Did 28 in ipython.

ipython/EulerProblem028.ipynb

@@ -27,39 +27,87 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "I would try to create a function $f(n)$ which yields the sum of the outmost ring of a n by n spiral.\n",
-    "\n",
-    "For example:\n",
-    "\n",
-    "$f(1) = 1$\n",
-    "\n",
-    "$f(3) = 3 + 5 + 7 + 9 = 24$\n",
-    "\n",
-    "$f(5) = 13 + 17 + 21 + 25 = 76$\n",
-    "\n",
-    "When we have this function we calculate the solution simply by\n",
-    "\n",
-    "~~~\n",
-    "s = sum([f(n) for n in range(1, 1002, 2)])\n",
-    "~~~\n",
-    "\n",
-    "For each outer ring there is an initial corner value c ($c_3 = 3, c_5 = 76$). Once we have this value we can caluclate f like $f(n) = c_{n} + (c_n + n - 1) + (c_n + 2(n-1)) + (c_n + 3(n-1)) = 4c_n + 6 (n-1)$"
+    "When we know the corner of a certain spiral we can calculate it's total like $f_n = 4 c_n + 6 (n - 1)$. We then only have to update the corner value for each spiral. "
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 1,
    "metadata": {
-    "collapsed": true
+    "collapsed": false
    },
    "outputs": [],
    "source": [
-    "def f(n):\n",
-    "    if n == 1:\n",
-    "        return 1\n",
-    "    return 0\n",
+    "total = 1\n",
+    "current_corner = 3\n",
     "\n",
-    "s = sum([f(n) for n in range(1, 1002, 2)])\n",
+    "for n in range(3, 1002, 2):\n",
+    "    total += 4 * current_corner  + 6 * (n - 1)\n",
+    "    current_corner += 4 * n - 2\n",
+    "\n",
+    "s = total"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "669171001"
+      ]
+     },
+     "execution_count": 2,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "assert(s == 669171001)\n",
+    "s"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "The only missing piece is how could we calculate the current corner value for a certain n. The series for this is as follows:\n",
+    "\n",
+    "$c = 1, 3, 13, 31, 57, 91, 133, 183, 241, \\dots$\n",
+    "\n",
+    "With some experimenting it can be seen that\n",
+    "\n",
+    "$c_n = (n - 1)^2 - (n - 2) = n^2 - 2n + 1 - n + 2 = n^2 - 3n + 3$.\n",
+    "\n",
+    "Now, we can insert $c_n$ into $f_n$ which gives as the sum of corners for the nth spiral:\n",
+    "\n",
+    "$f_n = 4n^2 - 12n + 12 + 6n - 6 = 4n^2 - 6n + 6$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "669171001"
+      ]
+     },
+     "execution_count": 3,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "s = 1 + sum([4 * n * n - 6 * n + 6 for n in range(3, 1002, 2)])\n",
     "assert(s == 669171001)\n",
     "s"
    ]
@@ -82,7 +130,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.6.3"
+   "version": "3.5.4"
   },
   "tags": [
    "spiral",