03_03_2023_express.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "1091bb02",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2023-03-15T13:08:39.661907Z",
     "start_time": "2023-03-15T13:08:39.656345Z"
    }
   },
   "outputs": [],
   "source": [
    "from collections import defaultdict\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "29cd87a9",
   "metadata": {},
   "source": [
    "# Riddler Express"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ca7c7ef7",
   "metadata": {},
   "source": [
    "From Oscar Lanzi comes a matter of tricky triangular numbers:\n",
    "\n",
    "Suppose you are adding up whole numbers in order. You start with 1, then add 2 to get 3. Then you add 3 to get 6. Then you add 4 to get 10, and so on.\n",
    "\n",
    "If you keep going — adding larger and larger numbers in this fashion, and looking at the resulting sums — you might notice a few patterns among the last two digits. One such pattern is that no sum ever ends with the digits “23.” Meanwhile, some pairs of final digits are more common than others.\n",
    "\n",
    "**Which pairs of final digits are the most common?** (Here, “23” is a distinct final pair of digits from “32.”)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6645d030",
   "metadata": {},
   "source": [
    "Let us get the final 2 digits of the first 1,000,000 numbers. They belong to the interval [0, 99]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "11f7f42c",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2023-03-15T13:14:51.741113Z",
     "start_time": "2023-03-15T13:14:51.735675Z"
    }
   },
   "outputs": [],
   "source": [
    "def get_numbers() -> int:\n",
    "    \"\"\"Generator that produces an infinite amount of ordered triangular numbers\"\"\"\n",
    "    s = 0\n",
    "    i = 1\n",
    "    while True:\n",
    "        s += i\n",
    "        yield s\n",
    "        i += 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "id": "a933a899",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2023-03-15T13:29:52.484012Z",
     "start_time": "2023-03-15T13:29:51.731963Z"
    }
   },
   "outputs": [],
   "source": [
    "digits_distrib = defaultdict(int)\n",
    "numbers = get_numbers()\n",
    "for _ in range(1_000_000):\n",
    "    digits_distrib[str(next(numbers))[-2:]] += 1"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "42ac8abd",
   "metadata": {},
   "source": [
    "Getting the last two digits that are the most recurring"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "678007ce",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2023-03-15T13:19:36.248844Z",
     "start_time": "2023-03-15T13:19:36.239829Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "['28', '78', '53']"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "maxes = [last_two_digits for last_two_digits in digits_distrib if digits_distrib[last_two_digits] == max(digits_distrib.values())]\n",
    "maxes"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "838d2357",
   "metadata": {},
   "source": [
    "Visual representation (ordering the dict items for the sake of readability)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "id": "9c1e763c",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2023-03-15T13:30:04.073633Z",
     "start_time": "2023-03-15T13:30:03.745983Z"
    },
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x720 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(10,10))\n",
    "plt.scatter((sdd := sorted(digits_distrib)), tuple(map(digits_distrib.get, sdd)));"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.10.0"
  },
  "toc": {
   "base_numbering": 1,
   "nav_menu": {},
   "number_sections": true,
   "sideBar": true,
   "skip_h1_title": false,
   "title_cell": "Table of Contents",
   "title_sidebar": "Contents",
   "toc_cell": false,
   "toc_position": {},
   "toc_section_display": true,
   "toc_window_display": false
  },
  "varInspector": {
   "cols": {
    "lenName": 16,
    "lenType": 16,
    "lenVar": 40
   },
   "kernels_config": {
    "python": {
     "delete_cmd_postfix": "",
     "delete_cmd_prefix": "del ",
     "library": "var_list.py",
     "varRefreshCmd": "print(var_dic_list())"
    },
    "r": {
     "delete_cmd_postfix": ") ",
     "delete_cmd_prefix": "rm(",
     "library": "var_list.r",
     "varRefreshCmd": "cat(var_dic_list()) "
    }
   },
   "types_to_exclude": [
    "module",
    "function",
    "builtin_function_or_method",
    "instance",
    "_Feature"
   ],
   "window_display": false
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}