From baf239682b10a5be50de62f18db1bf5be7e2a763 Mon Sep 17 00:00:00 2001
From: Peter Norvig <peter.norvig@gmail.com>
Date: Thu, 26 Dec 2024 09:36:20 -0800
Subject: [PATCH] Add files via upload

---
 ipynb/Advent-2024.ipynb | 498 +++++++++++++++++-----------------------
 1 file changed, 215 insertions(+), 283 deletions(-)

diff --git a/ipynb/Advent-2024.ipynb b/ipynb/Advent-2024.ipynb
index 4d9d4c3..49d4524 100644
--- a/ipynb/Advent-2024.ipynb
+++ b/ipynb/Advent-2024.ipynb
@@ -592,7 +592,7 @@
     {
      "data": {
       "text/plain": [
-       "Puzzle  4.1:   .033 seconds, answer 2401              ok"
+       "Puzzle  4.1:   .032 seconds, answer 2401              ok"
       ]
      },
      "execution_count": 35,
@@ -1032,7 +1032,7 @@
     {
      "data": {
       "text/plain": [
-       "Puzzle  6.2:  1.967 seconds, answer 2162              ok"
+       "Puzzle  6.2:  1.942 seconds, answer 2162              ok"
       ]
      },
      "execution_count": 58,
@@ -1178,7 +1178,7 @@
     {
      "data": {
       "text/plain": [
-       "Puzzle  7.1:   .013 seconds, answer 1985268524462     ok"
+       "Puzzle  7.1:   .014 seconds, answer 1985268524462     ok"
       ]
      },
      "execution_count": 67,
@@ -1220,7 +1220,7 @@
     {
      "data": {
       "text/plain": [
-       "Puzzle  7.2:   .811 seconds, answer 150077710195188   ok"
+       "Puzzle  7.2:   .788 seconds, answer 150077710195188   ok"
       ]
      },
      "execution_count": 70,
@@ -1268,7 +1268,7 @@
     {
      "data": {
       "text/plain": [
-       "Puzzle  7.2:   .660 seconds, answer 150077710195188   ok"
+       "Puzzle  7.2:   .646 seconds, answer 150077710195188   ok"
       ]
      },
      "execution_count": 73,
@@ -1552,7 +1552,7 @@
     {
      "data": {
       "text/plain": [
-       "Puzzle  9.1:   .019 seconds, answer 6332189866718     ok"
+       "Puzzle  9.1:   .020 seconds, answer 6332189866718     ok"
       ]
      },
      "execution_count": 88,
@@ -1632,7 +1632,7 @@
     {
      "data": {
       "text/plain": [
-       "Puzzle  9.2:  2.754 seconds, answer 6353648390778     ok"
+       "Puzzle  9.2:  2.680 seconds, answer 6353648390778     ok"
       ]
      },
      "execution_count": 91,
@@ -1934,7 +1934,7 @@
     {
      "data": {
       "text/plain": [
-       "Puzzle 11.1:   .068 seconds, answer 194482            ok"
+       "Puzzle 11.1:   .067 seconds, answer 194482            ok"
       ]
      },
      "execution_count": 107,
@@ -2017,7 +2017,7 @@
     {
      "data": {
       "text/plain": [
-       "Puzzle 11.2:   .061 seconds, answer 232454623677743   ok"
+       "Puzzle 11.2:   .060 seconds, answer 232454623677743   ok"
       ]
      },
      "execution_count": 112,
@@ -2222,7 +2222,7 @@
     {
      "data": {
       "text/plain": [
-       "Puzzle 12.1:   .052 seconds, answer 1402544           ok"
+       "Puzzle 12.1:   .051 seconds, answer 1402544           ok"
       ]
      },
      "execution_count": 123,
@@ -2878,42 +2878,42 @@
        "</style>\n",
        "\n",
        "<div class=\"animation\">\n",
-       "  <img id=\"_anim_img5d9cef03daf64662882705f9f9bba16b\">\n",
+       "  <img id=\"_anim_imgf59c28f9f5894974b4791611a2c64487\">\n",
        "  <div class=\"anim-controls\">\n",
-       "    <input id=\"_anim_slider5d9cef03daf64662882705f9f9bba16b\" type=\"range\" class=\"anim-slider\"\n",
+       "    <input id=\"_anim_sliderf59c28f9f5894974b4791611a2c64487\" type=\"range\" class=\"anim-slider\"\n",
        "           name=\"points\" min=\"0\" max=\"1\" step=\"1\" value=\"0\"\n",
-       "           oninput=\"anim5d9cef03daf64662882705f9f9bba16b.set_frame(parseInt(this.value));\">\n",
+       "           oninput=\"animf59c28f9f5894974b4791611a2c64487.set_frame(parseInt(this.value));\">\n",
        "    <div class=\"anim-buttons\">\n",
-       "      <button title=\"Decrease speed\" aria-label=\"Decrease speed\" onclick=\"anim5d9cef03daf64662882705f9f9bba16b.slower()\">\n",
+       "      <button title=\"Decrease speed\" aria-label=\"Decrease speed\" onclick=\"animf59c28f9f5894974b4791611a2c64487.slower()\">\n",
        "          <i class=\"fa fa-minus\"></i></button>\n",
-       "      <button title=\"First frame\" aria-label=\"First frame\" onclick=\"anim5d9cef03daf64662882705f9f9bba16b.first_frame()\">\n",
+       "      <button title=\"First frame\" aria-label=\"First frame\" onclick=\"animf59c28f9f5894974b4791611a2c64487.first_frame()\">\n",
        "        <i class=\"fa fa-fast-backward\"></i></button>\n",
-       "      <button title=\"Previous frame\" aria-label=\"Previous frame\" onclick=\"anim5d9cef03daf64662882705f9f9bba16b.previous_frame()\">\n",
+       "      <button title=\"Previous frame\" aria-label=\"Previous frame\" onclick=\"animf59c28f9f5894974b4791611a2c64487.previous_frame()\">\n",
        "          <i class=\"fa fa-step-backward\"></i></button>\n",
-       "      <button title=\"Play backwards\" aria-label=\"Play backwards\" onclick=\"anim5d9cef03daf64662882705f9f9bba16b.reverse_animation()\">\n",
+       "      <button title=\"Play backwards\" aria-label=\"Play backwards\" onclick=\"animf59c28f9f5894974b4791611a2c64487.reverse_animation()\">\n",
        "          <i class=\"fa fa-play fa-flip-horizontal\"></i></button>\n",
-       "      <button title=\"Pause\" aria-label=\"Pause\" onclick=\"anim5d9cef03daf64662882705f9f9bba16b.pause_animation()\">\n",
+       "      <button title=\"Pause\" aria-label=\"Pause\" onclick=\"animf59c28f9f5894974b4791611a2c64487.pause_animation()\">\n",
        "          <i class=\"fa fa-pause\"></i></button>\n",
-       "      <button title=\"Play\" aria-label=\"Play\" onclick=\"anim5d9cef03daf64662882705f9f9bba16b.play_animation()\">\n",
+       "      <button title=\"Play\" aria-label=\"Play\" onclick=\"animf59c28f9f5894974b4791611a2c64487.play_animation()\">\n",
        "          <i class=\"fa fa-play\"></i></button>\n",
-       "      <button title=\"Next frame\" aria-label=\"Next frame\" onclick=\"anim5d9cef03daf64662882705f9f9bba16b.next_frame()\">\n",
+       "      <button title=\"Next frame\" aria-label=\"Next frame\" onclick=\"animf59c28f9f5894974b4791611a2c64487.next_frame()\">\n",
