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pfmoore/Squares Problem.ipynb

Created February 26, 2021 16:23
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Squares problem
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Squares Problem.ipynb
{
"cells": [
{
"cell_type": "markdown",
"id": "funded-tongue",
"metadata": {},
"source": [
"# Solving the Guardian squares problem\n",
"\n",
"The Guardian on 22 Feb 2021 included a problem, stated as follows:\n",
"\n",
"> Arrange the numbers 1 to 15 in order so that each pair of neighbours adds up to a square number (for example 11+5=16).\n",
"\n",
"While this looks at first glance like a numerical problem, it can be expressed as a problem in graph theory, as follows.\n",
"\n",
"Create a graph containing the numbers 1-15 as nodes. Add an (undirected) edge between every pair of numbers that adds up to a square. Now, we are looking for a path through the graph that passes through every node once. Such a path is a solution to the problem.\n",
"\n",
"Python has a powerful graph library, NetworkX, so we'll use that to investigate this problem."
]
},
{
"cell_type": "markdown",
"id": "solid-underground",
"metadata": {},
"source": [
"First, we will import the various libraries we will be using. We'll use NetworkX obviously, and we'll also use the matplotlib plotting library to draw the graph. We'll also use the stdlib itertools utilities to streamline creating sets of numbers.\n",
"\n",
"We use a Jupyter directive to display matplotlib images inline."
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "catholic-ecology",
"metadata": {},
"outputs": [],
"source": [
"import networkx as nx\n",
"from itertools import *\n",
"\n",
"%matplotlib inline"
]
},
{
"cell_type": "markdown",
"id": "smart-german",
"metadata": {},
"source": [
"The first thing we want to do is to create the graph of numbers. We'll make this a function, taking the number range as parameters, so that we can explore variations of the problem."
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "listed-indication",
"metadata": {},
"outputs": [],
"source": [
"def make_graph(end, start=1):\n",
" # For speed, we'll create a set of all possible squares\n",
" # that can be totals of two numbers in the set.\n",
" squares = set(takewhile(lambda sq: sq <= 2 * end, map(lambda x: x * x, count())))\n",
"\n",
" G = nx.Graph()\n",
" G.add_nodes_from(range(start, end+1))\n",
" for i in range(start, end+1):\n",
" for j in range(i+1, end+1):\n",
" if i + j in squares:\n",
" G.add_edge(i, j)\n",
" return G"
]
},
{
"cell_type": "markdown",
"id": "egyptian-maker",
"metadata": {},
"source": [
"We'll now create a graph for the Guardian problem (1-15) and display it."
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "southern-creativity",
"metadata": {},
"outputs": [],
"source": [
"G = make_graph(15)"
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "employed-factor",
"metadata": {},
"outputs": [
{
"data": {
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6c/Tu3Ru7d+/G3XffDZ1Oh48//lii74bIfn57jq+tOoZyQQaFIINckEEQZFArBJitIib881som1Rb2fj8XXh9Sx6+zm95v491DKm9cUehcoVCgZdffhn33XcfHn/8cWzatAlLly5FeDhLm5Fn+W3ga6uO4Z/v6Y2pTepsjh4QhQWZ+deVcrKKIipqTahp5YAw6xhSeyZ1ofKbb74Zer0e//d//4d+/fphxYoVGDp0qCRjEdlCJoqOpHh4v+KyGqS+m+2WvTe1QsDO6XezpBP5vczMTDz99NMYPXo05s2bh8BA/psg9/PbjaeGOobuwDqGRJcNGTIER44cwYULF6DT6XDw4EFPT4n8kN8GPuByHUN7M9vsxTqGRFcLDw/H2rVrMWvWLNx///14/fXXYTY7dqaWyBF+Hfgkr2MIoItGjaEJWknHIPJG48ePh16vR3Z2NgYPHoxTp055ekrkJ/w68EldxzBAIWDJRF2rpdGI/FlUVBS2bduGCRMmYODAgfjwww/hp2kH5EZ+m9zS1MLMfGTsOu2yDuzA5ZJOacm9ruvATkTNO378OB5//HFERETgH//4B7p16+bpKZGPYuADIIoifrf4C3z9Uz1kSueXPhvqGF5b0omIWmc0GjF37lwsX74cy5Ytw6hRoxy6T3FZDXKKK3D4TBnyz1fBaLZCpRAQ11WD/t3DkRQViqjwIBfPnrwFAx+AI0eOIDU1FU/O/xhbTpuc6sgeoBAwKSUGk1NjGfSIHLR//3488cQTGDx4MBYuXIgOHTq0eY3JYsX2PAMysguRb6iEUi6gxmiGtclPOEEGBKkUMFmsiNOGIC0lBsMStNyO8DN+H/jOnj2LgQMHYt68eRg/fjxySyqQvlaP0qp6u7pWB6vk6KxRY8lEnU0lnYiodVVVVZg+fTq2b9+OlStXIjk5ucXXOvPvtpNGjaUTdEiM5L9bf+HXga+6uhrJyckYPXo0Xn755cbPmyxW7LjyzvGkoRJKuQw1Rksz7xzlMFlExF955ziU7xyJXO7zzz/Hc889hyeeeAKvvfYa1OpftyNEUcTCrAJkZBdypYZs5reBz2KxYPTo0ejYsSNWrFjR4v/sUtUxJCLbXbhwAc899xxOnz6N1atXo2/fvhBFEa9uzMV6fYlLEtO4N+8//Dbw/eUvf0FOTg62bt0KlUrl6ekQURtEUcTKlSvxwgsv4MUXX4Q86UF8uPsHZmOT3fwy8C1ZsgSLFy/Gvn37WCmeyMv88MMPGJ8+E4a+EyHKlS6/f4BCwLpJd3Kv3of53YbUF198gddffx1btmxh0CPyQlHde0B5zySIcmmay9SZrUhfo4fJwubRvsqvAt/hw4fx1FNP4bPPPkOvXq03jiWi9ml7ngEXq40ApNuHK62qx448g2T3J8/ym358JSUleOihh7BkyRIMHDjQ09MhIgdlZJ9q9sjCkwN7YKwuCvHdQrA55yxmrDsCABgQHYZpQ+PQNzIUFquIb364iNmbj+FCZX2LY1QbLcjILpSsRyF5ll888VVVVeG3v/0t0tPT8cgjj3h6OkTkoOKyGuQbqpr9muFSPRZ/dQqffl981edDA5X4+LszGDT/K9w1fyeq6814e2y/Nsc6aahESXmtS+ZN7YvPP/FZLBaMHz8eN998M1588UVPT4eInJBTXAGlXGi2gfS2Y+cAAH0jQxERGtD4+a/zL1z1upX7f8R/n2t71UcplyGnqJzHlXyQzz/xTZs2DbW1tVi2bBnP5hB5ucNnylBtdK533+09O6LAUNnm62qMFhwqKnNqLGqffPqJb9GiRdixYwf27dsHpdL1ac9E5F7556vgzAGsm7qFYPK9sXh21fdtvtYqAgUtLKuSd/PZwLd582bMmzcPe/fuRVhYmKenQ0QuYHSiLFmPTkH41+9uw5zP83DgR9ue5JpbUiXv55OBT6/X45lnnsHnn3+Onj17eno6ROQiKgebRkeGBWLN72/H+zsL8L9DJTZfp5aoSTV5VrsNfI720yoqKsJDDz2EZcuW4fbbb/fAzIlIKnFdNdhVcKHZ5U65IINCkEEuyCAIMqgVAsxWEZ01Kqz9w+1Yuf8nrPnujM1jCTIgVqtx4eypvWhXJcuc7adVWVmJQYMGYeLEiZg5c6bnvhEiksSWoz/jxfVHUFV/fYLL1NRYTL2mxuaCzHyIAP4yJA7V11zTZ/a2VsfSqOWYPyaJZ/l8ULsJfM7201r0aBJenvQEIiMj8cEHHzCDk8gHFZfVIPXdbLfsvakVAnZOv5vHGXyQx5c6ne2nVW20oPpiDUYv3YOwTgPwv8WzGPSIfFRUeBDitCE4WlIh+Vjx2hAGPR/l0Z3bhn5aH+w67VQTSQCwyuSoufEuvPZFPtrJQywRSSAtJQbBKrmkYwSp5EhLiZF0DPIcjwa+RVkFLmsiCQB1JivW64uxKKvAJfcjovZnWIIWnTTqtl/ohC4aNYYmaCUdgzzHY4Evt6QCy7ILXdpEEgBqTRYsyy5E7lnpl0KIyP2UcgFLJ+gQINFRgwCFgCUTdVDKeZTBV3nkb9ZksSJ9rd7p5c2WsJ8WkW9LjAzFpJQYBCpdu+QZqJRjUkoMm9D6OI8Evu15BpRWtdwSxBXYT4vIt01OjcUYXZTLgl+gUsAYXRQmp8a65H7Ufnkkq7OlflrXUskFzB2ZiLt6d0JYkApnSqsxf9vJ66qtN4f9tIh8m0wmw9yRfdBFo8IyB7PCG+9lNaF79U+YO/I+ZoX7Abc/8bXWT+tackGGnytq8diH36DvnG14e0c+Fk/QIcrGFGP20yLybTKZDFOGxGFd2p3o3jHI7mzPYJUcPToG4eNnbsHpTYuxatUqiWZK7Ynbn/ha66d1rVqTBQuaZGjuPHEeRRdrkBgZimIbAhr7aRH5h8TIUGRNS8GOK5WfThoqoZTLUGO0NFP5SQ6TRUT8lcpPQ69UftqwYQPuvvtuJCQk4JZbbvHcN0OSc3vgc6afVmeNCr06B6PgfNu9tIBf+2lxuZPI9ynlAkb0jcCIvhEoKa9FTlE5DhWVocBQhXqzFWqFgFitBgOiw5EUHXbdG+I+ffrggw8+wJgxY3DgwAF07drVQ98JSc3tgc/RfloKQYYFjw7Aen0xCi9U23QN+2kR+afIsEBEhgXa/aZ39OjR0Ov1GDduHDIzM9nH00e5fY/PkX5aMhnw3iP9YbJYMWvTMbuuZT8tIrLHnDlzoNFoMH36dE9PhSTi9sDnSD+t+aP7obNGjbTVB2G22ve4yH5aRGQPuVyONWvWYOvWrfjXv/7l6emQBNweFeK6amBPtvD/ezgRvbtq8Pt/H7D76Y39tIjIEWFhYdiwYQNeeOEFHDhwwNPTIRdz+x5f/+7hCD5Q1Gw/rWtFhgVi4u09UG+y4MBfhzR+/q8bjmLj4bNtXh+kkmNAdLhT8yUi/5SQkIAPP/ywMdlFq2XtTl/h9sCXFBVqcymxkvJa3PjSFofHMllEJEWHOXw9Efm3UaNGQa/X45FHHmGyiw9x+1JnQz8td2A/LSJy1pw5cxASEoJp06Z5eirkIh7J/GA/LSLyFoIgYPXq1di2bRuTXXyERwIf+2kRkTdhsotv8UjgYz8tIvI2CQkJWL58OcaMGQODgZ1fvJnHIgP7aRGRt3n44Yfxu9/9DuPGjYPRaPT0dMhBMlF0pICYa4iiiFc3HsN6fbFLOrEHKuUYo4vC3JF92FqEiCRhtVoxcuRI9OjRA4sXL/b0dMgBHl0LbOinlZbcy+llzwCFgLTkXgx6RCSphmSXHTt24KOPPvL0dMgBHn3iayq3pALpa/Uoraq3qUltg2CVHJ01aiyZqOPyJhG5zfHjx5GSkoLPP/8ct912m6enQ3ZoN4EPAEwWK/7x5Xd4c5Meam1Pu/tpERG508aNG/GnP/0JBw4cQLdu3Tw9HbKR2yu3tEYpF1DyzecYF1qPqdOftrufFhGRO40cObKxjVFWVhZUKpWnp0Q2aFdPfKIo4qabbsLq1atx6623eno6RERtslqtePjhhxEdHY0lS5Z4ejpkg3a1PnjkyBEYjUbccsstnp4KEZFNBEHAqlWrkJWVhRUrVnh6OmSDdrXU+cknn+CRRx5hViYReZXQ0FBs2LABycnJ6NOnD26//XZPT4la0W6WOkVRRHx8PNauXcsnPiLySps2bcLzzz/fYrJLcVkNcoorcPhMGfLPV8FotkKlEBDXVYP+3cORFBWKqPAgD8zcv7SbwHf48GGMHj0ahYWFfOIjIq81e/ZsZGZmYufOnVCpVDBZrNieZ0BGdiHyDZVQygXUGM3NZKsrYLJYEXclW30Ys9Ul024C31//+ldYrVbMmzfP01MhInKY1WrFqFGjEBkZifSX33D4fHInjRpLJ+iQGMnzya7WLgKfKIqIjY3FJ598Ap1O5+npEBE5paKiArqnXoEsYTjMouMrWAEKAZNSYjA5NZYrYS7ULp6jDx8+DFEUMWDAAE9PhYjIKaIo4q2vzkDR936ngh4A1JmtyNh1Gq9uPIZ28IziM9pF4GM2JxH5ikVZBVivL0G92TWBqtZkwXp9MRZlFbjkftQOAp8oio2Bj4jIm+WWVGBZdqFLus00VWuyYFl2IXLPVrj0vv7K44Hv0KFDkMlk6N+/v6enQkTkMJPFivS1etSZrZLcv85sRfoaPUwWae7vTzwe+LjMSUS+YHueAaVV9ZKOUVpVjx157P7uLI9WbmlY5vzss888OQ0iIqdlZJ9q8cjCkwN7YKwuCvHdQrA55yxmrDsCAOjdVYN3xyWhR6dgAMDRkgrM3nwMp85XNXufaqMFGdmFGNE3Qppvwk94NPAdPHgQCoUCSUlJnpwGEZFTistqkG9oPlgBgOFSPRZ/dQrJsV0QoPx1oe38pTqkr9GjuLwWggx4cuCNeP+xAbh/0e4W73XSUImS8lp2p3GCR5c6P/30U4wbN47LnETk1XKKK1qtsrLt2DlszzOgrMZ41ecv1ZlRXF4LAJDJZLBYRdx45emvJUq5DDlF5U7P2Z957ImvYZlzw4YNnpoCEZFLHD5Thmqj