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@@ -8,23 +8,33 @@
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"\n",
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"# RaceTrack\n",
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"\n",
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"From 538's [The Riddler](https://fivethirtyeight.com/features/can-you-zoom-around-the-race-track/) on 6 August 2021:\n",
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"From 538's [The Riddler](https://fivethirtyeight.com/features/can-you-zoom-around-the-race-track/) on 6 August 2021 (slightly edited):\n",
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"\n",
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"> The game of [RaceTrack](https://link.springer.com/chapter/10.1007/978-3-642-13122-6_26) was published by recreational mathematician Martin Gardner in 1973. There have been a few modifications and even some neat [digital versions](https://harmmade.com/vectorracer/#) since then, and it’s high time we had a race here on The Riddler. So without further ado, welcome to the Riddler-opolis 500!\n",
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"> The game of [RaceTrack](https://link.springer.com/chapter/10.1007/978-3-642-13122-6_26) was published by recreational mathematician [Martin Gardner](http://www.martin-gardner.org/) in [1973](https://www.scientificamerican.com/article/mathematical-games-1973-01/). There have been some neat [digital versions](https://harmmade.com/vectorracer/#) since then, and it’s high time we had a race here on The Riddler: the Riddler-opolis 500!\n",
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">\n",
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"> You begin at the midpoint of the starting line at the bottom (the thicker border), and your goal is to circumnavigate the race track shown below in a single counterclockwise loop. You’ll be moving from point to point on the grid, without ever venturing into the gray surrounding square or the central circle. If the center is at the origin (with radius 3), the starting point in blue is at (0, -5). The starting line is between (0, -3) and (0, -7).\n",
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"> You begin at the blue point on the starting line below the circle (diagram shown below) with a velocity vector of zero, and your goal is to circumnavigate the track in a counterclockwise loop. You’ll be moving from point to point on the grid, without ever venturing into the outer wall or the central circle. (Moving tangent to the circle is allowed, as is being on a grid point along a wall.)\n",
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">\n",
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"> For your first move, you can choose among nine grid points surrounding and including the starting position. From there, you have inertia. So for your second move, you again have nine possible destinations, but they are determined by your current velocity vector. That is, you can maintain your current direction and speed, or you can alter your destination by one point in any direction (horizontally, vertically or diagonally). For example, if your first move was up and to the right, then your possible second moves are shown below, although two of them will cause you to crash into the wall.\n",
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"> For each move you have a choice of nine possible destinations, corresponding to the 3x3 grid that surrounds the point formed by adding your current velocity vector to your current position. That is, you can maintain your current velocity, or you can alter your velocity (and hence your destination) by one grid point in any direction (horizontally, vertically or diagonally). For example, if your first move was up and to the right, then your possible second moves are shown below, although two of them will cause you to crash into the central circle.\n",
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">\n",
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"> \n",
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">\n",
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"> How quickly can you navigate the track? At no time can your path venture into the wall. (Being tangent to the wall is allowed, as is being on a grid point along a wall.)\n",
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"> How quickly can you navigate the track? \n",
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">\n",
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"\n",
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"\n",
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"# Defining Points and the Track\n",
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"\n",
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"First some imports: "
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"I will define the following:\n",
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"- `Point`: a point in the 2-D plane, implemented as a complex number. `X(p)` and `Y(p)` give coordinates\n",
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"- `Path`: a list of points representing the car's moves\n",
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"- `track`: all the legal grid points\n",