        "          <i class=\"fa fa-step-forward\"></i></button>\n",
-       "      <button title=\"Last frame\" aria-label=\"Last frame\" onclick=\"anim5d9cef03daf64662882705f9f9bba16b.last_frame()\">\n",
+       "      <button title=\"Last frame\" aria-label=\"Last frame\" onclick=\"animf59c28f9f5894974b4791611a2c64487.last_frame()\">\n",
        "          <i class=\"fa fa-fast-forward\"></i></button>\n",
-       "      <button title=\"Increase speed\" aria-label=\"Increase speed\" onclick=\"anim5d9cef03daf64662882705f9f9bba16b.faster()\">\n",
+       "      <button title=\"Increase speed\" aria-label=\"Increase speed\" onclick=\"animf59c28f9f5894974b4791611a2c64487.faster()\">\n",
        "          <i class=\"fa fa-plus\"></i></button>\n",
        "    </div>\n",
-       "    <form title=\"Repetition mode\" aria-label=\"Repetition mode\" action=\"#n\" name=\"_anim_loop_select5d9cef03daf64662882705f9f9bba16b\"\n",
+       "    <form title=\"Repetition mode\" aria-label=\"Repetition mode\" action=\"#n\" name=\"_anim_loop_selectf59c28f9f5894974b4791611a2c64487\"\n",
        "          class=\"anim-state\">\n",
-       "      <input type=\"radio\" name=\"state\" value=\"once\" id=\"_anim_radio1_5d9cef03daf64662882705f9f9bba16b\"\n",
+       "      <input type=\"radio\" name=\"state\" value=\"once\" id=\"_anim_radio1_f59c28f9f5894974b4791611a2c64487\"\n",
        "             >\n",
-       "      <label for=\"_anim_radio1_5d9cef03daf64662882705f9f9bba16b\">Once</label>\n",
-       "      <input type=\"radio\" name=\"state\" value=\"loop\" id=\"_anim_radio2_5d9cef03daf64662882705f9f9bba16b\"\n",
+       "      <label for=\"_anim_radio1_f59c28f9f5894974b4791611a2c64487\">Once</label>\n",
+       "      <input type=\"radio\" name=\"state\" value=\"loop\" id=\"_anim_radio2_f59c28f9f5894974b4791611a2c64487\"\n",
        "             checked>\n",
-       "      <label for=\"_anim_radio2_5d9cef03daf64662882705f9f9bba16b\">Loop</label>\n",
-       "      <input type=\"radio\" name=\"state\" value=\"reflect\" id=\"_anim_radio3_5d9cef03daf64662882705f9f9bba16b\"\n",
+       "      <label for=\"_anim_radio2_f59c28f9f5894974b4791611a2c64487\">Loop</label>\n",
+       "      <input type=\"radio\" name=\"state\" value=\"reflect\" id=\"_anim_radio3_f59c28f9f5894974b4791611a2c64487\"\n",
        "             >\n",
-       "      <label for=\"_anim_radio3_5d9cef03daf64662882705f9f9bba16b\">Reflect</label>\n",
+       "      <label for=\"_anim_radio3_f59c28f9f5894974b4791611a2c64487\">Reflect</label>\n",
        "    </form>\n",
        "  </div>\n",
        "</div>\n",
@@ -2923,9 +2923,9 @@
        "  /* Instantiate the Animation class. */\n",
        "  /* The IDs given should match those used in the template above. */\n",
        "  (function() {\n",
-       "    var img_id = \"_anim_img5d9cef03daf64662882705f9f9bba16b\";\n",
-       "    var slider_id = \"_anim_slider5d9cef03daf64662882705f9f9bba16b\";\n",
-       "    var loop_select_id = \"_anim_loop_select5d9cef03daf64662882705f9f9bba16b\";\n",
+       "    var img_id = \"_anim_imgf59c28f9f5894974b4791611a2c64487\";\n",
+       "    var slider_id = \"_anim_sliderf59c28f9f5894974b4791611a2c64487\";\n",
+       "    var loop_select_id = \"_anim_loop_selectf59c28f9f5894974b4791611a2c64487\";\n",
        "    var frames = new Array(3);\n",
        "    \n",
        "  frames[0] = \"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAA8AAAALQCAYAAABfdxm0AAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90\\\n",
@@ -5697,14 +5697,14 @@
        "    /* set a timeout to make sure all the above elements are created before\n",
        "       the object is initialized. */\n",
        "    setTimeout(function() {\n",
-       "        anim5d9cef03daf64662882705f9f9bba16b = new Animation(frames, img_id, slider_id, 200.0,\n",
+       "        animf59c28f9f5894974b4791611a2c64487 = new Animation(frames, img_id, slider_id, 200.0,\n",
        "                                 loop_select_id);\n",
        "    }, 0);\n",
        "  })()\n",
        "</script>\n"
       ],
       "text/plain": [
-       "<matplotlib.animation.FuncAnimation at 0x175fb8e00>"
+       "<matplotlib.animation.FuncAnimation at 0x150cc4f50>"
       ]
      },
      "execution_count": 147,
@@ -5937,42 +5937,42 @@
        "</style>\n",
        "\n",
        "<div class=\"animation\">\n",
-       "  <img id=\"_anim_imgc28e0e9e1b144da299de29e5ae4bfa26\">\n",
+       "  <img id=\"_anim_img341b36c396fc4a0cb12fe324af161d71\">\n",
        "  <div class=\"anim-controls\">\n",
-       "    <input id=\"_anim_sliderc28e0e9e1b144da299de29e5ae4bfa26\" type=\"range\" class=\"anim-slider\"\n",
+       "    <input id=\"_anim_slider341b36c396fc4a0cb12fe324af161d71\" type=\"range\" class=\"anim-slider\"\n",
        "           name=\"points\" min=\"0\" max=\"1\" step=\"1\" value=\"0\"\n",
-       "           oninput=\"animc28e0e9e1b144da299de29e5ae4bfa26.set_frame(parseInt(this.value));\">\n",
+       "           oninput=\"anim341b36c396fc4a0cb12fe324af161d71.set_frame(parseInt(this.value));\">\n",
        "    <div class=\"anim-buttons\">\n",
-       "      <button title=\"Decrease speed\" aria-label=\"Decrease speed\" onclick=\"animc28e0e9e1b144da299de29e5ae4bfa26.slower()\">\n",
+       "      <button title=\"Decrease speed\" aria-label=\"Decrease speed\" onclick=\"anim341b36c396fc4a0cb12fe324af161d71.slower()\">\n",
        "          <i class=\"fa fa-minus\"></i></button>\n",
-       "      <button title=\"First frame\" aria-label=\"First frame\" onclick=\"animc28e0e9e1b144da299de29e5ae4bfa26.first_frame()\">\n",
+       "      <button title=\"First frame\" aria-label=\"First frame\" onclick=\"anim341b36c396fc4a0cb12fe324af161d71.first_frame()\">\n",
        "        <i class=\"fa fa-fast-backward\"></i></button>\n",
-       "      <button title=\"Previous frame\" aria-label=\"Previous frame\" onclick=\"animc28e0e9e1b144da299de29e5ae4bfa26.previous_frame()\">\n",
+       "      <button title=\"Previous frame\" aria-label=\"Previous frame\" onclick=\"anim341b36c396fc4a0cb12fe324af161d71.previous_frame()\">\n",