2eHrj8wahiCVHIJMhncz81t9bY3RgkNFZVzudIKkga+1gqwhxlIoQ7Xo16+flFMgIpJc/vkqOHO+vN9r2xGolGPMzVEoKatt9bVWEShoZVmV2ubywGdLQdbdBRcgWM0Q7/8rHlqylwVZicirGV1whKHWZMGab3+C/pWhGPJuNkqrjS2+tl6iIxP+wqWBL7ek4rqCrM39BVlFwCq7PPTRkgrMXJeDN1mQlYi8lErhmjftgkyGQKUc3ToEtBr41C4az1+55E9PFEUsyMzH2Ix9OHOxxq4q5MDlFN0zF2swNmMfFmbmsyYdEXmVuK4atJajJxdkUCsEyAUZhCa/H9S7M/pEdIAgAzRqBV554DeoqDXh1IWWlzIFGRCr1UjwXfgPp5/4RFHEqxtzsV5f4nTFgoaCrBeqjJg7sg+zPYnIK/TvHo7gA0Woqm8+weXP9/TG1CFxjR+PHhCFBZn5yDdUYs6DfdAtNAB1Jgtyiivw1EfftbqUGaSSY0B0uMu/B3/idFuihZn5yNh12qW16QKVcqQl98KUJv+jEBG1V8VlNUh9N9ste29qhYCd0+/mOT4nOLXUyYKsRERAVHgQ4rQhbhkrXhvCoOckhwMfC7ISEf0qLSUGwSq5pGMEqeRIS4mRdAx/4HDgY0FWIqJfDUvQopNGLekYXTRqDE3QSjqGP3B4j+/BxbtxtORSm687Nnv4VR8HKOVY9c1PmL35mE3j9IsMxaY/DXJkikREbpVbUoGxGfskWQkLUAhYN+lOJN7AI1/Ociirs62CrE31mb2t8fdBKjkO/HUIvjj6s81jsSArEXmLxMhQTEqJkSzhj0HPNRxa6myrIGtL7k/shtJqI7778aLN17AgKxF5k8mpsRiji0Kg0jX7fYFKOcboojA5NdYl9yMHA5+jBVnH6KLwmb7YrmsaCrISEXkDmUyGuSP7IC25FwKcrLASoBCQltyL55pdzKG/FUcKskaGBeL2np2wzs7Ax4KsRORtZDIZpgyJw7q0O9G9Y5D92Z6mOnQNurynN2VIHIOeizkU+BwpyDpqQCS+//EiituoPN4cFmQlIm+UGBmKrGkpeGtsEvpFhkKtEKBRyyFcE8culyyTQ60Q0C8yFGOj6xC48y30iejgmYn7OIeSWxwpyDpaF4llXxc6MhwLshKR11LKBYzoG4ERfSNQUl6LnKJyHCoqQ4GhCvVmK9QKAbFaDQZEhyMpOgyRYYEwm+9AwpI52LlzJ1JTUz39LfgchwJfXFcNdhVcsHm5U9c9HN06BNiVzdmABVmJyFdEhgUiMiywzSayCoUCc+bMwcsvv4x7772XS50u5tCjVP/u4QhW2R4zx+oisfXYObu7NgAsyEpE/unRRx9FdXU1tmzZ4ump+ByHDrCzICsRkfQ2bNiA2bNnQ6/XQxC45eMqDv1JsiArEZH0Ro4cCaVSifXr13t6Kj7F4bcQLMhKRCQtmUyG119/HbNmzYLF4touOP7M4cDHgqxERNIbNmwYOnfujDVr1nh6Kj7DqUa0LMhKRCS97OxsPP300zhx4gRUKpWnp+P1nNotbSjI6qqadA0ClXJMSolh0CMiApCSkoLevXvjo48+8vRUfIJTT3wAIIoiXt14DOv1xS6pRt5QkJW16YiIfvXdd99hzJgxKCgoQEBAgKen49Wczo9lQVYiIunddttt0Ol0yMjI8PRUvJ7TT3xN5ZZUIH2tHqVV9XYdVg9WydFZo8aSiToubxIRtSAnJwfDhw/HqVOnoNGwopWjXBr4AMBksWJHngEZ2YU4aaiEUi5DjdECa5NRZACC1XKYLCLitSFIS4nB0AStQz3+iIj8yWOPPYakpCS89NJLnp6K13J54GuquYKsR3P0uDU2CuPuvbWxICsREdnmxIkTGDx4MAoKChAWFubp6XglSQNfc6ZOnYro6GhMnz7dncMSEfmMp59+GtHR0Xjttdc8PRWv5Pa1xZiYGBQWOtaeiIiIgFmzZmHJkiX45ZdfPD0Vr+T2wNerVy+cPn3a3cMSEfmMnj174tFHH8Wbb77p6al4JbcvdZ44cQIPPvggCgoK3DksEZFPKSkpQd++fXHs2DFERLTe34+u5vbAV1dXh9DQUFRXV0OhcKgPLhERAZg+fTqMRiPef/99T0/Fq7g98AFAdHQ0du/ejRtvvNHdQxMR+Yzz58/jN7/5DfR6PXr06OHp6XgNjxycY4ILEZHzunbtirS0NMydO9fTU/EqHgl8vXr1YuAjInKBGTNmYMOGDcybsIPHnviY2UlE5Lzw8HBMnToVs2fP9vRUvAaXOomIvNyUKVOQmZmJ3NxcT0/FK3hsqZNPfERErhESEoKZM2di1qxZnp6KV3B7VmdxWQ32HC/G1LkLMHzc72C0WKFSCIjrqkH/7uFIigpFVHiQO6dEROT1amtr0bt3b2zcuBG33HKLp6fTrrkl8JksVmy/0rEh31AJpVxAZW09ZMKvndsFGRCkUsBksSLuSseGYezYQERks6VLl2Lz5s348ssvPT2Vdk3ywOdMj75OGjWWTtAhMZI9+oiI2mI0GhEfH49Vq1Zh0KBBnp5OuyVZ4BNFEQuzCpCRXYg6s9Xh+wQoBExKicHk1Fh2ZSciasNHH32ElStX4quvvoJMJkNxWQ1yiitw+EwZ8s9XwWjm9pIkgU8URby6MRfr9SWoNdn+lNeSQKUcY3RRmDuyD4MfEVErzGYzEhL74plXF2BvWXDj9lKN0XxVQ3B/3l6SJPAtzMxHxq7TLgl6DQKVcqQl98KUIXEuuycRka/JLanAkx/uwsUaE6BQ23ydP20vuTzw5ZZUYGzGPqeWN1sSoBCwbtKdSLzBt/9SiIjsxe0l27n0mdZksSJ9rV6SoAcAdWYr0tfoYbJIc38iIm/UsL30wa7TTv/8rTNbkbHrNF7deAwe6GHgFi4NfNvzDCitqrfptTFdNFj7h9tx5G/D8PWMuzE8QWvTdaVV9diRZ3BmmkREPmVRVoHLcioAoNZkwXp9MRZl+Wb9T5cGvozsUzYdWZALMix/8mZknTiP/q9tx0ufHcV7j/ZHz87BbV5bbbQgI5vlzoiIgMvbS8uyC12aUwFcDn7LsguRe7bCpfdtD1wW+IrLapBvqLLptTFdgqENCcA/9/wAqwjsP12K738qw6gBkTZdf9JQiZLyWmemS0Tk9bi95BiXBb6c4gqn0mBlAOK1ITa9VimXIaeo3OGxiIh8gT3bS47yxe0llwW+w2fKUG002/Ta0xeqUVptxB+Te0EhyDA4tjNu79kJgUp52xcDqDFacKiozJnpEhF5PVu3lwDgvUf647uXUnH0b8Owc3oKHr0l2qbrfHF7SeGqG+Wfr4KtCUBmq4jnVn2POQ/2QVpKDI4WV2DL0Z9htPFx3SoCBTYuqxIR+SJ7tpcAYOnXp/Di+iMwWqyI6RKM/zx7B46drUDu2UttXtuwvRQZFujMlNsNlwU+W4NWgxPnKvHo8m8aP16fdifW64ttvr5eojVtIiJv0LC9ZOvPwoLzvwZJUbz8X49OwTYFvobtJQa+a6gU9q2a3tQtBD/8Ug2ZDHjijhvRNUSNdQdtD3xqO8cjIvIl9mwvNZg7MhFjdVEIVMmRW1KBr06et+m6hu2lEX0jHJlqu+OywBfXVYNdBRdsXu4cNSASj93aHQpBhgM/XsTjK76F0cbMIUEGxGo1TsyWiMi72bO91ODVjbn426Zc6LqH445enfx2e8llga9/93AEHyhCVb1t70De+PIE3vjyhENjBankGBAd7tC1RES+wN7tpQZWEY3Hxx6/owf+te9Hm67zpe0ll60XJkWFuu2sh8kiIik6zC1jERG1R/ZuL11LLsjQo6Pt7Yh8aXvJZd9JVHgQ4mw8h+eseG2Iz2yyEhE5Iq6rBrbWkO4UrMKD/SIQpJJDkAHJsZ3xUNIN2Fv4i03X+9r2ksuWOgEgLSUGM9fl2NVp3V5BKjnSUmIkuz8RkTewZ3tJBPD47T3w/x7uC5kMKCmvxWuf5yHzuG3JLVJtL3mqSa5LA9+wBC3e