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@@ -12,9 +12,9 @@
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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](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: the Riddler-opolis 500!\n",
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">\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 crashing 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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"> 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 in straight line segments from point to point on the grid, without ever crashing 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 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 accelerating one grid point in any direction (horizontally, vertically or diagonally). For example, if your first move was up and to the right, then your nine possible second moves are shown below, although two of them will cause you to crash into the central circle.\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 accelerating one grid point in any direction (horizontally, vertically or diagonally). For example, if your first move was up and to the right, then your nine possible second moves are shown below, although two of them will cause you to crash into the central circle and one of them is standing still.\n",
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">\n",
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"> \n",
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">\n",
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@@ -31,9 +31,8 @@
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"- `start`: the single starting point.\n",
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"- `zero`: a zero 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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"- `accelerations`: the nine possible acceleration vectors, to changes the car's velocity.\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 current velocity when it has just made the last move in a path."
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"- `accelerations`: the nine possible acceleration vectors (to change the car's velocity), mapped to the digit number that The Riddler uses to identify them.\n"
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]
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},
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{
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@@ -43,22 +42,22 @@
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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\n",
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"from typing import List, Tuple, Dict, Set\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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"track = {Point(x, y) for x in range(-7, 8) for y in range(-7, 8) \n",
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" if abs(Point(x, y)) >= 3}\n",
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"start = Point(0, -5)\n",
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"zero = Vector(0, 0)\n",
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"radius = 3\n",
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"accelerations = [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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"def X(point) -> float: \"x-coordinate\"; return point.real\n",
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"def Y(point) -> float: \"y-coordinate\"; return point.imag\n",
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"\n",
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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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"accelerations = {Vector(-1, 1): 1, Vector(0, 1): 2, Vector(1, 1): 3, \n",
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" Vector(-1, 0): 4, Vector(0, 0): 5, Vector(1, 0): 6, \n",
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" Vector(-1, -1):7, Vector(0, -1):8, Vector(1, -1):9}"
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]
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},
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{
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@@ -67,7 +66,7 @@
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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 now to introduce a way to visualize the racetrack and the path(s) on it:"
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"I think it will be helpful to be able to visualize paths on the racetrack, so I'll define a function to do that now, even before attempting to find solutions:"
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]
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},
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{
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@@ -81,7 +80,7 @@
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" fig, ax = plt.subplots()\n",
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" plt.xticks([]); plt.yticks([])\n",
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" ax.set_aspect('equal');\n",
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" ax.add_artist(plt.Circle((0, 0), 3, alpha=0.1, color='k', ec='w'))\n",
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" ax.add_artist(plt.Circle((0, 0), 3, alpha=1/3, color='k', ec='w'))\n",
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" ax.plot(*XY(track), 'k,' ) # grid points\n",
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" ax.plot(*XY([start]), 'ks', ms=10) # start point\n",
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" ax.plot(*XY(finish), 'k:') # finish line\n",
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@@ -98,7 +97,11 @@
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Below is my replication of The Riddler's diagram above: the first move goes up and to the right, and there are nine paths for the second move (two of which crash into the central circle)."
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"Below is my recreation of The Riddler's diagram of the first two moves. Here and throughout I'll use the convention:\n",
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"- `p` is a point indicating a move\n",
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" - In this example, `p` is up and right from the start to `Point(1, -4)`, so the velocity is `Vector(1, 1)`.\n",
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"- `p1` is where the next point would be if the velocity remains constant.\n",
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"- `p2` is where the next point actually is with acceleration applied to change the velocity."
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]
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},
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{
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@@ -108,7 +111,7 @@
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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
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||||
"image/png": "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 truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
@@ -118,10 +121,11 @@
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"path1 = [start, Point(1, -4)]\n",
|
||||
"paths = [path1 + [Point(path1[-1] + velocity(path1) + a)] \n",
|
||||
" for a in accelerations]\n",
|
||||
"plot(paths)"
|
||||
"p = Point(1, -4)\n",
|
||||
"p1 = p + Vector(1, 1)\n",
|
||||
"p2s = [p1 + a for a in accelerations]\n",
|
||||
"\n",
|
||||
"plot([[start, p, p2] for p2 in p2s])"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -130,9 +134,12 @@
|
||||
"source": [
|
||||
"# Circumnavigating the Track\n",
|
||||
"\n",
|
||||
"The goal of the race is to circumnavigate the center circle. I can test for that by seeing if the path goes from the fourth [quadrant](https://mathworld.wolfram.com/Quadrant.html) to the first, second, third, and back to the fourth. The function `is_complete(path)` does that.\n",
|
||||
"The goal of the race is to circumnavigate the center circle. I can test for that by checking if the path goes from the fourth [quadrant](https://mathworld.wolfram.com/Quadrant.html) to the first, second, third, and back to the fourth. The predicate `is_complete(path)` does that.\n",
|
||||
"\n",
|
||||
"If you want to generalize this to other race tracks, you can change the definition of `quadrant` (there could be more than 4 quadrants/regions, and they could have any shape), and change the `regions` in the parameter list of `is_complete`."