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"- `start`: the single starting point\n",
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"- `finish`: a Path defining the finish line\n",
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"- `zero`: a vector (your starting velocity vector)\n",
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"- `radius`: the radius of the central circle, whose center is defined as the origin\n",
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"- `deltas`: the nine possible changes to the car's velocity on each move\n",
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"- `X(point)`, `Y(point)`: the x- and y-coordinates of a point, respectively\n",
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"- `velocity(path)`: the car's velocity when it has just made the last move in a path"
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]
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},
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{
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@@ -34,33 +44,23 @@
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"outputs": [],
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"source": [
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"import matplotlib.pyplot as plt\n",
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"from typing import List, Tuple, Iterator\n",
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"import math"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Now define a `Point` in the 2-D plane as a complex number (with functions to extract the `X` and `Y` components), and a `Path` as a list of points. Then define some global variables: the points in the `track`; the `finish` line; the `start` point; the `zero` vector; and the allowable `deltas` (the nine possible changes on each move)."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"metadata": {},
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"outputs": [],
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"source": [
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"Point = Vector = complex\n",
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"Path = List[Point]\n",
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"from typing import List, Tuple\n",
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"\n",
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"Point = Vector = complex\n",
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"Path = List[Point]\n",
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"track = {Point(x, y) for x in range(-7, 8) for y in range(-7, 8) if abs(Point(x, y)) >= 3}\n",
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"start = Point(0, -5)\n",
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"finish = [Point(0, -3), Point(0, -7)]\n",
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"zero = Vector(0, 0)\n",
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"radius = 3\n",
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"deltas = [Vector(x, y) for y in (1, 0, -1) for x in (-1, 0, 1)]\n",
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"\n",
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"def X(point) -> float: return point.real\n",
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"def Y(point) -> float: return point.imag\n",
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"\n",
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"track = {Point(x, y) for x in range(-7, 8) for y in range(-7, 8) if abs(Point(x, y)) >= 3}\n",
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"finish = [Point(0, -3), Point(0, -7)]\n",
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"start = sum(finish) / 2 # midpoint of finish line\n",
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"zero = Vector(0, 0)\n",
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"deltas = [Vector(x, y) for y in (1, 0, -1) for x in (-1, 0, 1) ]"
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"def velocity(path) -> Vector:\n",
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" \"\"\"Velocity of the car at the end of the path.\"\"\"\n",
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" return zero if len(path) == 1 else path[-1] - path[-2] "
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]
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},
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{
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@@ -69,16 +69,16 @@
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"source": [
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"# Visualization\n",
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"\n",
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"I think it will be helpful to introduce a way to visualize the track and the path(s) on it:"
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"I think it will be helpful now to introduce a way to visualize the racetrack and the path(s) on it:"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"execution_count": 2,
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"metadata": {},
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"outputs": [],