        "          <i class=\"fa fa-step-backward\"></i></button>\n",
-       "      <button title=\"Play backwards\" aria-label=\"Play backwards\" onclick=\"animc28e0e9e1b144da299de29e5ae4bfa26.reverse_animation()\">\n",
+       "      <button title=\"Play backwards\" aria-label=\"Play backwards\" onclick=\"anim341b36c396fc4a0cb12fe324af161d71.reverse_animation()\">\n",
        "          <i class=\"fa fa-play fa-flip-horizontal\"></i></button>\n",
-       "      <button title=\"Pause\" aria-label=\"Pause\" onclick=\"animc28e0e9e1b144da299de29e5ae4bfa26.pause_animation()\">\n",
+       "      <button title=\"Pause\" aria-label=\"Pause\" onclick=\"anim341b36c396fc4a0cb12fe324af161d71.pause_animation()\">\n",
        "          <i class=\"fa fa-pause\"></i></button>\n",
-       "      <button title=\"Play\" aria-label=\"Play\" onclick=\"animc28e0e9e1b144da299de29e5ae4bfa26.play_animation()\">\n",
+       "      <button title=\"Play\" aria-label=\"Play\" onclick=\"anim341b36c396fc4a0cb12fe324af161d71.play_animation()\">\n",
        "          <i class=\"fa fa-play\"></i></button>\n",
-       "      <button title=\"Next frame\" aria-label=\"Next frame\" onclick=\"animc28e0e9e1b144da299de29e5ae4bfa26.next_frame()\">\n",
+       "      <button title=\"Next frame\" aria-label=\"Next frame\" onclick=\"anim341b36c396fc4a0cb12fe324af161d71.next_frame()\">\n",
        "          <i class=\"fa fa-step-forward\"></i></button>\n",
-       "      <button title=\"Last frame\" aria-label=\"Last frame\" onclick=\"animc28e0e9e1b144da299de29e5ae4bfa26.last_frame()\">\n",
+       "      <button title=\"Last frame\" aria-label=\"Last frame\" onclick=\"anim341b36c396fc4a0cb12fe324af161d71.last_frame()\">\n",
        "          <i class=\"fa fa-fast-forward\"></i></button>\n",
-       "      <button title=\"Increase speed\" aria-label=\"Increase speed\" onclick=\"animc28e0e9e1b144da299de29e5ae4bfa26.faster()\">\n",
+       "      <button title=\"Increase speed\" aria-label=\"Increase speed\" onclick=\"anim341b36c396fc4a0cb12fe324af161d71.faster()\">\n",
        "          <i class=\"fa fa-plus\"></i></button>\n",
        "    </div>\n",
-       "    <form title=\"Repetition mode\" aria-label=\"Repetition mode\" action=\"#n\" name=\"_anim_loop_selectc28e0e9e1b144da299de29e5ae4bfa26\"\n",
+       "    <form title=\"Repetition mode\" aria-label=\"Repetition mode\" action=\"#n\" name=\"_anim_loop_select341b36c396fc4a0cb12fe324af161d71\"\n",
        "          class=\"anim-state\">\n",
-       "      <input type=\"radio\" name=\"state\" value=\"once\" id=\"_anim_radio1_c28e0e9e1b144da299de29e5ae4bfa26\"\n",
+       "      <input type=\"radio\" name=\"state\" value=\"once\" id=\"_anim_radio1_341b36c396fc4a0cb12fe324af161d71\"\n",
        "             >\n",
-       "      <label for=\"_anim_radio1_c28e0e9e1b144da299de29e5ae4bfa26\">Once</label>\n",
-       "      <input type=\"radio\" name=\"state\" value=\"loop\" id=\"_anim_radio2_c28e0e9e1b144da299de29e5ae4bfa26\"\n",
+       "      <label for=\"_anim_radio1_341b36c396fc4a0cb12fe324af161d71\">Once</label>\n",
+       "      <input type=\"radio\" name=\"state\" value=\"loop\" id=\"_anim_radio2_341b36c396fc4a0cb12fe324af161d71\"\n",
        "             checked>\n",
-       "      <label for=\"_anim_radio2_c28e0e9e1b144da299de29e5ae4bfa26\">Loop</label>\n",
-       "      <input type=\"radio\" name=\"state\" value=\"reflect\" id=\"_anim_radio3_c28e0e9e1b144da299de29e5ae4bfa26\"\n",
+       "      <label for=\"_anim_radio2_341b36c396fc4a0cb12fe324af161d71\">Loop</label>\n",
+       "      <input type=\"radio\" name=\"state\" value=\"reflect\" id=\"_anim_radio3_341b36c396fc4a0cb12fe324af161d71\"\n",
        "             >\n",
-       "      <label for=\"_anim_radio3_c28e0e9e1b144da299de29e5ae4bfa26\">Reflect</label>\n",
+       "      <label for=\"_anim_radio3_341b36c396fc4a0cb12fe324af161d71\">Reflect</label>\n",
        "    </form>\n",
        "  </div>\n",
        "</div>\n",
@@ -5982,9 +5982,9 @@
        "  /* Instantiate the Animation class. */\n",
        "  /* The IDs given should match those used in the template above. */\n",
        "  (function() {\n",
-       "    var img_id = \"_anim_imgc28e0e9e1b144da299de29e5ae4bfa26\";\n",
-       "    var slider_id = \"_anim_sliderc28e0e9e1b144da299de29e5ae4bfa26\";\n",
-       "    var loop_select_id = \"_anim_loop_selectc28e0e9e1b144da299de29e5ae4bfa26\";\n",
+       "    var img_id = \"_anim_img341b36c396fc4a0cb12fe324af161d71\";\n",
+       "    var slider_id = \"_anim_slider341b36c396fc4a0cb12fe324af161d71\";\n",
+       "    var loop_select_id = \"_anim_loop_select341b36c396fc4a0cb12fe324af161d71\";\n",
        "    var frames = new Array(1);\n",
        "    \n",
        "  frames[0] = \"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAA8AAAALQCAYAAABfdxm0AAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90\\\n",
@@ -6662,14 +6662,14 @@
        "    /* set a timeout to make sure all the above elements are created before\n",
        "       the object is initialized. */\n",
        "    setTimeout(function() {\n",
-       "        animc28e0e9e1b144da299de29e5ae4bfa26 = new Animation(frames, img_id, slider_id, 200.0,\n",
+       "        anim341b36c396fc4a0cb12fe324af161d71 = new Animation(frames, img_id, slider_id, 200.0,\n",
        "                                 loop_select_id);\n",
        "    }, 0);\n",
        "  })()\n",
        "</script>\n"
       ],
       "text/plain": [
-       "<matplotlib.animation.FuncAnimation at 0x175fe3d10>"
+       "<matplotlib.animation.FuncAnimation at 0x1508c92b0>"
       ]
      },
      "execution_count": 150,
@@ -6701,7 +6701,7 @@
     {
      "data": {
       "text/plain": [
-       "Puzzle 14.2:  1.869 seconds, answer 6876              ok"
+       "Puzzle 14.2:  1.825 seconds, answer 6876              ok"
       ]
      },
      "execution_count": 152,
@@ -6897,7 +6897,7 @@
     {
      "data": {
       "text/plain": [
-       "Puzzle 15.1:   .029 seconds, answer 1563092           ok"
+       "Puzzle 15.1:   .028 seconds, answer 1563092           ok"
       ]
      },
      "execution_count": 160,
@@ -7080,7 +7080,7 @@
     {
      "data": {
       "text/plain": [
-       "Puzzle 16.1:   .147 seconds, answer 103512            ok"
+       "Puzzle 16.1:   .144 seconds, answer 103512            ok"
       ]
      },
      "execution_count": 169,
@@ -7144,7 +7144,7 @@
     {
      "data": {
       "text/plain": [
-       "Puzzle 16.2:   .852 seconds, answer 554               ok"
+       "Puzzle 16.2:   .846 seconds, answer 554               ok"
       ]
      },
      "execution_count": 172,
@@ -7450,7 +7450,7 @@
     {
      "data": {
       "text/plain": [
-       "Puzzle 17.2:   .023 seconds, answer 267265166222235   ok"
+       "Puzzle 17.2:   .024 seconds, answer 267265166222235   ok"
       ]
      },
      "execution_count": 187,
@@ -7761,7 +7761,7 @@
     {
      "data": {
       "text/plain": [
-       "Puzzle 19.2:   .187 seconds, answer 595975512785325   ok"
+       "Puzzle 19.2:   .184 seconds, answer 595975512785325   ok"
       ]
      },
      "execution_count": 204,
@@ -7957,7 +7957,7 @@
     {
      "data": {
       "text/plain": [
-       "Puzzle 20.2:   .770 seconds, answer 982891            ok"
+       "Puzzle 20.2:   .756 seconds, answer 982891            ok"
       ]
      },
      "execution_count": 215,
@@ -8002,7 +8002,7 @@
     {
      "data": {
       "text/plain": [
-       "Puzzle 20.1:   .023 seconds, answer 1343              ok"
+       "Puzzle 20.1:   .022 seconds, answer 1343              ok"
       ]
      },
      "execution_count": 217,
@@ -8057,7 +8057,7 @@
    "source": [
     "### Part 1: What is the sum of the complexities of the five codes on your list?\n",
     "\n",
-    "The problem is that the numeric keypad is inaccessible, so we have to get a robot to press the buttons, and we have to command the robot's arm by pressing some buttons on a **remote control directional keypad**: arrow keys to move the arm, and the \"`A`\" (activate) button to make the arm press the button it is currently pointing at. Here are the two keypads:\n",
+    "We know the codes, so what is the difficulty? Unfortunately, the numeric keypad is inaccessible, so we have to get a **robot** to press the buttons, and we have to command the robot's arm by pressing some buttons on a **remote control directional keypad**: arrow keys to move the arm, and the \"`A`\" (activate) button to make the arm press the button it is currently pointing at. Here are the two keypads:\n",
     "\n",
     "        +---+---+---+            +---+---+\n",
     "        | 7 | 8 | 9 |            | ^ | A |\n",
@@ -8069,28 +8069,28 @@
     "            | 0 | A |\n",
     "            +---+---+\n",
     "        \n",
-    "Long story short: it turns out the directional keypad is *also* inaccessible, and we actually need **three robots**, each pressing buttons on the next keypad, so there are four levels of button pressing all together:\n",
+    "Long story short: it turns out the robot's remote keypad is *also* inaccessible, and we end up needing **three robots**, each pressing buttons on the next keypad, so there are four levels of button pressing all together:\n",
     "\n",
-    "- Level 4: You press some buttons on Robot 3's directional keypad (e.g. \"<vA<<A\"...)\n",
-    "- Level 3: Robot 3 presses some buttons on Robot 2's directional keypad (e.g. \"v<<A>>\"...)\n",
-    "- Level 2: Robot 2 presses some buttons on Robot 1's directional keypad (e.g. \"<A^A^^AvvvA\")\n",
-    "- Level 1: Robot 1 presses some buttons on the numeric keypad (e.g. \"029A\").\n",
+    "- You press some buttons on Robot 3's remote keypad (e.g. \"<vA<<A\"...)\n",
+    "- Robot 3 presses some buttons on Robot 2's remote keypad (e.g. \"v<<A>>\"...)\n",
+    "- Robot 2 presses some buttons on Robot 1's remote keypad (e.g. \"<A^A^^AvvvA\")\n",
+    "- Robot 1 presses some buttons on the door's numeric keypad (e.g. \"029A\").\n",
     "\n",
     "We'd like to find the shortest possible sequence of button presses at the top level. Once we've figured out the shortest button sequence to enter each of the codes, we have to compute the **complexity** of each code, which is defined as the integer part of the code times the length of the shortest possible sequence.\n",
     "\n",
     "Here are some considerations:\n",
     "- To get from one button to the next there can be several paths.\n",
-    "- For example, to get from \"A\" to \"1\" the shortest paths are \"^<<\" or \"<^<\" (but not \"<<^\", because that would hit the \".\" position).\n",
+    "- For example, to get from \"A\" to \"7\" the shortest paths include \"^^^<<\" and \"^<^<^\" and \"<^<^^\" (but not \"<<^^^\", because that would hit the empty position, which is forbidden.\n",
     "- Each of these paths have the same length, so if we were directly pressing Robot 1's remote control, any path would do.\n",
     "- However, it might take the robot at the next level a different number of presses to command each of these paths.\n",
-    "- So finding the shortest path at one level depends on the levels below it.\n",
-    "- Fortunately, whenever there is a button press on the numeric keypad, we know that the robots on all the higher-level directional keypads are at \"A\", because each one must have activated a button press on the level below.\n",
-    "- Therefore, the parts of the problem are decomposible and commutative: we can optimize \"get the robot arm from the current position to the desired position\" and that will not depend on the solutions to other subparts of the problem.\n",
-    "- I'm guessing that in Part 2 the button sequences will be ridiculously long, so I'll look for efficient representations.\n",
-    "- If we only need to know the total number of button presses (not the order), then we can aggregate them:\n",
-    "    - For example, if the task is to press \"0909A\", and we always start at \"A\", then we can represent this as:\n",
-    "    - {A-to-0: once, 0-to-9: twice, 9-to-0: twice, 9-to-A: once}\n",
-    "    - In other words, a sequence of button presses is implemented as a Counter of start-to-destination pairs."
