{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Problem 15:\n",
    " \n",
    "### [Euler Project #15](https://projecteuler.net/problem=15)\n",
    " \n",
    "### Starting in the top left corner of a 2×2 grid, and only being able to move to the right and down, there are exactly 6 routes to the bottom right corner.\n",
    "\n",
    "### How many such routes are there through a 20×20 grid?\n",
    "\n",
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Let's try to map the example cases, by assigning a binary value to RIGHT and DOWN moves.\n",
    "\n",
    " 0 = R <br>\n",
    " 1 = D\n",
    "\n",
    " 0 0 1 1 <br>\n",
    " 0 1 0 1 <br>\n",
    " 0 1 1 0 <br>\n",
    " 1 0 0 1 <br>\n",
    " 1 0 1 0 <br>\n",
    " 1 1 0 0 <br>\n",
    "\n",
    "### A couple things that shake out of this,\n",
    "1. The total length of a sequence of moves is the equal to Length + Width of the grid.<br>\n",
    "    1. So, for a 20x20, each sequence will be 40 moves in length.\n",
    "    1. That could 2**40 possible combinations, most of which will not be valid.\n",
    "2. Each unique sequence of moves has a twin which is a mirror across the diagonal.\n",
    "3. Each valid sequence has an equal number of RIGHT and DOWN moves.\n",
    "\n",
    "### I'm sure there is some combinatorial mathematics that describes how to do this analytically. That will require some research, so I'll save this problem for another day.\n",
    "\n",
    "---"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "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": "3.8.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}