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"source": [
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"def plot(paths=(), track=track, start=start, finish=finish):\n",
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"def plot(paths=()):\n",
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" \"\"\"Plot the track and any paths, along with the start and finish line.\"\"\"\n",
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" fig, ax = plt.subplots()\n",
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" plt.xticks([]); plt.yticks([])\n",
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@@ -87,29 +87,30 @@
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" ax.plot(*XY(track), 'k,' ) # grid points\n",
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" for path in paths:\n",
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" ax.plot(*XY(path), 'o-') # paths\n",
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" ax.plot(*XY([start]), 'bD') # start\n",
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" ax.plot(*XY(finish), 'b:') # finish\n",
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" lengths = {len(path) - 1 for path in paths}\n",
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" plt.title(f'{len(paths)} paths, lengths {lengths}')\n",
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"\n",
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"def XY(points): return [X(p) for p in points], [Y(p) for p in points]"
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" ax.plot(*XY([start]), 'bD') # start point\n",
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" ax.plot(*XY(finish), 'b:') # finish line\n",
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" plt.title(f'{len(paths)} paths of {set(len(path) - 1 for path in paths)} moves')\n",
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" \n",
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"def XY(points) -> Tuple[List[float], List[float]]: \n",
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" \"\"\"A tuple of: (the x-coordinates of points, the y-coordinates of points).\"\"\"\n",
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" return [X(p) for p in points], [Y(p) for p in points]"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Here is a single path consisting of the first move from the start point:"
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"Below is my replication of the diagram above: nine paths for the second move (two of which crash)."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 4,
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"execution_count": 3,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"image/png": 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truncated
|
||||
"image/png": "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 truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
@@ -120,14 +121,53 @@
|
||||
],
|
||||
"source": [
|
||||
"path = [start, Point(1, -4)]\n",
|
||||
"plot([path])"
|
||||
"paths = [path + [Point(path[-1] + velocity(path) + d)] \n",
|
||||
" for d in deltas]\n",
|
||||
"plot(paths)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"Here are nine paths for the second move (two of which crash into the circle in the center):"
|
||||
"# Navigating Regions\n",
|
||||
"\n",
|
||||
"The goal of the race is to circumnavigate the center circle. I can describe that in a way that's generalizable to different tracks by saying that `is_complete(path)` is true if the path enters *regions* of points in a prescribed order. For our track a path is complete if it travels from the fourth [quadrant](https://mathworld.wolfram.com/Quadrant.html) to the first, second, third, and back to the fourth. A *region* is defined by a predicate that is true of any point in the region."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 4,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"def Q1(p: Point) -> bool: \"upper right quadrant\"; return X(p) >= 0 and Y(p) >= 0\n",
|
||||
"def Q2(p: Point) -> bool: \"upper left quadrant\"; return X(p) < 0 and Y(p) >= 0\n",
|
||||
"def Q3(p: Point) -> bool: \"lower left quadrant\"; return X(p) < 0 and Y(p) < 0 \n",
|
||||
"def Q4(p: Point) -> bool: \"lower right quadrant\"; return X(p) >= 0 and Y(p) < 0\n",
|
||||
"\n",
|
||||
"circumnavigate = (Q4, Q1, Q2, Q3, Q4)\n",
|
||||
" \n",
|
||||
"def is_complete(path, regions=circumnavigate) -> bool:\n",
|
||||
" \"\"\"Does the path go through all the regions in the prescribed order?\"\"\"\n",
|
||||
" current = 0\n",
|
||||
" for p in path:\n",
|
||||
" if regions[current](p):\n",
|
||||
" current += 1\n",
|
||||
" if current == len(regions):\n",
|
||||
" return True\n",
|
||||
" return False"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"# Possible and Legal Moves\n",
|
||||
"\n",
|
||||
"I'm thinking about how to solve for our specific track, and also how to make the code general enough that it could handle other tracks. I'll do that by splitting up responsibility:\n",
|
||||
"- `possible_moves(path)` is general; it returns up to 9 points that are on the track and have the right velocity. \n",
|
||||