+    "- To command \"^^^\" the next robot has to command this robot to the \"^\" key and then press \"AAA\". If the \"^\" and \"<\" keys are interspersed, the commands will take more buttons, because there will be navigation between each \"A\". Therefore, I will consider at most two paths: all the horizontal arrows first, then all the veertical, or vice versa.\n",
+    "- Fortunately, whenever there is a button press on the numeric keypad, we know that the robots on all the higher-level remote keypads are at \"A\", because each one must have activated a button press to make it all happen.\n",
+    "- Therefore, the parts of the problem are *decomposible*: we can optimize \"get the robot arm from the current position to the desired position\" and that will not depend on the solutions to other subparts of the problem.\n",
+    "- I'm guessing that in Part 2 the button sequences will be ridiculously long, so I'll look for an efficient representations.\n",
+    "    - If Part 2 needs to know something about the actual path of presses, we could represent it as a `Counter` of start+destination subtasks. For example, if the task is to press \"0909A\", and we always start at \"A\", then we can represent this {'A0': 1, '09: 2, '90': 2, '9A': 1}.\n",
+    "    - If Part 2 only needs to know the number of button presses, but adds more robots and/or more keyboards, then we don't need to explicitly represent the button presses, we just need to count the length. I'm going to guess that that will be the case.\n",
+    "\n",
+    "I could represent a keypad with my `Grid` class, but I don't need what the Grid provides, all I need is a way to look up the (x, y) location of each key. I'll also explicitly save the (x, y) position of the forbidden empty spot as `keypad.avoid`."
    ]
   },
   {
@@ -8111,6 +8111,14 @@
     "remote_keypad  = Keypad(['_^A', '<v>'])"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "id": "486ae303-36cf-4750-965b-3402eec2b625",
+   "metadata": {},
+   "source": [
+    "Now I can compute the possible paths from any starting key to any destination key on a keypad, and press the destination key. There will be either one or two paths, depending on whether one of them hits the `avoid` spot."
+   ]
+  },
   {
    "cell_type": "code",
    "execution_count": 222,
@@ -8119,7 +8127,7 @@
    "outputs": [],
    "source": [
     "def possible_paths(key1, key2, keypad) -> Iterable[str]:\n",
-    "    \"\"\"Should we try, e.g. \">>^\" or \"^>>\", or both, to get from pos1 to pos2 on keypad?\"\"\"\n",
+    "    \"\"\"Should we try, e.g. \">>^\" or \"^>>\", or both, to get from key1 to key2 on keypad?\"\"\"\n",
     "    pos1, pos2 = keypad[key1], keypad[key2]\n",
     "    (dx, dy) = sub(pos2, pos1)\n",
     "    horizontal = abs(dx) * ('>' if dx > 0 else '<')\n",
@@ -8132,223 +8140,147 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 223,
+   "execution_count": 446,
    "id": "391f6120-6373-43f2-b697-585c90e45e54",
    "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "text/plain": [
-       "['>>vvA', 'vv>>A']"
-      ]
-     },
-     "execution_count": 223,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
+   "outputs": [],
    "source": [
-    "list(possible_paths('7', '3', numeric_keypad))"
+    "assert list(possible_paths('A', '7', numeric_keypad)) == ['^^^<<A']\n",
+    "assert list(possible_paths('8', 'A', numeric_keypad)) == ['>vvvA', 'vvv>A']\n",
+    "assert list(possible_paths('v', 'A', remote_keypad))  == ['>^A', '^>A']"
    ]
   },
   {
-   "cell_type": "code",
-   "execution_count": 224,
-   "id": "aec8849e-5fe7-47ab-9569-7fcf4f3ede17",
+   "cell_type": "markdown",
+   "id": "74f1f872-5c0c-4baa-a941-d910dc9c1048",
    "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "text/plain": [
-       "['>>vvvA']"
-      ]
-     },
-     "execution_count": 224,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
    "source": [
-    "list(possible_paths('7', 'A', numeric_keypad))"
+    "We need to compute the complexity:"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 225,
-   "id": "9f5fe5bb-de7c-435d-a912-d80a6c4e5de2",
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "text/plain": [
-       "'ok'"
-      ]
-     },
-     "execution_count": 225,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "def min_keypresses(code, num_remotes, remote_keypad=remote_keypad, numeric_keypad=numeric_keypad):\n",
-    "    \"\"\"The number of keypresses to type `code`, going through `num_remotes`.\"\"\"\n",
-    "    def total_presses(level: int, keys: str) -> int:\n",
-    "        return sum(cached_lengths[(level, key1, key2)] \n",
-    "                   for key1, key2 in sliding_window('A' + keys, 2))\n",
-    "    ## Initializing the cached_lengths[level, position1, position2] dict\n",
-    "    cached_lengths = {(0, key1, key2): 1 for key1 in remote_keypad for key2 in remote_keypad}\n",
-    "    for level in range(1, num_remotes + 1):\n",
-    "        keypad = numeric_keypad if level == num_remotes else remote_keypad\n",
-    "        for key1 in keypad:\n",
-    "            for key2 in keypad:\n",
-    "                paths = possible_paths(key1, key2, keypad)\n",
-    "                cached_lengths[(level, key1, key2)] = min(total_presses(level - 1, path) for path in paths)\n",
-    "    ## Go through the cache, finding the shortest total number of presses\n",
-    "    return total_presses(level, code)\n",
-    "\n",
-    "part_1 = sum(min_keypresses(code, 3)  * int(code[:-1]) for code in codes)\n",
-    "assert part_1 == 205160\n",
-    "part_2 = sum(min_keypresses(code, 26) * int(code[:-1]) for code in codes)\n",
-    "assert part_2 == 252473394928452\n",
-    "'ok'"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 226,
+   "execution_count": 449,
    "id": "1e2e8c59-744f-4f36-9251-c8ea5fd60aed",
    "metadata": {},
    "outputs": [],
    "source": [
-    "def complexity(code, presses: int): \n",
-    "    \"\"\"The integer part of the code times the number of keypresses.\"\"\"\n",
-    "    return int(cat(digits(code))) * presses"
+    "def complexity(code, num_presses: int): \n",
+    "    \"\"\"The integer part of the code times the (niminimum) number of key presses.\"\"\"\n",
+    "    return int(cat(digits(code))) * num_presses"
    ]
   },
   {
-   "cell_type": "code",
-   "execution_count": 227,
-   "id": "cfd67700-2b5c-4f2f-a631-382816a30ebf",
+   "cell_type": "markdown",
+   "id": "975b5719-9afc-4b0a-9e07-c1c2b38059f7",
    "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "text/plain": [
-       "'ok'"
-      ]
-     },
-     "execution_count": 227,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
    "source": [
-    "def min_keypresses(code: str, levels: int, remote_keypad=remote_keypad, numeric_keypad=numeric_keypad):\n",
