diff --git a/Exercise sheet 4/exercise_sheet_04.ipynb b/Exercise sheet 4/exercise_sheet_04.ipynb
index 3843be8..e8644ae 100644
--- a/Exercise sheet 4/exercise_sheet_04.ipynb	
+++ b/Exercise sheet 4/exercise_sheet_04.ipynb	
@@ -609,7 +609,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 13,
    "id": "683d2de3",
    "metadata": {
     "deletable": false,
@@ -625,7 +625,18 @@
      "task": false
     }
    },
-   "outputs": [],
+   "outputs": [
+    {
+     "data": {
+      "image/png": "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\n",
+      "text/plain": [
+       "<Figure size 432x288 with 1 Axes>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
    "source": [
     "def plot_disk_configuration(positions,L):\n",
     "    fig,ax = plt.subplots()\n",
@@ -671,7 +682,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 14,
    "id": "19c010eb",
    "metadata": {
     "deletable": false,
@@ -690,18 +701,30 @@
    "source": [
     "def two_disks_overlap(x1,x2,L):\n",
     "    '''Return True if the disks centered at x1 and x2 (represented as 2-element arrays) overlap in [0,L)^2.'''\n",
-    "    # YOUR CODE HERE\n",
-    "    raise NotImplementedError()\n",
+    "    # To take into account all overlap with the boundaries, we will also\n",
+    "    # test shifted disks. As d(x1 + d, x2) = d(x1, x2 - d), we do not have\n",
+    "    # to shift both disks explicitly.\n",
+    "    for dx in [0, -L, L]:\n",
+    "        for dy in [0, -L, L]:\n",
+    "            if (x1[0] - x2[0] + dx)**2 + (x1[1] - x2[1] + dy)**2 <= 4:\n",
+    "                return True\n",
+    "    return False\n",
     "    \n",
     "def disk_config_valid(x,L):\n",
     "    '''Return True if the configuration x (as two-dimensional array) is non-overlapping in [0,L)^2.'''\n",
-    "    # YOUR CODE HERE\n",
-    "    raise NotImplementedError()"
+    "    n = len(x)\n",
+    "    for idx_x1 in range(n):\n",
+    "        # We should not compare x1 with itself, and the previous disks have\n",
+    "        # already been compared.\n",
+    "        for idx_x2 in range(idx_x1 + 1, n):\n",
+    "            if two_disks_overlap(x[idx_x1], x[idx_x2], L):\n",
+    "                return False\n",
+    "    return True"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 15,
    "id": "e0f2024a",
    "metadata": {
     "deletable": false,
@@ -731,7 +754,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 16,
    "id": "68e57e57",
    "metadata": {
     "deletable": false,