"- `legal_moves(path)` is specific; it won't cross into the center circle, and it won't pass from the fourth quadrant backwards into the third (while that is not technically illegal, it was causing problems, and no fast path should do that)."
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -136,19 +176,120 @@
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"def velocity(path) -> Vector:\n",
|
||||
" \"\"\"Velocity of the car at the end of the path.\"\"\"\n",
|
||||
" return zero if len(path) == 1 else path[-1] - path[-2] "
|
||||
"def possible_moves(path) -> List[Point]:\n",
|
||||
" \"\"\"All moves with the right velocity that end up on the track.\"\"\"\n",
|
||||
" p, v = path[-1], velocity(path)\n",
|
||||
" return {(p + v + Δ) for Δ in deltas} & track\n",
|
||||
" \n",
|
||||
"def legal_moves(path) -> List[Point]:\n",
|
||||
" \"\"\"What points can be reached in one step from this path?\"\"\"\n",
|
||||
" p = path[-1]\n",
|
||||
" return [p2 for p2 in possible_moves(path)\n",
|
||||
" if not (Q4(p) and Q3(p2)) # Don't go backwards over the finish line\n",
|
||||
" and not intersects_circle(p, p2)]\n",
|
||||
"\n",
|
||||
"def intersects_circle(p: Point, p2: Point, radius=radius, N=20):\n",
|
||||
" \"\"\"Are any of N+1 points on the line segment [p, p2] inside the circle's radius?\"\"\"\n",
|
||||
" linepoints = (p + (i / N) * (p2 - p) for i in range(N + 1))\n",
|
||||
" return any(abs(q) < radius for q in linepoints)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"The math for solving for the intersection of a line segment and a circle looked [complicated](https://stackoverflow.com/questions/6091728/line-segment-circle-intersection), so instead I just define N+1 points along the line from point *p* to point *p2* and ask if any of those points is within the radius of the circle. If N is too small this could miss an intersection, but on the small grid for this race track, it shouldn't be a problem."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"# A Frontier of Possible Paths\n",
|
||||
"\n",
|
||||
"Now I want to expand paths by adding legal moves to the end until I find complete path(s). I could keep all my paths in a `list`, but there will be an exponential number of them (up to 9<sup><i>n<i/></sup> paths of length *n* and it looks like *n* will be around 10 to 13).\n",
|
||||
"\n",
|
||||
"Instead I will keep track of a `frontier`, which I define as a mapping of `{(endpoint, velocity): path}`. The idea is that there may be many paths that arrive at the some endpoint with the same velocity in the same number of moves, but if so, the best continuation of all of them will be the same, so I only need to keep one of them. "
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 6,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"def expand(frontier) -> dict:\n",
|
||||
" \"\"\"The {(endpoint, velocity): path} mapping extended one move in all legal ways.\"\"\"\n",
|
||||
" return {(p, velocity([path[-1], p])): path + [p] \n",
|
||||
" for path in frontier.values() \n",
|
||||
" for p in legal_moves(path)}"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"I will show how the frontier expands over the first few steps:"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 7,
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
"output_type": "display_data"
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
"output_type": "display_data"
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
"output_type": "display_data"
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
"output_type": "display_data"
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
"output_type": "display_data"
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": "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 truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
@@ -158,46 +299,23 @@
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"plot([path + [Point(path[-1] + velocity(path) + d)] for d in deltas])"
|
||||
"frontier = {(start, zero): [start]}\n",
|
||||
"\n",
|
||||
"for i in range(6):\n",
|
||||
" frontier = expand(frontier)\n",
|
||||
" plot(frontier.values())\n",
|
||||
" plt.show()"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"# Complete Paths and Legal Moves\n",
|
||||
"(Note that the growth is much slower then 9<sup><i>n</i></sup>.)\n",
|
||||
"\n",
|
||||
"For our circle-in-the-middle track, I can say that a path `is_complete` if the path goes from the fourth quadrant to the first, second, third, and back to the fourth:"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 7,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"def Q1(p) -> bool: return X(p) >= 0 and Y(p) >= 0 # upper right\n",
|
||||
"def Q2(p) -> bool: return X(p) < 0 and Y(p) >= 0 # upper left\n",
|
||||
"def Q3(p) -> bool: return X(p) < 0 and Y(p) < 0 # lower left \n",
|
||||
"def Q4(p) -> bool: return X(p) >= 0 and Y(p) < 0 # lower right\n",
|
||||
" \n",
|
||||
"def is_complete(path, regions=(Q4, Q1, Q2, Q3, Q4)) -> bool:\n",
|
||||
" \"\"\"Does the path complete a tour of the regions, in order?\"\"\"\n",
|
||||
" for p in path:\n",
|
||||
" if regions[0](p):\n",
|
||||
" regions = regions[1:]\n",
|
||||