1KhRfbHGlbe9SheNGkNtLGhNROSr7NleulhtvOr4mL1cub1kslixPc+AjOzCFpvk7i64gKADRZI1yWU/PiIiL/Xg4j04WiJ9EemgugvIGNULgwcPdqpHX25JBdLX6lFaVW/XyqCrm+S6fLcyMTIUk1JibC4/ZqtApRyTUmIY9IiIrkhLiUGwyrU/a68VpJIj9QYrnn32WQwYMAArVqxAba19TQJEUcSCzHyMzdiHMxdr7N4OqzZacOZiDcZm7MPCzHyn+wRKkqYzOTUWY3RRLgt+gUo5xuiiMDk11iX3IyLyBcMStOikUUs6RheNGu9O+x2OHz+ON998E+vXr0ePHj3w0ksvoaioqM3r22OTXEkCn0wmw9yRfZCW3AsBTqbABigEpCX3wtyRfZx6xCYi8jVKuYClE3RO/5xtSYBCwJKJOijlAgRBwPDhw7Flyxbs3bsXtbW1SEpKwrhx47Br164WA1F7bJLr8j2+azmzpttZo8aSiToubxIRtWJhZj4ydp12aTPaQKUcacm9MGVIXIuvqaysxMqVK/H+++8jMDAQkydPxvjx4xEYePm4WXvN+ZA88AGXs3h2XMniOWmohFIuQ43RclUWjyC7vJZssoiIv5LFM9SFWTxERL7q8nLiMazXF7sk+DVsL9m60ma1WrFjxw4sWrQIBw4cwB/+8Ac8+1wanvq0EGckzPLv3jEIWdNS7I4Tbgl8TZWU1yKnqByHispQYKhCvdkKtUJArFaDAdHhSIoO4+F0IiI7iaKIRVkFWJZd6NQTVoBCwKSUGExOjXVoe6mgoACLFy/Gx3tPQpOaBqugdHgubQlWyfHW2CS7i2e7PfAREZF02sv20oiF2cg7Z3th6wf7RWBKahxuCAvAhcp6zFiXgwM/tt1wvF9kKDb9aZBdc3PpAXYiIvKsxMhQZE1LaXN7CVYrNIFKSbaXistqUPiL7Uucg3p3xov33YQ/f3wIh4vL0TXE9kxVR5rkMvAREfkYpVzAiL4RGNE3otntJaupHvu+XI/5f39Jku0le5vk/mVIHBbtPIVDReUAAMOlepvHcqRJLgMfEZEPiwwLRGRY4FX7YFarFZoZ9yO559+h0bg+p8KeJrmCDOgbGYrM4wZ8PeNuqBUCtucZ8PcvjtsUOB1pksuUSSIiPyMIAnr37o2CAsfPwrXGnia5nTVqqBQC7k/shnEf7MeIRbvRJ6ID/nyvbQVLHGmSy8BHROSH4uLikJ+fL8m97WmSW3fl+MXK/T/iQmU9ympM+MeeH3BPfBeb72Fvk1wGPiIiPyRl4LOnSe6lOjPOltde9YRo71EDe5vkMvAREfmh2NhYyQKfPU1yAeDTg8V46s4b0SlYhQ4BCvz+rp7IOmFbr0BHmuQy8BER+aG4uDjJ9vj6dw9HsMr23Mn3dxbgSHEFvpp+N7KmpeDYzxVY8tUpm651pEkuD7ATEfmhCxcuIC4uDhcvXnR5A4Dishqkvptt996bI9QKATun323XcQY+8RER+aHOnTsDAEpLS11+76jwIMRpQ1x+3+bEa0PsPofIwEdE5IdkMpmkCS7uapKblhJj93UMfEREfkrKwDfkpi4QTDUQRemWO7to1BiaoLX7OgY+IiI/JVVmZ1lZGcaOHoWAg6sRoJDmqa9pk1x7MfAREfkpKTI79Xo9br75ZvTu3Rv7Pv8v0u/ujUCla4NfoFKOSSkxDneRYOAjIvJTrl7q/Oc//4nhw4dj3rx5eO+996BUKjE5NRZjdFEuC34NTXInp9pW0qw5LFJNRORnistqkFNcgW9/EnA+YSzGL98PlUKOuK4a9O8ejqSoUESFB9l8v9raWjz//PP49ttvsXv3btx0002NX5PJZJg7sg+6aFQuaZKbltzL4Sa5jXPiOT4iIt9nslix/UqPvnxDJZRyATVG81U9+gQZEKRSwGSxIu5Kj75hbfToKywsxJgxY5CQkIAPP/wQGk3LVVTaS5NcBj4iIh/nTMDppFFj6QQdEiOvDzgbN27Es88+i1mzZuH555+36SnMZLG22ST3cgCWS9IkF2DgIyLyWaIoYmFWATJcsMQ4KSWmcYnRbDbjlVdewdq1a/HJJ5/gjjvucOi+zTXJVSsExGo1GBAdLkmTXICBj4jIJ4miiFc35mK9vgS1Jtuf8lrSkFTypzs6Y/z48VAqlVizZg26dLG9fVB7wcBHROSDFmbmI2PXaZcEvQZqOVBzcBOeua0bZs2aBblc2sosUmHgIyLyMbklFRibsc+p5c2WKAXgf88PckmSiafwHB8RkQ8xWaxIX6uXJOgBgMkKpK/Rw2SRvvOCVBj4iIh8yPY8A0qr6iUdo7SqHjvyDJKOISUeYCci8iEZ2adaPLLw5MAeGKuLQny3EGzOOYsZ6440fu3OmE6Y+1AibggLxOGicsxYl4OS8tpm71NttCAjuxAj+kZI8j1IjU98REQ+orisBvmGqha/brhUj8VfncKn3xdf9fnwICUyHr8Z7+w4if5zt+NISTkWjx/Q6lgnDZUtBsb2joGPiMhH5BRXtHrIe9uxc9ieZ0BZjfGqz9/XpxsKDFX4Ivcc6s1WLMgswG8iOiCmS3CL91LKZcgpKnfV1N2KgY+IyEccPlOGaqPZ7uvitCE4/vOlxo9rTRb8VFqD2K4td1GvMVpwqKjMoXl6GgMfEZGPyD9fBUcOqAWpFKisM131uco6EzTqltNArCJQ0MqyanvGwEdE5COMDh5hqDGaoQlQXvU5TYACVfWtPz3WS3RkQmoMfEREPkKlcOxHer6hEr/p9uuyZqBSjh4dg1FwvrLV69QOjudp3jlrIiK6TlxXDVprkCAXZFArBMgFGYQmv9+WZ0BctxDc16cb1AoBU1JjceLcJRReqG7xXoIMiNW23IKoPeM5PiIiH9G/eziCDxS1uET553t6Y+qQuMaPRw+IwoLMfCzIKsCk1Qfx2kOJWPBofxwuKsefPz7U6lhBKjkGRIe7dP7uwlqdREQ+orisBqnvZrtl702tELBz+t2StA2SGpc6iYh8RFR4EOK0LR9BcKV4bYhXBj2AgY+IyKekpcQgWCVtu6AglRxpKTGSjiElBj4iIh8yLEGLThq1pGN0CFRiaIJW0jGkxMBHRORDlHIBSyfoECDhUYOLVfVY+tUpeGuKCAMfEZGPSYwMxaSUGAQqpVnyNFpEZOw6jVc3HvPK4MfAR0TkgyanxmKMLgoBSml+zNeaLFivL8airAJJ7i8lBj4iIh8kk8nw2kMJCCv5DoLYfH8+Z9WaLFiWXYjcsxWS3F8qDHxERD5qwYIFsB7bii4dgiQbo85sRfoaPUwW76nbycBHROSDvvnmG7z55ptI/38ZbRabdlZpVT125BkkHcOVGPiIiHxMaWkpHn30USxfvhzrj19CtdG2pc4nB/bApufvwsm59+Htsf1sHq/aaEFGdqGj03U7Bj4iIh9itVrx1FNPYdy4cdANHoJ8O3rmGS7VY/FXp/Dp98V2j3vSUImS8lq7r/MEBj4iIh/yzjvvoLS0FG+88QZyiiuglNv+Y37bsXPYnmdAWY3R7nGVchlyisrtvs4TGPiIiHzE3r178c477+C///0vlEolDp8pQ7VR2v29BjVGCw4VlbllLGcx8BER+YBffvkF48ePxz//+U90794dAJB/vgruOl9uFYECO5ZVPYn9+IiI2pHishrkFFfg8Jky5J+vgtFshUohIK6rBv27hyMpKhRR4VcfT7BarXjiiScwfvx4PPDAA42fN7qhPVFT7miH5AoMfEREHmayWLE9z4CM7ELkGyqhlAuoMZphbfK0trvgAoIOFMFksSJOG4K0lBgMS9BCKRfw5ptvorKyEq+//vpV91VJWK+zOWo3j+coBj4iIg/KLalA+lo9SqvqG48dNPfkZBXReB7vaEkFZq7LwZsaNf5wE7Bw4UJ8//33UCqVV10T11WDXQUXbF7ulAsyKAQZ5IIMgiCDWiHAbBVhsbZ9A0EGxGo1tg3kYQx8REQeIIoiFmYVICO7EHUOLBFWGy2ovliDV3cZMXb2CkRGRl73mv7dwxF8oMjmA+x/vqc3pg6Ja/x49IAoLMjMxwIb6nEGqeQYEB1u+zfgQTLRG0trExF5MVEU8erGXKzXl6DW5HwdzUClHGN0UZg7sg9kMlnj54vLapD6brZb9t7UCgE7p9/tFV3ZvWNBlojIhyzKKnBZ0ANa7pSg1SgRLtS5ZIy2xGtDvCLoAQx8RERulVtSgWXZhS4Leg2adkowm83417/+hfj4eAgnMhEg8aZWkEqOtJQYaQdxIQY+IiI3MVmsSF+rd2hPzxZ1Ziue/CAbCYl98dFHH2HlypXIXvs+ukrYnQEAumjUGJqglXQMV2LgIyJyk+15BpRW1Us6RlmNGc+8ugBff/01Bg8eDKVcwNIJOgRIdNQgQCFgyUSdXaXRPI1ZnUREbpKRfarFTglPDuyBsbooxHcLweacs5ix7ggAICosEHtevBfVTTIzM3YV4v2dp5q9j6hQY19Z8FVJLomRoZiUEoOMXaddusQaqJQjLbkXEm8Iddk93YGBj4jIDYrLalrtlNDQGSE5tgsClNc/PfV7bbtN5+mAXzslNE02mZwaiwtVRqzXF7s0k3RyaqzT93I373k2JSLyYm11SnCmM8K1muuUIJPJMHdkH6Ql93J62TNAISAtudd1xye8BZ/4iIjcwNlOCXtn3gsRIvYU/IK/f3kcZTWmFl/b0ClhRN+Iqz4vk8kwZUgcUn+jva5ajC2CVXJ01qixZKLO65Y3m+ITHxGRGzjaKeFijREPLt6Du+bvxIOL9yBYrcDCRwe0ek1bnRISI0ORNS0Fb41NQr/IUKgVAjRqOYRrHt4EGaBRy6EUAMWls3hrbBIyp6V4ddAD+MRHROQWjnZKqDFacLSkAgDwS5URf9t0DAdeHoJglbzVp7W2qrUo5QJG9I3AiL4RKCmvRU5ROQ4VlaHAUIV6sxVqhYBYrQYDosORGKHBbYmx6D0lBUp5RKv39QYMfEREbuCqTgkiLj82Cm3srdnTKSEyLBCRYYHXLY02NX78eKxevRqvvfaazfdtr7jUSUTkBnFdNWgtVsmvdENo2hlBLsjQPzoMvToHQyYDwoKUmP1gH+wvLEVlK4WnpeiU8MQTT2D16tXwhfLOfOIjInKDtjoltNQZ4fQv1XhhWDw6aVSoqjNj96lfMPk/h1odS4pOCQMGDEBgYCD27t2LQYMGufTe7sbuDEREbuALnRLmzZuHH374AR988IFL7+tuXOokInKDqPAgxGlD3DKWVJ0SJk6ciHXr1qGuzj0dH6TCwEdE5CZpKTEIVsklHUPKTgnR0dFISkrCli1bJLm/uzDwERG5ybAELTpp1JKOIXWnhCeeeAKrVq2S7P7uwD0+IiI3yi2pwNiMfZK0JgpQCFg36U5JD5hfunQJ3bt3R2FhITp16iTZOFLiEx8RkRs1dEoIVLp2yTNQKceklBjJq6p06NAB999/P/773/9KOo6UGPiIiNxscmosxuiiXBb83N0pwduXO7nUSUTkAaIoYlFWAZZlFzq17BmgEDApJQaTU2Pd1inBbDYjKioKu3fvRmys97UlYuAjIvKg3JIKr+yUMHXqVISGhmLOnDluH9tZDHxERB5mslixI8+AjOxCnDRUQimXocZoQdO+s4Ls8lEFk0VEvDYEaSkxGJqgbbXHn5QOHjyIRx55BKdOnfK6nnwMfERE7UhbnRKSosMkOZxuL1EU0adPHyxfvhx33XWXp6djFwY+IiJyyBtvvIGffvoJr7zxLnKKK3D4TBnyz1fBaLZCpRAQ11WD/t3DkRQViqjwIE9PtxEDHxER2c1ksWLtrmN4ZU02giJioJQLqDGam1meVcBksSLuyvLsMA8uzzZg4CMiIrs4k5DTSaPG0gk6JEZ6ros7Ax8REdlEFEUszCpAhhcewWiKgY+IiNokiiJe3ZiL9foS1Jpsf8prScOh+7kj+7g9+LFyCxERtWlRVoHLgh4A1JosWK8vxqKsApfczx4MfERE1Krckgosyy50WdBrUGuyYFl2IXLPVrj0vm1h4CMiohaZLFakr9VL0k0CAOrMVqSv0cNkkb4zfQMGPiIiatH2PANKq+olHaO0qh478gySjtGUwm0jERGR18nIPmXzkYWosEDMfTgRuu7hMJot+CL3HF77PA8Wa+s5lNVGCzKyCzGib4QrptwmPvEREVGzistqkG+osvn1cx9ORGlVPW77eyZGLNqD23t2xBN39LDp2pOGSpSU1zo6Vbsw8BERUbNyiivsqrISHR6Ez4/+jHqzFReq6pGdfwFxXTU2XauUy5BTVO7gTO3DwEdERM06fKYM1Uazza9fsfcHPNjvBgQoBWg7qHF3XFdk51+w6doaowWHisocnapduMdHRETNyj9fBXtKnHz7w0WMv607cv82HAq5gHUHi7DNxqQVqwgU2LGs6gw+8RERUbOMdhxhkMmAlU/fiq2555Dwt23oP3c7QgOV+L/7brL5HvUSHZm4FgMfERE1S6WwPUSEBSoRFR6Ef+//EUaLFeU1Jnx6sBj3xHe1+R5qO8ZzBgMfERE1K66rBraW0SyrMeHMxRo8fkcPyAUZOgQoMEYXhRPnLtl0vSADYrW2JcI4i4GPiIia1b97OIJVtqeCpK0+iJS4LtC/MhRfz7gHJosVc7fk2XRtkEqOAdHhjk7VLkxuISKiZiVFhdpVSizv50t4bPk3Do1lsohIig5z6Fp78YmPiIiaFRUehDhtiFvGiteGIDIs0C1jMfAREVGL0lJiEKySSzpGkEqOtJQYScdoioGPiIhaNCxBi04ataRjdNGoMTRBK+kYTTHwERFRi5RyAUsn6BAg0VGDAIWAJRN1dpVGcxYDHxERtSoxMhSTUmIQqHTtkmegUo5JKTFIvCHUpfdtCwMfERG1aXJqLMboolwW/AKVcozRRWFyaqxL7mcPmSjaU4mNiIj8lSiKWJRVgGXZhU51ZA9QCJiUEoPJqbGQ2XpC3oUY+IiIyC65JRVIX6tHaVW9zU1qASBYJUdnjRpLJurcvrzZFAMfERHZzWSxYkeeARnZhThpqIRSLkON0YKmzdYF2eWjCiaLiHhtCNJSYjA0QevWRJbmMPAREZFfYXILERH5FQY+IiLyKwx8RETkVxj4iIjIrzDwERGRX/n/OVpq47/HBG0AAAAASUVORK5CYII=\n",
"text/plain": [
"<Figure size 432x288 with 1 Axes>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"nx.draw_spring(G, with_labels=True, node_size=750, font_color=\"w\")"
]
},
{
"cell_type": "markdown",
"id": "digital-selection",
"metadata": {},
"source": [
"From this, it's easy to see that there is a solution, and what the problem will be with finding it - there is a \"short cut\" from 1 to 3, and if you take that, you don't get the full solution. So we can't search for \"shortest\" paths, we need to search for \"covering\" paths (in graph theoretical terminology, a covering path includes every node).\n",
"\n",
"To be more specific, we are looking for [Hamiltonian path](https://en.wikipedia.org/wiki/Hamiltonian_path) - a path that visits each vertex exactly once. As with many graph problems, the Hamiltonian path problem is NP-complete."
