "There are four stubborn endings of length 1: '0', '1', '5', and '6'. So far, every time we try, each of the four can be extended from length *d* to length *d*+1 in exactly one way. Will that always be the case? I leave it up to you to provide a proof, but I can show that it is true for every *d* up to 2000:"
-    "\n",
-    "I can show that for all lengths *d* up to 2000, the result of `stubborn_endings(d)` is always four strings that end with '0', '1', '5', and '6' in that order:"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 18,
+   "execution_count": 9,
    "id": "654d53af-e58f-4ed2-874b-c4573f3a9a7a",
    "metadata": {},
-   "outputs": [
+   "outputs": [],
-    {
-     "data": {
-      "text/plain": [
-       "Counter({('0', '1', '5', '6'): 2000})"
-      ]
-     },
-     "execution_count": 18,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
    "source": [
-    "Counter(tuple(ending[-1] for ending in stubborn_endings(d))\n",
+    "def last_digits(endings) -> list[str]: \n",
-    "        for d in range(1, 2001))"
+    "    \"\"\"A list of the last digits of these endings.\"\"\"\n",
+    "    return [e[-1] for e in endings]\n",
+    "    \n",
+    "for d in range(1, 2000):\n",
+    "    assert last_digits(stubborn_endings(d)) == ['0', '1', '5', '6']"
    ]
   },
   {
@@ -378,8 +345,6 @@
    "id": "537556a3-d570-4a50-93bf-9632b40f9b47",
    "metadata": {},
    "source": [
-    "I leave it up to the reader to find a proof ofr numbers beyond 2000.\n",
-    "\n",
     "## What digits are used to extend each ending?\n",
     "\n",
     "I don't have a theory of which digit is used to extend each stubborn ending, but I can count:"
@@ -387,7 +352,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 11,
+   "execution_count": 10,
    "id": "4dbd7ace-8b41-4ea7-b71f-8f7e318243e1",
    "metadata": {},
    "outputs": [
@@ -418,7 +383,7 @@
        "          '6': 173})}"
       ]
      },
-     "execution_count": 11,
+     "execution_count": 10,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -426,7 +391,7 @@
    "source": [
     "from collections import Counter\n",
     "\n",
-    "{e[-1]: Counter(e[:-1]) for e in stubborn_endings(2000)}"
+    "{end: Counter(start) for (*start, end) in stubborn_endings(2000)}"
    ]
   },
   {