" return not regions"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"Now I'll define the `legal_moves` that can be made. Given a path, `legal_moves(path)` yields all points that can follow on to the end of the path. Each point must be a point on the track (not inside the circle), the velocity vector can differ from the previous velocity vector only by an allowable delta, and the line from the previous final point on the path to the new point must not intersect the circle. \n",
|
||||
"# Searching for Fastest Complete Paths\n",
|
||||
"\n",
|
||||
"I have inserted a hack here: to make sure that paths start off going counterclockwise, for the first two steps I insist that the deltas be either up or to the right or zero.\n",
|
||||
"\n",
|
||||
"I got the calculation for `intersects_circle` from several posts on math overflow."
|
||||
"Now all I have to do to solve the problem is repeatedly expand the frontier until some path(s) are complete. The first completed paths are guaranteed to be the fastest (i.e., have the fewest moves), because we are expanding one move at a time. In the end, I return a list of all the completed paths in the frontier, but there may be many other paths that are not returned, because I only kept one path for each `(endpoint, velocity)` pair along the way. "
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -205,42 +323,6 @@
|
||||
"execution_count": 8,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"def legal_moves(path, track=track, deltas=deltas) -> Iterator[Point]:\n",
|
||||
" \"\"\"What points can be reached in one step from this path?\"\"\"\n",
|
||||
" if len(path) <= 2: # Make sure we start counterclockwwise\n",
|
||||
" deltas = [Point(x, y) for y in (0, 1) for x in (0, 1)]\n",
|
||||
" p, v = path[-1], velocity(path)\n",
|
||||
" for d in deltas:\n",
|
||||
" new_p = p + v + d\n",
|
||||
" if new_p in track and not intersects_circle(p, new_p, 3, zero):\n",
|
||||
" yield new_p\n",
|
||||
" \n",
|
||||
"def intersects_circle(p, q, r, c):\n",
|
||||
" \"\"\"Does a line from p to q intersect a circle of radius r at center c?\"\"\"\n",
|
||||
" if p == q:\n",
|
||||
" return abs(c - p) < r\n",
|
||||
" norm = (q - p) / abs(q - p)\n",
|
||||
" v = abs(X(norm) * Y(c - p) - Y(norm) * X(c - p)) # Norm of the 2D cross-product\n",
|
||||
" return (v < r)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"# Finding a Fastest Path\n",
|
||||
"\n",
|
||||
"I can use a breadth-first search, adding each legal move to each path until I find path(s) that are complete. \n",
|
||||
"\n",
|
||||
"I keep track of a `frontier`, which is a mapping of `{(point, velocity): path}` for all paths of the longest length explored so far. The reason for this is that there may be an exponential number of paths that arrive at the some position with the same velocity in the same number of moves. But none of them will have any advantage in reaching the finish line ahead of the others, so I only need to keep one of them. In the end, I return the completed paths I have found, but there may be many other paths that are not returned, because I only kept one for each `(point, velocity)` pair along the way. "
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 9,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"def search(path=[start]) -> Path:\n",
|
||||
" \"\"\"Find the shortest possible complete path(s).\"\"\"\n",
|
||||
@@ -248,24 +330,18 @@
|
||||
" completed = []\n",
|
||||
" while not completed:\n",
|
||||
" frontier = expand(frontier)\n",
|
||||
" completed = [path for path in frontier.values() if is_complete(path)]\n",
|
||||
" return completed\n",
|
||||
"\n",
|
||||
"def expand(frontier) -> dict:\n",
|
||||
" \"\"\"The {(point, velocity): path} mapping extended one step in all legal ways.\"\"\"\n",
|
||||
" return {(p, velocity(path + [p])): path + [p] \n",
|
||||
" for path in frontier.values() \n",
|
||||
" for p in legal_moves(path)}"
|
||||
" completed = list(filter(is_complete, frontier.values()))\n",
|
||||
" return completed"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 10,
|
||||
"execution_count": 9,
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"image/png": "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 truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
@@ -282,36 +358,78 @@
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"We see that the fastest path takes 12 moves, and there are several ways to get there.\n",
|
||||
"\n",
|
||||
"# Submitting your Answer\n",
|
||||
"\n",
|
||||
"I'll need some help to put my answers into the form requested by The Riddler:\n",
|
||||
"I'll need a bit of work to put my answers into the form requested by The Riddler:\n",
|
||||
"\n",
|
||||
"> Finally, submitting your answer can be tricky work. Please be sure to submit both your total time, as well as your sequence of moves. Each move should be assigned a digit from 1 through 9, corresponding to the nine possible destinations of the move:\n",
|
||||
">\n",
|
||||
"> 1 2 3\n",
|
||||
"> 4 5 6\n",
|
||||
"> 7 8 9"
|
||||
"> 7 8 9\n",
|
||||
"\n",
|
||||
"I will define two functions to translate back and forth between my path and The Riddler's list of digits: "
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 10,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"Digits = List[int]\n",
|
||||
"\n",
|
||||
"def digits_from_path(path) -> Digits:\n",
|
||||
" \"\"\"The digits that the path represents.\"\"\"\n",
|
||||
" return [deltas.index(velocity(path[:i + 1]) - velocity(path[:i])) + 1\n",