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "humanitarian-spectacular",
"metadata": {},
"outputs": [],
"source": [
"# Brute force solution, from https://groups.google.com/g/networkx-discuss/c/xYqNH86hhBU\n",
"\n",
"def hamiltonian_paths(G):\n",
" for i in G:\n",
" for j in G:\n",
" if j == i: continue\n",
" for p in nx.all_simple_paths(G, i, j):\n",
" if len(p) == len(G):\n",
" yield p"
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "latin-shoot",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[8, 1, 15, 10, 6, 3, 13, 12, 4, 5, 11, 14, 2, 7, 9]\n",
"[9, 7, 2, 14, 11, 5, 4, 12, 13, 3, 6, 10, 15, 1, 8]\n"
]
}
],
"source": [
"for p in hamiltonian_paths(G):\n",
" print(p)"
]
},
{
"cell_type": "code",
"execution_count": 10,
"id": "mathematical-carry",
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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\n",
"text/plain": [
"<IPython.core.display.Image object>"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"import networkx.drawing.nx_pydot\n",
"from IPython.display import Image\n",
"d = networkx.drawing.nx_pydot.to_pydot(G)\n",
"Image(d.create_png())"
]
},
{
"cell_type": "code",
"execution_count": 17,
"id": "artificial-firewall",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"(15, 2, 0.01600000000325963)\n",
" [8, 1, 15, 10, 6, 3, 13, 12, 4, 5, 11, 14, 2, 7, 9]\n",
" [9, 7, 2, 14, 11, 5, 4, 12, 13, 3, 6, 10, 15, 1, 8]\n",
"(16, 2, 0.014999999955762178)\n",
" [8, 1, 15, 10, 6, 3, 13, 12, 4, 5, 11, 14, 2, 7, 9, 16]\n",
" [16, 9, 7, 2, 14, 11, 5, 4, 12, 13, 3, 6, 10, 15, 1, 8]\n",
"(17, 2, 0.03200000000651926)\n",
" [16, 9, 7, 2, 14, 11, 5, 4, 12, 13, 3, 6, 10, 15, 1, 8, 17]\n",
" [17, 8, 1, 15, 10, 6, 3, 13, 12, 4, 5, 11, 14, 2, 7, 9, 16]\n",
"(18, 0, 0.015000000013969839)\n",
"(19, 0, 0.03200000000651926)\n",
"(20, 0, 0.045999999972991645)\n",
"(21, 0, 0.0940000000409782)\n",
"(22, 0, 0.25)\n",
"(23, 6, 0.46899999998277053)\n",
" [2, 23, 13, 12, 4, 21, 15, 10, 6, 19, 17, 8, 1, 3, 22, 14, 11, 5, 20, 16, 9, 7, 18]\n",
" [9, 16, 20, 5, 11, 14, 22, 3, 1, 8, 17, 19, 6, 10, 15, 21, 4, 12, 13, 23, 2, 7, 18]\n",
" [18, 7, 9, 16, 20, 5, 11, 14, 22, 3, 1, 8, 17, 19, 6, 10, 15, 21, 4, 12, 13, 23, 2]\n",
" [18, 7, 2, 23, 13, 12, 4, 21, 15, 10, 6, 19, 17, 8, 1, 3, 22, 14, 11, 5, 20, 16, 9]\n",
" [18, 7, 9, 16, 20, 5, 11, 14, 2, 23, 13, 12, 4, 21, 15, 10, 6, 19, 17, 8, 1, 3, 22]\n",
" [22, 3, 1, 8, 17, 19, 6, 10, 15, 21, 4, 12, 13, 23, 2, 14, 11, 5, 20, 16, 9, 7, 18]\n",
"(24, 0, 1.0310000000172295)\n",
"(25, 20, 2.280999999959022)\n",
" [2, 23, 13, 12, 24, 25, 11, 14, 22, 3, 1, 8, 17, 19, 6, 10, 15, 21, 4, 5, 20, 16, 9, 7, 18]\n",
" [3, 22, 14, 2, 23, 13, 12, 4, 21, 15, 10, 6, 19, 17, 8, 1, 24, 25, 11, 5, 20, 16, 9, 7, 18]\n",
" [4, 21, 15, 10, 6, 19, 17, 8, 1, 3, 22, 14, 2, 23, 13, 12, 24, 25, 11, 5, 20, 16, 9, 7, 18]\n",
" [8, 17, 19, 6, 10, 15, 21, 4, 12, 13, 23, 2, 14, 22, 3, 1, 24, 25, 11, 5, 20, 16, 9, 7, 18]\n",
" [9, 16, 20, 5, 4, 21, 15, 10, 6, 19, 17, 8, 1, 3, 22, 14, 11, 25, 24, 12, 13, 23, 2, 7, 18]\n",
" [10, 15, 21, 4, 12, 13, 23, 2, 14, 22, 3, 6, 19, 17, 8, 1, 24, 25, 11, 5, 20, 16, 9, 7, 18]\n",
" [11, 25, 24, 12, 13, 23, 2, 14, 22, 3, 1, 8, 17, 19, 6, 10, 15, 21, 4, 5, 20, 16, 9, 7, 18]\n",
" [13, 23, 2, 14, 22, 3, 1, 8, 17, 19, 6, 10, 15, 21, 4, 12, 24, 25, 11, 5, 20, 16, 9, 7, 18]\n",
" [18, 7, 9, 16, 20, 5, 4, 21, 15, 10, 6, 19, 17, 8, 1, 3, 22, 14, 11, 25, 24, 12, 13, 23, 2]\n",
" [18, 7, 9, 16, 20, 5, 11, 25, 24, 1, 8, 17, 19, 6, 10, 15, 21, 4, 12, 13, 23, 2, 14, 22, 3]\n",
" [18, 7, 9, 16, 20, 5, 11, 25, 24, 12, 13, 23, 2, 14, 22, 3, 1, 8, 17, 19, 6, 10, 15, 21, 4]\n",
" [18, 7, 9, 16, 20, 5, 11, 25, 24, 1, 3, 22, 14, 2, 23, 13, 12, 4, 21, 15, 10, 6, 19, 17, 8]\n",
" [18, 7, 2, 23, 13, 12, 24, 25, 11, 14, 22, 3, 1, 8, 17, 19, 6, 10, 15, 21, 4, 5, 20, 16, 9]\n",
" [18, 7, 9, 16, 20, 5, 11, 25, 24, 1, 8, 17, 19, 6, 3, 22, 14, 2, 23, 13, 12, 4, 21, 15, 10]\n",
" [18, 7, 9, 16, 20, 5, 4, 21, 15, 10, 6, 19, 17, 8, 1, 3, 22, 14, 2, 23, 13, 12, 24, 25, 11]\n",
" [18, 7, 9, 16, 20, 5, 11, 25, 24, 12, 4, 21, 15, 10, 6, 19, 17, 8, 1, 3, 22, 14, 2, 23, 13]\n",
" [18, 7, 9, 16, 20, 5, 11, 25, 24, 12, 4, 21, 15, 10, 6, 19, 17, 8, 1, 3, 13, 23, 2, 14, 22]\n",
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" [22, 14, 2, 23, 13, 3, 1, 8, 17, 19, 6, 10, 15, 21, 4, 12, 24, 25, 11, 5, 20, 16, 9, 7, 18]\n",
" [23, 2, 14, 22, 3, 13, 12, 4, 21, 15, 10, 6, 19, 17, 8, 1, 24, 25, 11, 5, 20, 16, 9, 7, 18]\n",
"(26, 24, 4.329000000027008)\n",
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" [4, 21, 15, 1, 8, 17, 19, 6, 10, 26, 23, 2, 14, 22, 3, 13, 12, 24, 25, 11, 5, 20, 16, 9, 7, 18]\n",
" [8, 17, 19, 6, 10, 26, 23, 2, 14, 22, 3, 13, 12, 4, 21, 15, 1, 24, 25, 11, 5, 20, 16, 9, 7, 18]\n",
" [9, 16, 20, 5, 11, 25, 24, 12, 4, 21, 15, 1, 8, 17, 19, 6, 10, 26, 23, 13, 3, 22, 14, 2, 7, 18]\n",