|
||||
"<center><div><img src=\"https://upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Cartesian-coordinate-system-with-quadrant.svg/242px-Cartesian-coordinate-system-with-quadrant.svg.png\" width=120></div></center>\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"If you wanted to generalize this to other race tracks, you could change the definition of `quadrant` (there could be more than 4 \"quadrants,\" and they could have any shape), and change the `goals` in `is_complete`."
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -141,13 +148,13 @@
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"def is_complete(path, regions=(4, 1, 2, 3, 4)) -> bool:\n",
|
||||
" \"\"\"Does the path go through all the regions in the prescribed order?\"\"\"\n",
|
||||
" i = 0\n",
|
||||
"def is_complete(path, goals=(4, 1, 2, 3, 4)) -> bool:\n",
|
||||
" \"\"\"Does the path go through all the goal quadrants in the prescribed order?\"\"\"\n",
|
||||
" i = 0 # Index of next goal to be achieved\n",
|
||||
" for p in path:\n",
|
||||
" if quadrant(p) == regions[i]:\n",
|
||||
" i += 1\n",
|
||||
" if i == len(regions):\n",
|
||||
" if quadrant(p) == goals[i]:\n",
|
||||
" i += 1 # Start looking for the next goal\n",
|
||||
" if i == len(goals):\n",
|
||||
" return True\n",
|
||||
" return False\n",
|
||||
"\n",
|
||||
@@ -167,12 +174,10 @@
|
||||
"\n",
|
||||
"I'm considering how to solve for our specific central-circle-track, and also make the code general enough that it can handle other tracks. I'll do that by splitting up responsibility for determining legal moves:\n",
|
||||
"- `all_moves(path)` is general; it considers points with all 9 accelerations and returns the points that are on the track.\n",
|
||||
"- `circle_track_moves(path)` is specific; it limits moves in the following ways:\n",
|
||||
" - It won't let a move intersect the center circle (even if the start and end points of a move are outside the circle).\n",
|
||||
"- `circular_track_moves(path)` is specific; it limits moves in the following ways:\n",
|
||||
" - It won't make moves that stay in one place (they are legal, but wasteful and not needed on this track).\n",
|
||||
" - It won't pass from the fourth quadrant backwards into the third (such moves are legal, but wasteful).\n",
|
||||
" \n",
|
||||
"In the code below `p` is the previous point on a path, `p1` is where the next point would be if the velocity remained constant, and `p2` is the point we arrive at when acceleration is applied."