-    "    \"\"\"The number of keypresses to type `code`, going through `levels`.\"\"\"\n",
-    "    def total_presses(keys: str, level: int) -> int:\n",
-    "        return sum(cached_lengths[(level, key1, key2)] \n",
-    "                   for key1, key2 in sliding_window('A' + keys, 2))\n",
-    "    ## Initializing the cached_lengths[level, position1, position2] dict\n",
-    "    cached_lengths = {(0, key1, key2): 1 for key1 in remote_keypad for key2 in remote_keypad}\n",
-    "    for level in range(1, levels):\n",
-    "        keypad = numeric_keypad if level == levels - 1 else remote_keypad\n",
-    "        for key1 in keypad:\n",
-    "            for key2 in keypad:\n",
-    "                paths = possible_paths(key1, key2, keypad)\n",
-    "                cached_lengths[(level, key1, key2)] = min(total_presses(path, level - 1) for path in paths)\n",
-    "    ## Go through the cache, finding the shortest total number of presses\n",
-    "    return total_presses(code, levels - 1)\n",
-    "\n",
-    "part_1 = sum(min_keypresses(code, 4)  * int(code[:-1]) for code in codes)\n",
-    "assert part_1 == 205160\n",
-    "part_2 = sum(min_keypresses(code, 27) * int(code[:-1]) for code in codes)\n",
-    "assert part_2 == 252473394928452\n",
-    "'ok'"
+    "And here's how we compute the minimum number of keypresses. Normally I would do this with a `@cache` decorator, but I was encountering **bugs** doing that and got confused, so I refactored to explicitly create a cache of path lengths saying how many presses it takes at each level to command a single button press, getting you from the start button to the destination button. This made debugging slightly easier for me, or maybe it was just starting over that made it easier."
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 228,
+   "execution_count": 489,
    "id": "704ade67-31e6-4f39-b48c-0a7185c42c48",
    "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "text/plain": [
-       "'ok'"
-      ]
-     },
-     "execution_count": 228,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
+   "outputs": [],
    "source": [
-    "def min_keypresses(code: str, levels: int) -> int:\n",
+    "def min_keypresses(code: str, levels: int, numeric_keypad=numeric_keypad, remote_keypad=remote_keypad) -> int:\n",
     "    \"\"\"The number of keypresses to type `code`, going through `levels`.\"\"\"\n",
     "    def total_presses(keys: str, level: int) -> int:\n",
-    "        return sum(cached_lengths[(level, key1, key2)] \n",
+    "        return sum(one_button_path_lengths[level, key1, key2] \n",
     "                   for key1, key2 in sliding_window('A' + keys, 2))\n",
-    "    ## Initializing the cached_lengths[level, position1, position2] dict\n",
-    "    cached_lengths = {(0, key1, key2): 1 for key1 in remote_keypad for key2 in remote_keypad}\n",
+    "    ## Initializing the one_button_path_lengths[level, key1, key2] dict\n",
+    "    one_button_path_lengths = {(0, key1, key2): 1\n",
+    "                               for key1 in remote_keypad for key2 in remote_keypad}\n",
     "    for level in range(1, levels):\n",
     "        keypad = numeric_keypad if level == levels - 1 else remote_keypad\n",
     "        for key1 in keypad:\n",
     "            for key2 in keypad:\n",
     "                paths = possible_paths(key1, key2, keypad)\n",
-    "                cached_lengths[(level, key1, key2)] = min(total_presses(path, level - 1) for path in paths)\n",
-    "    ## Go through the cache, finding the shortest total number of presses\n",
-    "    return total_presses(code, levels - 1)\n",
-    "\n",
-    "part_1 = sum(min_keypresses(code, 4)  * int(code[:-1]) for code in codes)\n",
-    "assert part_1 == 205160\n",
-    "part_2 = sum(min_keypresses(code, 27) * int(code[:-1]) for code in codes)\n",
-    "assert part_2 == 252473394928452\n",
-    "'ok'"
+    "                one_button_path_lengths[level, key1, key2] = min(total_presses(path, level - 1) for path in paths)\n",
+    "    ## The cache is built, so just add up the one_button_path_lengths.\n",
+    "    return total_presses(code, levels - 1)"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 229,
+   "execution_count": 491,
+   "id": "4d8c8659-52d3-4349-8465-7003e880502f",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "assert list(sliding_window('A029A', 2)) == ['A0', '02', '29', '9A'] # All the start+destination pairs from the code."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "696ad2eb-244a-4ed1-ac0b-d40a0556090f",
+   "metadata": {},
+   "source": [
+    "Here is the answer, which turns out to be a fairly small number:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 493,
    "id": "393a8c6b-6a1c-4495-aad5-92f48711fbf1",
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/plain": [
-       "Puzzle 21.1:   .002 seconds, answer 205160            ok"
+       "Puzzle 21.1:   .004 seconds, answer 205160            ok"
       ]
      },
-     "execution_count": 229,
+     "execution_count": 493,
      "metadata": {},
      "output_type": "execute_result"
     }
    ],
    "source": [
     "answer(21.1, 205160, lambda:\n",
-    "       sum(complexity(code, min_keypresses(code, 4)) for code in codes))"
+    "       sum(complexity(code, min_keypresses(code, levels=4)) for code in codes))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "2a16c8f8-e1e5-47f8-b6c6-4a7500a457e6",
+   "metadata": {},
+   "source": [
+    "### Part 2: Now, what is the sum of the complexities of the five codes on your list?\n",
+    "\n",
+    "As I suspected, Part 2 gets us into the ridiculous number of button presses. The problem is the same, but now there are **27** levels, not 4. No new code, just sum the complexity of the `min_keypresses` again:"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 230,
+   "execution_count": 495,
    "id": "563c5959-692d-4368-8c0b-469fa0142678",
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/plain": [
-       "Puzzle 21.2:   .008 seconds, answer 252473394928452   ok"
+       "Puzzle 21.2:   .016 seconds, answer 252473394928452   ok"
       ]
      },
-     "execution_count": 230,
+     "execution_count": 495,
      "metadata": {},
      "output_type": "execute_result"
     }
    ],
    "source": [
     "answer(21.2, 252473394928452, lambda:\n",
-    "       sum(complexity(code, min_keypresses(code, 27)) for code in codes))"
+    "       sum(complexity(code, min_keypresses(code, levels=27)) for code in codes))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1cf2d848-2f43-48dc-8862-493cebd94558",
+   "metadata": {},
+   "source": [
+    "If we could press one button per microsecond, it would only take 8 years to do this number of button presses."