|
||||
" for i in range(1, len(path))]\n",
|
||||
"\n",
|
||||
"def path_from_digits(digits, start=start) -> Path:\n",
|
||||
" \"\"\"The path that the digits represent.\"\"\"\n",
|
||||
" velo = zero # Velocity\n",
|
||||
" path = [start]\n",
|
||||
" for d in digits:\n",
|
||||
" velo += deltas[d - 1]\n",
|
||||
" path.append(velo + path[-1])\n",
|
||||
" return path"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 11,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"outputs": [
|
||||
{
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"[3, 3, 1, 7, 7, 8, 4, 8, 9, 9, 6, 3]\n",
|
||||
"[3, 3, 1, 7, 7, 5, 7, 8, 9, 9, 6, 6]\n",
|
||||
"[3, 3, 1, 7, 7, 5, 7, 8, 9, 9, 6, 3]\n",
|
||||
"[3, 3, 1, 7, 7, 8, 4, 8, 9, 9, 3, 3]\n",
|
||||
"[3, 3, 1, 7, 7, 5, 7, 8, 9, 9, 3, 9]\n",
|
||||
"[3, 3, 1, 7, 7, 5, 7, 8, 9, 9, 3, 6]\n",
|
||||
"[3, 3, 1, 7, 7, 5, 7, 8, 9, 9, 3, 3]\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"def encode_path(path) -> List[int]:\n",
|
||||
" \"\"\"The digits representing the moves (as coded changes in acceleration).\"\"\"\n",
|
||||
" return [deltas.index(velocity(path[:i + 1]) - velocity(path[:i])) + 1\n",
|
||||
" for i in range(1, len(path))]\n",
|
||||
"for path in search([start]):\n",
|
||||
" digits = digits_from_path(path)\n",
|
||||
" assert path == path_from_digits(digits_from_path(path)) # Inverses\n",
|
||||
" assert digits == digits_from_path(path_from_digits(digits)) # Inverses\n",
|
||||
" print(digits)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"# Alternative Paths\n",
|
||||
"\n",
|
||||
"def decode_path(digits, start=start) -> Path:\n",
|
||||
" \"\"\"The points that the digits represent.\"\"\"\n",
|
||||
" velocity = zero\n",
|
||||
" path = [start]\n",
|
||||
" for d in digits:\n",
|
||||
" velocity += deltas[d - 1]\n",
|
||||
" path.append(velocity + path[-1])\n",
|
||||
" return path"
|
||||
"Here we search from several different opening move sequences and get different results, but all of 12 moves:"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -320,38 +438,48 @@
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"[3, 3, 1, 7, 7, 5, 7, 8, 9, 9, 3, 6]\n",
|
||||
"[3, 3, 1, 7, 7, 5, 7, 8, 9, 9, 3, 9]\n",
|
||||
"[3, 3, 1, 7, 7, 5, 7, 8, 9, 9, 3, 3]\n",
|
||||
"[3, 3, 1, 7, 7, 5, 7, 8, 9, 9, 6, 3]\n",
|
||||
"[3, 3, 1, 7, 7, 5, 7, 8, 9, 9, 6, 6]\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"for path in search():\n",
|
||||
" print(encode_path(path))\n",
|
||||
" assert path == decode_path(encode_path(path)) "
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"To get an idea of some of the paths that were discarded by the `frontier`, here we start with a path that is tighter around the circle, and search for completions:"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 13,
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
"output_type": "display_data"
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
"output_type": "display_data"
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
"output_type": "display_data"
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
"output_type": "display_data"
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": "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 truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
@@ -361,7 +489,80 @@
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"plot(search(decode_path([3, 5, 2, 4, 4])))"
|
||||
"for opening in ([3, 3], [3, 5], [3, 6], [6, 2], [6, 3]):\n",
|
||||
" plot(search(path_from_digits(opening)))"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"# Tests\n",
|
||||
"\n",
|
||||
"The code that I have the least confidence in is `intersects_circle`. Here are some tests and visualizations for it:"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 13,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"def test_intersects(expected: bool, segments: List[Path]):\n",
|
||||
" \"\"\"Test if segments do the expected thing with `intersects_circle`.\"\"\"\n",
|
||||
" plot(segments)\n",
|
||||
" for p, q in segments:\n",
|
||||
" assert intersects_circle(p, q) is expected"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 14,
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
"output_type": "display_data"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"test_intersects(True, [ # Line segments that intersect the circle\n",
|
||||
" [Point(-2, 3), Point(5, 2)], \n",
|
||||
" [Point(-1, 5), Point(4, 0)], \n",
|
||||
" [Point(2, 4), Point(2, -4)],\n",
|
||||
" [Point(-3, -1),Point(1, -4)],\n",
|
||||
" [Point(-6, 0), Point(7, 7)]])"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 15,
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": "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 truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
"output_type": "display_data"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"test_intersects(False, [ # Line segments that do not intersect the circle\n",
|
||||
" [Point(-2, 3), Point(5, 3)], \n",
|
||||
" [Point(-1, 5), Point(4, 1)], \n",
|
||||
" [Point(3, 4), Point(3, -4)],\n",
|
||||
" [Point(-3, -2),Point(1, -4)],\n",
|
||||
" [Point(-7, 0), Point(7, 7)]])"
|
||||
]
|
||||
}
|
||||
],
|
||||
|
||||
Reference in new issue
Block a user