|
||||
" - It won't pass from the fourth quadrant backwards over the finish line into the third (legal, but wasteful).\n",
|
||||
" - It won't let a move intersect the central circle (even if the start and end points of the move are outside the circle).\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -181,56 +186,69 @@
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"def all_moves(path) -> List[Point]:\n",
|
||||
"def all_moves(path, track=track) -> Set[Point]:\n",
|
||||
" \"\"\"Moves with any allowable acceleration that end up on the track.\"\"\"\n",
|
||||
" p1: Point = path[-1] + velocity(path)\n",
|
||||
" p1 = path[-1] + velocity(path)\n",
|
||||
" return {(p1 + a) for a in accelerations} & track\n",
|
||||
" \n",
|
||||
"def circle_track_moves(path) -> List[Point]:\n",
|
||||
" \"\"\"Reasonable moves on the circle track.\"\"\"\n",
|
||||
"def circular_track_moves(path) -> Set[Point]:\n",
|
||||
" \"\"\"Reasonable moves on the circular track.\"\"\"\n",
|
||||
" p = path[-1]\n",
|
||||
" return [p2 for p2 in all_moves(path)\n",
|
||||
" return {p2 for p2 in all_moves(path)\n",
|
||||
" if p2 != p\n",
|
||||
" and not (quadrant(p) == 4 and quadrant(p2) == 3) # Don't go backwards over finish line\n",
|
||||
" and not intersects_circle(p, p2)]\n",
|
||||
" and not (quadrant(p) == 4 and quadrant(p2) == 3) # Don't go backwards\n",
|
||||
" and not intersects_circle(p, p2)}\n",
|
||||
"\n",
|
||||
"def intersects_circle(p: Point, p2: Point, center=zero, radius=radius, N=20) -> bool:\n",
|
||||
" \"\"\"Does a line from p to p2 intersect a circle with given center and radius?\"\"\"\n",
|
||||
" return any(abs(q - center) < radius for q in waypoints(p, p2, N))\n",
|
||||
"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] "
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"The math for solving for the intersection of a line segment and a circle looks [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 check if any of those points is within the radius of the circle (which by definition is centered on the origin). If N is too small this could miss an intersection, but that shouldn't be a problem for our uses."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 6,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"def intersects_circle(p: Point, p2: Point, radius=radius) -> bool:\n",
|
||||
" \"\"\"Does any point w on the line from p to p2 fall within radius of center?\"\"\"\n",
|
||||
" return any(abs(w) < radius for w in waypoints(p, p2))\n",
|
||||
"\n",
|
||||
"def waypoints(p, p2, N) -> List[Point]:\n",
|
||||
"def waypoints(p, p2, N=20) -> List[Point]:\n",
|
||||
" \"\"\"All the points that are i/N of the way from p to p2\"\"\"\n",
|
||||
" return [p + (i / N) * (p2 - p)\n",
|
||||
" for i in range(N + 1)]"
|
||||
]
|
||||
},
|
||||
{
|
||||
"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",
|
||||
"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 I estimate *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. \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. I only need to keep one of them, because they can all be continued in exactly the same ways.\n",
|
||||
" \n",
|
||||
"*Note:* Before I added the limitation that moves can't pass backwards from the fourth quadrant to the third, I had a bug: sometimes the frontier would capture a `(endpoint, velocity)` pair for a path that was in the third quadrant but hadn't circumnavigated and was replacing a path to the same point that had circumnavigated. This meant that I was discarding some valid paths."
|
||||
"*Note:* Before I added the limitation that moves can't pass backwards from the fourth quadrant to the third, I had **a bug**: sometimes the frontier would capture a `(endpoint, velocity)` pair for a path that was in the third quadrant but hadn't circumnavigated, and was replacing a path to the same point that had circumnavigated. That meant I was discarding some valid paths."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 6,
|
||||
"execution_count": 7,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"def expand(frontier, legal_moves=circle_track_moves) -> dict:\n",
|
||||
"Frontier = Dict[Tuple[Point, Vector], Path]\n",
|
||||
"\n",
|
||||
"def expand(frontier, legal_moves=circular_track_moves) -> Frontier:\n",
|
||||
" \"\"\"The {(endpoint, velocity): path} frontier extended one move in all legal ways.\"\"\"\n",
|
||||
" return {(p, velocity([path[-1], p])): path + [p] \n",
|
||||
" for path in frontier.values() \n",
|
||||
@@ -243,16 +261,16 @@
|
||||
"source": [
|
||||
"# Searching for Fastest Complete Paths\n",
|
||||
"\n",
|
||||
"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 found 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 the frontier only keeps one path for each `(endpoint, velocity)` pair along the way. "
|
||||
"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 found 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 the frontier only keeps one path for each `(endpoint, velocity)` pair, and there are only so many ways to cross the finish line. "
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 7,
|
||||
"execution_count": 8,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"def search(path=[start], legal_moves=circle_track_moves) -> List[Path]:\n",
|
||||
"def search(path=[start], legal_moves=circular_track_moves) -> List[Path]:\n",
|
||||
" \"\"\"Find the shortest possible complete path(s).\"\"\"\n",
|
||||
" frontier = {(path[-1], velocity(path)): path}\n",
|
||||
" complete_paths = []\n",
|
||||
@@ -266,17 +284,19 @@
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"# Solutions"
|
||||
"# Solutions\n",
|
||||
"\n",
|
||||