    ]
   },
   {
@@ -8363,7 +8295,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 232,
+   "execution_count": 230,
    "id": "2c1e7612-4ec5-4ce1-b591-5c3a14f8ea61",
    "metadata": {},
    "outputs": [
@@ -8414,7 +8346,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 234,
+   "execution_count": 232,
    "id": "3807adbc-261a-4108-88ba-6e2b061526b5",
    "metadata": {},
    "outputs": [],
@@ -8430,17 +8362,17 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 235,
+   "execution_count": 233,
    "id": "fce754cd-f6da-4e7d-b406-3314c9c28ca8",
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/plain": [
-       "Puzzle 22.1:   .321 seconds, answer 14273043166       ok"
+       "Puzzle 22.1:   .314 seconds, answer 14273043166       ok"
       ]
      },
-     "execution_count": 235,
+     "execution_count": 233,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -8484,7 +8416,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 237,
+   "execution_count": 235,
    "id": "cbbb4793-b13d-4a95-ba03-673b3ebaa229",
    "metadata": {},
    "outputs": [],
@@ -8517,17 +8449,17 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 238,
+   "execution_count": 236,
    "id": "34eaf7fe-fe9b-4858-b1e8-e4c3713ea093",
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/plain": [
-       "Puzzle 22.2:  1.265 seconds, answer 1667              ok"
+       "Puzzle 22.2:  1.259 seconds, answer 1667              ok"
       ]
      },
-     "execution_count": 238,
+     "execution_count": 236,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -8547,7 +8479,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 240,
+   "execution_count": 238,
    "id": "a2da8c78-d240-4c52-a83a-f3599f4f69a4",
    "metadata": {},
    "outputs": [
@@ -8557,7 +8489,7 @@
        "(-3, 2, -1, 2)"
       ]
      },
-     "execution_count": 240,
+     "execution_count": 238,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -8578,7 +8510,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 242,
+   "execution_count": 240,
    "id": "9de3f365-c546-494a-a18c-2ae58d88792b",
    "metadata": {},
    "outputs": [
@@ -8608,7 +8540,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 244,
+   "execution_count": 242,
    "id": "85f5b145-8c5e-448c-8b45-ad5750252ff2",
    "metadata": {},
    "outputs": [
@@ -8659,7 +8591,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 246,
+   "execution_count": 244,
    "id": "289d2325-1e58-41f5-b4b2-b90ae26e7887",
    "metadata": {},
    "outputs": [],
@@ -8684,7 +8616,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 247,
+   "execution_count": 245,
    "id": "6425577d-4ca9-45de-9698-cd9b026f7ce6",
    "metadata": {},
    "outputs": [
@@ -8694,7 +8626,7 @@
        "Puzzle 23.1:   .001 seconds, answer 1170              ok"
       ]
      },
-     "execution_count": 247,
+     "execution_count": 245,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -8716,7 +8648,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 249,
+   "execution_count": 247,
    "id": "88811073-22b2-4378-afa5-9d38462c63b5",
    "metadata": {},
    "outputs": [],
@@ -8739,7 +8671,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 250,
+   "execution_count": 248,
    "id": "0b5f08ac-18e2-4933-9737-cdbc842c5809",
    "metadata": {},
    "outputs": [
@@ -8749,7 +8681,7 @@
        "Puzzle 23.2:   .003 seconds, answer bo,dd,eq,ik,lo,lu,ph,ro,rr,rw,uo,wx,yg ok"
       ]
      },
-     "execution_count": 250,
+     "execution_count": 248,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -8773,7 +8705,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 252,
+   "execution_count": 250,
    "id": "47421581-71df-4c72-a62e-c40d0596fdbb",
    "metadata": {},
    "outputs": [
@@ -8836,7 +8768,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 254,
+   "execution_count": 252,
    "id": "a9a4780f-6033-452f-b49c-74b97c9e2440",
    "metadata": {},
    "outputs": [],
@@ -8858,7 +8790,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 255,
+   "execution_count": 253,
    "id": "72437439-dddf-4202-9944-36e796800304",
    "metadata": {},
    "outputs": [
@@ -8868,7 +8800,7 @@
        "Puzzle 24.1:   .001 seconds, answer 36035961805936    ok"
       ]
      },
-     "execution_count": 255,
+     "execution_count": 253,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -8915,7 +8847,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 258,
+   "execution_count": 256,
    "id": "3b41a166-d33b-402b-971b-1dc1d33135ec",
    "metadata": {},
    "outputs": [
@@ -8925,7 +8857,7 @@
        "Counter({'AND': 89, 'XOR': 89, 'OR': 44})"
       ]
      },
-     "execution_count": 258,
+     "execution_count": 256,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -8946,7 +8878,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 260,
+   "execution_count": 258,
    "id": "ec8d86d4-8caf-40f2-b516-c7da4acbda19",
    "metadata": {},
    "outputs": [],
@@ -8978,7 +8910,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 262,
+   "execution_count": 260,
    "id": "2a06af7f-2b9a-4b1f-b9dd-9ec6fb94a1e6",
    "metadata": {},
    "outputs": [],
@@ -9019,7 +8951,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 263,
+   "execution_count": 261,
    "id": "e9b87c38-f67c-4948-9af6-251021747e9d",
    "metadata": {},
    "outputs": [
@@ -9029,7 +8961,7 @@
        "Puzzle 24.2:   .000 seconds, answer jqf,mdd,skh,wpd,wts,z11,z19,z37 ok"
       ]
      },
-     "execution_count": 263,
+     "execution_count": 261,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -9053,7 +8985,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 265,
+   "execution_count": 263,
    "id": "f5971853-7139-4f17-bdc5-6c51e12a928d",
    "metadata": {},
    "outputs": [
@@ -9112,7 +9044,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 267,