" [11, 25, 24, 12, 13, 3, 22, 14, 2, 23, 26, 10, 6, 19, 17, 8, 1, 15, 21, 4, 5, 20, 16, 9, 7, 18]\n",
" [13, 3, 22, 14, 2, 23, 26, 10, 6, 19, 17, 8, 1, 15, 21, 4, 12, 24, 25, 11, 5, 20, 16, 9, 7, 18]\n",
" [13, 12, 4, 21, 15, 10, 26, 23, 2, 14, 22, 3, 6, 19, 17, 8, 1, 24, 25, 11, 5, 20, 16, 9, 7, 18]\n",
" [15, 21, 4, 12, 13, 3, 22, 14, 2, 23, 26, 10, 6, 19, 17, 8, 1, 24, 25, 11, 5, 20, 16, 9, 7, 18]\n",
" [18, 7, 9, 16, 20, 5, 11, 25, 24, 12, 4, 21, 15, 1, 8, 17, 19, 6, 10, 26, 23, 13, 3, 22, 14, 2]\n",
" [18, 7, 9, 16, 20, 5, 11, 25, 24, 12, 13, 3, 22, 14, 2, 23, 26, 10, 6, 19, 17, 8, 1, 15, 21, 4]\n",
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" [18, 7, 2, 14, 22, 3, 13, 23, 26, 10, 6, 19, 17, 8, 1, 15, 21, 4, 12, 24, 25, 11, 5, 20, 16, 9]\n",
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" [18, 7, 9, 16, 20, 5, 11, 25, 24, 12, 4, 21, 15, 1, 8, 17, 19, 6, 10, 26, 23, 2, 14, 22, 3, 13]\n",
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" [18, 7, 9, 16, 20, 5, 4, 21, 15, 1, 8, 17, 19, 6, 10, 26, 23, 2, 14, 11, 25, 24, 12, 13, 3, 22]\n",
" [18, 7, 9, 16, 20, 5, 11, 25, 24, 1, 8, 17, 19, 6, 3, 13, 12, 4, 21, 15, 10, 26, 23, 2, 14, 22]\n",
" [18, 7, 9, 16, 20, 5, 11, 25, 24, 1, 8, 17, 19, 6, 3, 22, 14, 2, 23, 13, 12, 4, 21, 15, 10, 26]\n",
" [18, 7, 9, 16, 20, 5, 11, 25, 24, 1, 8, 17, 19, 6, 10, 15, 21, 4, 12, 13, 3, 22, 14, 2, 23, 26]\n",
" [22, 3, 13, 12, 24, 25, 11, 14, 2, 23, 26, 10, 6, 19, 17, 8, 1, 15, 21, 4, 5, 20, 16, 9, 7, 18]\n",
" [22, 14, 2, 23, 26, 10, 15, 21, 4, 12, 13, 3, 6, 19, 17, 8, 1, 24, 25, 11, 5, 20, 16, 9, 7, 18]\n",
" [26, 10, 15, 21, 4, 12, 13, 23, 2, 14, 22, 3, 6, 19, 17, 8, 1, 24, 25, 11, 5, 20, 16, 9, 7, 18]\n",
" [26, 23, 2, 14, 22, 3, 13, 12, 4, 21, 15, 10, 6, 19, 17, 8, 1, 24, 25, 11, 5, 20, 16, 9, 7, 18]\n",
"(27, 70, 9.26500000001397)\n",
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" [1, 8, 17, 19, 6, 3, 22, 27, 9, 16, 20, 5, 4, 21, 15, 10, 26, 23, 13, 12, 24, 25, 11, 14, 2, 7, 18]\n",
" [3, 13, 12, 24, 25, 11, 14, 22, 27, 9, 16, 20, 5, 4, 21, 15, 1, 8, 17, 19, 6, 10, 26, 23, 2, 7, 18]\n",
" [3, 13, 23, 26, 10, 6, 19, 17, 8, 1, 15, 21, 4, 12, 24, 25, 11, 5, 20, 16, 9, 27, 22, 14, 2, 7, 18]\n",
" [3, 22, 27, 9, 16, 20, 5, 4, 21, 15, 1, 8, 17, 19, 6, 10, 26, 23, 13, 12, 24, 25, 11, 14, 2, 7, 18]\n",
" [5, 20, 16, 9, 27, 22, 3, 13, 23, 26, 10, 6, 19, 17, 8, 1, 15, 21, 4, 12, 24, 25, 11, 14, 2, 7, 18]\n",
" [5, 20, 16, 9, 27, 22, 14, 11, 25, 24, 1, 8, 17, 19, 6, 3, 13, 12, 4, 21, 15, 10, 26, 23, 2, 7, 18]\n",
" [6, 19, 17, 8, 1, 3, 13, 12, 24, 25, 11, 14, 22, 27, 9, 16, 20, 5, 4, 21, 15, 10, 26, 23, 2, 7, 18]\n",
" [6, 19, 17, 8, 1, 3, 13, 23, 26, 10, 15, 21, 4, 12, 24, 25, 11, 5, 20, 16, 9, 27, 22, 14, 2, 7, 18]\n",
" [6, 19, 17, 8, 1, 3, 22, 27, 9, 16, 20, 5, 4, 21, 15, 10, 26, 23, 13, 12, 24, 25, 11, 14, 2, 7, 18]\n",
" [8, 17, 19, 6, 10, 26, 23, 13, 3, 1, 15, 21, 4, 12, 24, 25, 11, 5, 20, 16, 9, 27, 22, 14, 2, 7, 18]\n",
" [10, 26, 23, 13, 3, 6, 19, 17, 8, 1, 15, 21, 4, 12, 24, 25, 11, 5, 20, 16, 9, 27, 22, 14, 2, 7, 18]\n",
" [12, 4, 21, 15, 10, 26, 23, 13, 3, 6, 19, 17, 8, 1, 24, 25, 11, 5, 20, 16, 9, 27, 22, 14, 2, 7, 18]\n",
" [12, 13, 3, 6, 19, 17, 8, 1, 24, 25, 11, 14, 22, 27, 9, 16, 20, 5, 4, 21, 15, 10, 26, 23, 2, 7, 18]\n",
" [12, 13, 23, 26, 10, 15, 21, 4, 5, 20, 16, 9, 27, 22, 3, 6, 19, 17, 8, 1, 24, 25, 11, 14, 2, 7, 18]\n",
" [14, 11, 25, 24, 12, 13, 3, 22, 27, 9, 16, 20, 5, 4, 21, 15, 1, 8, 17, 19, 6, 10, 26, 23, 2, 7, 18]\n",
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" [15, 21, 4, 5, 20, 16, 9, 27, 22, 3, 1, 8, 17, 19, 6, 10, 26, 23, 13, 12, 24, 25, 11, 14, 2, 7, 18]\n",
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" [16, 20, 5, 4, 21, 15, 1, 8, 17, 19, 6, 10, 26, 23, 2, 14, 11, 25, 24, 12, 13, 3, 22, 27, 9, 7, 18]\n",
" [16, 20, 5, 11, 25, 24, 1, 8, 17, 19, 6, 3, 13, 12, 4, 21, 15, 10, 26, 23, 2, 14, 22, 27, 9, 7, 18]\n",
" [18, 7, 2, 14, 11, 25, 24, 12, 13, 23, 26, 10, 15, 21, 4, 5, 20, 16, 9, 27, 22, 3, 6, 19, 17, 8, 1]\n",
" [18, 7, 2, 14, 22, 27, 9, 16, 20, 5, 11, 25, 24, 12, 4, 21, 15, 10, 26, 23, 13, 3, 6, 19, 17, 8, 1]\n",
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" [18, 7, 2, 23, 26, 10, 6, 19, 17, 8, 1, 15, 21, 4, 5, 20, 16, 9, 27, 22, 14, 11, 25, 24, 12, 13, 3]\n",
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" [18, 7, 2, 23, 26, 10, 15, 21, 4, 12, 13, 3, 6, 19, 17, 8, 1, 24, 25, 11, 14, 22, 27, 9, 16, 20, 5]\n",
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" [18, 7, 2, 14, 22, 27, 9, 16, 20, 5, 11, 25, 24, 1, 8, 17, 19, 6, 3, 13, 23, 26, 10, 15, 21, 4, 12]\n",
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" [18, 7, 2, 14, 11, 25, 24, 12, 13, 23, 26, 10, 6, 19, 17, 8, 1, 3, 22, 27, 9, 16, 20, 5, 4, 21, 15]\n",
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" [18, 7, 9, 27, 22, 3, 13, 12, 24, 25, 11, 14, 2, 23, 26, 10, 6, 19, 17, 8, 1, 15, 21, 4, 5, 20, 16]\n",