"We are ready to solve the problem:"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 8,
|
||||
"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>"
|
||||
]
|
||||
@@ -295,11 +315,11 @@
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"We see that the fastest path takes 12 moves, and there are 7 ways to get there (with other ways discarded).\n",
|
||||
"We see that the fastest solution is 12 moves, and there are at least 7 ways to get there (with other ways discarded).\n",
|
||||
"\n",
|
||||
"# Submitting your Answer\n",
|
||||
"\n",
|
||||
"I'll need a bit of work to put my answers into the form requested by The Riddler:\n",
|
||||
"I'll need a bit of \"tricky 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",
|
||||
@@ -312,7 +332,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 9,
|
||||
"execution_count": 10,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
@@ -320,22 +340,22 @@
|
||||
"\n",
|
||||
"def digits_from_path(path) -> Digits:\n",
|
||||
" \"\"\"The digits that the path represents.\"\"\"\n",
|
||||
" return [1 + accelerations.index(velocity(path[:i + 1]) - velocity(path[:i]))\n",
|
||||
" return [accelerations[velocity(path[:i + 1]) - velocity(path[:i])]\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",
|
||||
" v = zero # Velocity\n",
|
||||
" v = zero # Starting velocity\n",
|
||||
" path = [start]\n",
|
||||
" for d in digits:\n",
|
||||
" v += accelerations[d - 1]\n",
|
||||
" v += next(a for a in accelerations if accelerations[a] == d)\n",
|
||||
" path.append(v + path[-1])\n",
|
||||
" return path"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 10,
|
||||
"execution_count": 11,
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
@@ -350,7 +370,7 @@
|
||||
" [3, 3, 1, 7, 7, 5, 7, 8, 9, 9, 3, 3]]"
|
||||
]
|
||||
},
|
||||
"execution_count": 10,
|
||||
"execution_count": 11,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
@@ -365,17 +385,17 @@
|
||||
"source": [
|
||||
"# Alternative Solutions\n",
|
||||
"\n",
|
||||
"We can search from five different two-move opening sequences and one five-move sequence, and get different solution paths. Every solution is 12 moves. I don't understand why every plot has exactly 7 solution paths; I suppose it is because the finish line is crowded and there are only so many ways to get there."
|
||||
"I have identified six different opening sequences of moves that each lead to solutions with the fastest possible time of 12 moves. I don't understand why every plot has exactly 7 solution paths; I suppose it is just because the finish line is crowded and there are only so many ways to get there."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 11,
|
||||
"execution_count": 12,
|
||||
"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>"
|
||||
]
|
||||
@@ -385,7 +405,7 @@
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
@@ -395,7 +415,7 @@
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
@@ -405,7 +425,7 @@
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
@@ -415,7 +435,7 @@
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
@@ -425,7 +445,7 @@
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
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|
||||
"image/png": "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 truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
@@ -436,6 +456,7 @@
|
||||
],
|
||||
"source": [
|
||||
"openings = [3, 3], [3, 5], [3, 6], [6, 2], [6, 3], [3, 5, 2, 4, 4]\n",
|
||||
"\n",
|
||||
"for digits in openings:\n",
|
||||
" plot(search(path_from_digits(digits)))"
|
||||
]
|
||||
@@ -446,19 +467,17 @@
|
||||
"source": [
|
||||
"# Tests and Visualizations\n",
|
||||
"\n",
|
||||
"Below are a few unit tests; there should be more.\n",
|
||||
"\n",
|
||||
"Here are three tests/visualizations for `intersects_circle`:"
|
||||
"To gain more confidence in the code, here are some unit tests and visualizations:"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 12,
|
||||
"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",
|
||||
" \"\"\"Test if [p, q] segments get the expected result from `intersects_circle`.\"\"\"\n",
|
||||
" plot(segments, finish=[])\n",
|
||||
" for p, q in segments:\n",
|
||||
" assert intersects_circle(p, q) is expected"
|
||||
@@ -466,12 +485,12 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 13,
|
||||
"execution_count": 14,
|
||||
"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>"
|
||||
]
|
||||
@@ -485,18 +504,18 @@
|
||||
" [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(-3, -1),Point(1, -3)],\n",
|
||||
" [Point(-6, 0), Point(7, 7)]])"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 14,
|
||||
"execution_count": 15,
|
||||
"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>"
|
||||
]
|
||||
@@ -516,12 +535,12 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 15,
|
||||
"execution_count": 16,
|
||||
"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>"
|
||||
]
|
||||
@@ -531,9 +550,9 @@
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"# The waypoints that are checked to see if they are outside the circle\n",
|
||||
"p, q = Point(-7, 0), Point(5, -6)\n",
|
||||
"plot([waypoints(p, q, 20)], finish=[])\n",
|
||||
"# Waypoints on a line segment that are checked to see if they are outside the circle\n",
|
||||
"p, q = Point(-7, 7), Point(7, 0)\n",
|
||||
"plot([waypoints(p, q)], finish=[])\n",
|
||||
"assert not intersects_circle(p, q)"
|
||||
]
|
||||
},
|
||||
@@ -541,18 +560,18 @@
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"Here's a visualization for `expand(frontier)` showing the first few expansions.\n",
|
||||
"Note that the growth in number of paths is much slower then 9<sup><i>n</i></sup>."