+   "execution_count": 265,
    "id": "5302ac58-91fc-475a-83d9-cea91457df3b",
    "metadata": {},
    "outputs": [],
@@ -9134,7 +9066,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 268,
+   "execution_count": 266,
    "id": "c415f5c1-76c3-486c-9075-c46a5a8c5bcb",
    "metadata": {},
    "outputs": [],
@@ -9144,17 +9076,17 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 269,
+   "execution_count": 267,
    "id": "89c28b74-ed31-4bb5-b463-7177952a95ae",
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/plain": [
-       "Puzzle 25.1:   .022 seconds, answer 3196              ok"
+       "Puzzle 25.1:   .021 seconds, answer 3196              ok"
       ]
      },
-     "execution_count": 269,
+     "execution_count": 267,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -9176,7 +9108,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 271,
+   "execution_count": 269,
    "id": "34813fc9-a000-4cd8-88ae-692851b3242c",
    "metadata": {},
    "outputs": [
@@ -9190,53 +9122,53 @@
       "Puzzle  2.2:   .002 seconds, answer 328               ok\n",
       "Puzzle  3.1:   .001 seconds, answer 156388521         ok\n",
       "Puzzle  3.2:   .000 seconds, answer 75920122          ok\n",
-      "Puzzle  4.1:   .033 seconds, answer 2401              ok\n",
+      "Puzzle  4.1:   .032 seconds, answer 2401              ok\n",
       "Puzzle  4.2:   .026 seconds, answer 1822              ok\n",
       "Puzzle  5.1:   .001 seconds, answer 5762              ok\n",
       "Puzzle  5.2:   .001 seconds, answer 4130              ok\n",
       "Puzzle  6.1:   .002 seconds, answer 5329              ok\n",
-      "Puzzle  6.2:  1.967 seconds, answer 2162              ok\n",
-      "Puzzle  7.1:   .013 seconds, answer 1985268524462     ok\n",
-      "Puzzle  7.2:   .660 seconds, answer 150077710195188   ok\n",
+      "Puzzle  6.2:  1.942 seconds, answer 2162              ok\n",
+      "Puzzle  7.1:   .014 seconds, answer 1985268524462     ok\n",
+      "Puzzle  7.2:   .646 seconds, answer 150077710195188   ok\n",
       "Puzzle  8.1:   .003 seconds, answer 220               ok\n",
       "Puzzle  8.2:   .003 seconds, answer 813               ok\n",
-      "Puzzle  9.1:   .019 seconds, answer 6332189866718     ok\n",
-      "Puzzle  9.2:  2.754 seconds, answer 6353648390778     ok\n",
+      "Puzzle  9.1:   .020 seconds, answer 6332189866718     ok\n",
+      "Puzzle  9.2:  2.680 seconds, answer 6353648390778     ok\n",
       "Puzzle 10.1:   .005 seconds, answer 744               ok\n",
       "Puzzle 10.2:   .006 seconds, answer 1651              ok\n",
       "Puzzle 11.1:   .002 seconds, answer 194482            ok\n",
-      "Puzzle 11.2:   .061 seconds, answer 232454623677743   ok\n",
-      "Puzzle 12.1:   .052 seconds, answer 1402544           ok\n",
+      "Puzzle 11.2:   .060 seconds, answer 232454623677743   ok\n",
+      "Puzzle 12.1:   .051 seconds, answer 1402544           ok\n",
       "Puzzle 12.2:   .042 seconds, answer 862486            ok\n",
       "Puzzle 13.1:   .000 seconds, answer 29598             ok\n",
       "Puzzle 13.2:   .000 seconds, answer 93217456941970    ok\n",
       "Puzzle 14.1:   .000 seconds, answer 216027840         ok\n",
-      "Puzzle 14.2:  1.869 seconds, answer 6876              ok\n",
-      "Puzzle 15.1:   .029 seconds, answer 1563092           ok\n",
+      "Puzzle 14.2:  1.825 seconds, answer 6876              ok\n",
+      "Puzzle 15.1:   .028 seconds, answer 1563092           ok\n",
       "Puzzle 15.2:   .041 seconds, answer 1582688           ok\n",
-      "Puzzle 16.1:   .147 seconds, answer 103512            ok\n",
-      "Puzzle 16.2:   .852 seconds, answer 554               ok\n",
+      "Puzzle 16.1:   .144 seconds, answer 103512            ok\n",
+      "Puzzle 16.2:   .846 seconds, answer 554               ok\n",
       "Puzzle 17.1:   .000 seconds, answer 2,1,0,1,7,2,5,0,3 ok\n",
-      "Puzzle 17.2:   .023 seconds, answer 267265166222235   ok\n",
+      "Puzzle 17.2:   .024 seconds, answer 267265166222235   ok\n",
       "Puzzle 18.1:   .014 seconds, answer 344               ok\n",
       "Puzzle 18.2:   .032 seconds, answer 46,18             ok\n",
       "Puzzle 19.1:   .004 seconds, answer 242               ok\n",
-      "Puzzle 19.2:   .187 seconds, answer 595975512785325   ok\n",
-      "Puzzle 20.1:   .023 seconds, answer 1343              ok\n",
-      "Puzzle 20.2:   .770 seconds, answer 982891            ok\n",
+      "Puzzle 19.2:   .184 seconds, answer 595975512785325   ok\n",
+      "Puzzle 20.1:   .022 seconds, answer 1343              ok\n",
+      "Puzzle 20.2:   .756 seconds, answer 982891            ok\n",
       "Puzzle 21.1:   .002 seconds, answer 205160            ok\n",
       "Puzzle 21.2:   .008 seconds, answer 252473394928452   ok\n",
-      "Puzzle 22.1:   .321 seconds, answer 14273043166       ok\n",
-      "Puzzle 22.2:  1.265 seconds, answer 1667              ok\n",
+      "Puzzle 22.1:   .314 seconds, answer 14273043166       ok\n",
+      "Puzzle 22.2:  1.259 seconds, answer 1667              ok\n",
       "Puzzle 23.1:   .001 seconds, answer 1170              ok\n",
       "Puzzle 23.2:   .003 seconds, answer bo,dd,eq,ik,lo,lu,ph,ro,rr,rw,uo,wx,yg ok\n",
       "Puzzle 24.1:   .001 seconds, answer 36035961805936    ok\n",
       "Puzzle 24.2:   .000 seconds, answer jqf,mdd,skh,wpd,wts,z11,z19,z37 ok\n",
-      "Puzzle 25.1:   .022 seconds, answer 3196              ok\n",
+      "Puzzle 25.1:   .021 seconds, answer 3196              ok\n",
       "\n",
       "Correct: 49/49\n",
       "\n",
-      "Time in seconds: 0.008 median, 0.230 mean, 11.268 total.\n"
+      "Time in seconds: 0.008 median, 0.226 mean, 11.069 total.\n"
      ]
     }
    ],