|
||||
"Here's a visualization for `expand(frontier)` showing the first six expansions.\n",
|
||||
"Note that the growth in number of paths is much slower then O(9<sup><i>n</i></sup>). After 6 moves there are only 1,107 paths, or just 0.2% of 9<sup>6</sup> = 531,441."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 16,
|
||||
"execution_count": 17,
|
||||
"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>"
|
||||
]
|
||||
@@ -562,7 +581,7 @@
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
@@ -572,7 +591,7 @@
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
@@ -582,7 +601,27 @@
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"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>"
|
||||
]
|
||||
@@ -594,7 +633,7 @@
|
||||
"source": [
|
||||
"frontier = {(start, zero): [start]}\n",
|
||||
"\n",
|
||||
"for i in range(4):\n",
|
||||
"for i in range(6):\n",
|
||||
" frontier = expand(frontier)\n",
|
||||
" plot(frontier.values())"
|
||||
]
|
||||
@@ -608,7 +647,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 17,
|
||||
"execution_count": 18,
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
@@ -617,15 +656,15 @@
|
||||
"'tests pass'"
|
||||
]
|
||||
},
|
||||
"execution_count": 17,
|
||||
"execution_count": 18,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"assert start == Point(0, -5) == 0-5j\n",
|
||||
"assert path1 == [start, Point(1, -4)] == [0-5j, 1-4j]\n",
|
||||
"path1 = [start, Point(1, -4)] # First move\n",
|
||||
"\n",
|
||||
"assert start == Point(0, -5) == 0-5j\n",
|
||||
"assert X(start) == 0 and Y(start) == -5 \n",
|
||||
"assert velocity([start]) == 0\n",
|
||||
"assert velocity(path1) == Vector(1, 1)\n",
|
||||
@@ -646,7 +685,7 @@
|
||||
" 1-4j, 1-5j, 1-6j}\n",
|
||||
"\n",
|
||||
"# But 3 moves are backwards and 1 stays in the same place; those are disallowed\n",
|
||||
"assert circle_track_moves([start]) == [0-6j, 1-6j, 1-5j, 0-4j, 1-4j]\n",
|
||||
"assert circular_track_moves([start]) == {0-6j, 1-6j, 1-5j, 0-4j, 1-4j}\n",
|
||||
"assert expand({(start, zero): [start]}) == {\n",
|
||||
" (0-6j, 0-1j): [0-5j, 0-6j],\n",
|
||||
" (1-6j, 1-1j): [0-5j, 1-6j],\n",
|
||||
@@ -654,7 +693,16 @@
|
||||
" (0-4j, 0+1j): [0-5j, 0-4j],\n",
|
||||
" (1-4j, 1+1j): [0-5j, 1-4j]}\n",
|
||||
"\n",
|
||||
"def is_valid_path(path, legal_moves=circular_track_moves) -> bool:\n",
|
||||
" \"\"\"A valid path is complete, starts at start, stays on track, and has all legal moves.\"\"\"\n",
|
||||
" return (is_complete(path) \n",
|
||||
" and path[0] == start\n",
|
||||
" and set(path) <= track\n",
|
||||
" and all(path[i] in legal_moves(path[:i]) \n",
|
||||
" for i in range(1, len(path))))\n",
|
||||
"\n",
|
||||
"for path in solutions:\n",
|
||||
" assert is_valid_path(path)\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",
|
||||
|
||||
Reference in new issue
Block a user