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" [18, 7, 2, 14, 11, 25, 24, 12, 13, 23, 26, 10, 6, 3, 22, 27, 9, 16, 20, 5, 4, 21, 15, 1, 8, 17, 19]\n",
" [18, 7, 2, 23, 26, 10, 6, 3, 13, 12, 24, 25, 11, 14, 22, 27, 9, 16, 20, 5, 4, 21, 15, 1, 8, 17, 19]\n",
" [18, 7, 2, 14, 11, 25, 24, 1, 8, 17, 19, 6, 3, 22, 27, 9, 16, 20, 5, 4, 12, 13, 23, 26, 10, 15, 21]\n",
" [18, 7, 2, 14, 11, 25, 24, 12, 4, 5, 20, 16, 9, 27, 22, 3, 13, 23, 26, 10, 6, 19, 17, 8, 1, 15, 21]\n",
" [18, 7, 2, 14, 11, 25, 24, 12, 13, 23, 26, 10, 15, 1, 8, 17, 19, 6, 3, 22, 27, 9, 16, 20, 5, 4, 21]\n",
" [18, 7, 2, 23, 26, 10, 15, 1, 8, 17, 19, 6, 3, 13, 12, 24, 25, 11, 14, 22, 27, 9, 16, 20, 5, 4, 21]\n",
" [18, 7, 2, 14, 11, 25, 24, 12, 13, 3, 22, 27, 9, 16, 20, 5, 4, 21, 15, 1, 8, 17, 19, 6, 10, 26, 23]\n",
" [18, 7, 2, 14, 22, 27, 9, 16, 20, 5, 11, 25, 24, 1, 8, 17, 19, 6, 3, 13, 12, 4, 21, 15, 10, 26, 23]\n",
" [18, 7, 2, 14, 11, 5, 20, 16, 9, 27, 22, 3, 13, 23, 26, 10, 6, 19, 17, 8, 1, 15, 21, 4, 12, 24, 25]\n",
" [18, 7, 2, 14, 22, 27, 9, 16, 20, 5, 11, 25, 24, 12, 4, 21, 15, 10, 6, 19, 17, 8, 1, 3, 13, 23, 26]\n",
" [18, 7, 2, 14, 22, 3, 13, 23, 26, 10, 6, 19, 17, 8, 1, 15, 21, 4, 12, 24, 25, 11, 5, 20, 16, 9, 27]\n",
" [18, 7, 9, 16, 20, 5, 4, 21, 15, 1, 8, 17, 19, 6, 10, 26, 23, 2, 14, 11, 25, 24, 12, 13, 3, 22, 27]\n",
" [18, 7, 9, 16, 20, 5, 11, 25, 24, 1, 8, 17, 19, 6, 3, 13, 12, 4, 21, 15, 10, 26, 23, 2, 14, 22, 27]\n",
" [19, 17, 8, 1, 15, 21, 4, 5, 20, 16, 9, 27, 22, 3, 6, 10, 26, 23, 13, 12, 24, 25, 11, 14, 2, 7, 18]\n",
" [19, 17, 8, 1, 15, 21, 4, 5, 20, 16, 9, 27, 22, 14, 11, 25, 24, 12, 13, 3, 6, 10, 26, 23, 2, 7, 18]\n",
" [21, 4, 5, 20, 16, 9, 27, 22, 3, 6, 19, 17, 8, 1, 15, 10, 26, 23, 13, 12, 24, 25, 11, 14, 2, 7, 18]\n",
" [21, 4, 5, 20, 16, 9, 27, 22, 14, 11, 25, 24, 12, 13, 3, 6, 19, 17, 8, 1, 15, 10, 26, 23, 2, 7, 18]\n",
" [21, 15, 1, 8, 17, 19, 6, 10, 26, 23, 13, 3, 22, 27, 9, 16, 20, 5, 4, 12, 24, 25, 11, 14, 2, 7, 18]\n",
" [21, 15, 10, 26, 23, 13, 12, 4, 5, 20, 16, 9, 27, 22, 3, 6, 19, 17, 8, 1, 24, 25, 11, 14, 2, 7, 18]\n",
" [23, 26, 10, 6, 19, 17, 8, 1, 15, 21, 4, 5, 20, 16, 9, 27, 22, 3, 13, 12, 24, 25, 11, 14, 2, 7, 18]\n",
" [23, 26, 10, 15, 21, 4, 12, 13, 3, 6, 19, 17, 8, 1, 24, 25, 11, 5, 20, 16, 9, 27, 22, 14, 2, 7, 18]\n",
" [25, 24, 12, 4, 21, 15, 1, 8, 17, 19, 6, 10, 26, 23, 13, 3, 22, 27, 9, 16, 20, 5, 11, 14, 2, 7, 18]\n",
" [26, 23, 13, 3, 1, 8, 17, 19, 6, 10, 15, 21, 4, 12, 24, 25, 11, 5, 20, 16, 9, 27, 22, 14, 2, 7, 18]\n",
" [27, 9, 16, 20, 5, 11, 25, 24, 12, 4, 21, 15, 1, 8, 17, 19, 6, 10, 26, 23, 13, 3, 22, 14, 2, 7, 18]\n",
" [27, 22, 3, 13, 12, 24, 25, 11, 14, 2, 23, 26, 10, 6, 19, 17, 8, 1, 15, 21, 4, 5, 20, 16, 9, 7, 18]\n",
" [27, 22, 14, 2, 23, 26, 10, 15, 21, 4, 12, 13, 3, 6, 19, 17, 8, 1, 24, 25, 11, 5, 20, 16, 9, 7, 18]\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"(28, 104, 18.65600000001723)\n",
" [1, 15, 10, 26, 23, 13, 3, 6, 19, 17, 8, 28, 21, 4, 12, 24, 25, 11, 5, 20, 16, 9, 27, 22, 14, 2, 7, 18]\n",
" [1, 15, 21, 28, 8, 17, 19, 6, 10, 26, 23, 13, 3, 22, 27, 9, 16, 20, 5, 4, 12, 24, 25, 11, 14, 2, 7, 18]\n",
" [1, 24, 25, 11, 14, 22, 27, 9, 16, 20, 5, 4, 12, 13, 3, 6, 19, 17, 8, 28, 21, 15, 10, 26, 23, 2, 7, 18]\n",
" [3, 6, 19, 17, 8, 28, 21, 4, 12, 13, 23, 26, 10, 15, 1, 24, 25, 11, 5, 20, 16, 9, 27, 22, 14, 2, 7, 18]\n",
" [3, 13, 12, 4, 5, 20, 16, 9, 27, 22, 14, 11, 25, 24, 1, 15, 21, 28, 8, 17, 19, 6, 10, 26, 23, 2, 7, 18]\n",
" [3, 22, 27, 9, 16, 20, 5, 4, 12, 13, 23, 26, 10, 6, 19, 17, 8, 28, 21, 15, 1, 24, 25, 11, 14, 2, 7, 18]\n",
" [4, 5, 20, 16, 9, 27, 22, 3, 1, 15, 21, 28, 8, 17, 19, 6, 10, 26, 23, 13, 12, 24, 25, 11, 14, 2, 7, 18]\n",
" [4, 5, 20, 16, 9, 27, 22, 14, 11, 25, 24, 12, 13, 3, 1, 15, 21, 28, 8, 17, 19, 6, 10, 26, 23, 2, 7, 18]\n",
" [4, 12, 13, 23, 26, 10, 15, 21, 28, 8, 17, 19, 6, 3, 1, 24, 25, 11, 5, 20, 16, 9, 27, 22, 14, 2, 7, 18]\n",
" [4, 21, 28, 8, 17, 19, 6, 3, 1, 15, 10, 26, 23, 13, 12, 24, 25, 11, 5, 20, 16, 9, 27, 22, 14, 2, 7, 18]\n",
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]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"(29, 38, 36.29700000002049)\n",
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]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"(31, 698, 109.07799999997951)\n",
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]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
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]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"(32, 784, 210.4220000000205)\n",
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]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
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" [32, 4, 12, 13, 3, 1, 24, 25, 11, 5, 31, 18, 7, 29, 20, 16, 9, 27, 22, 14, 2, 23, 26, 10, 15, 21, 28, 8, 17, 19, 6, 30]\n",
" [32, 4, 12, 13, 3, 6, 10, 26, 23, 2, 14, 22, 27, 9, 16, 20, 29, 7, 18, 31, 5, 11, 25, 24, 1, 15, 21, 28, 8, 17, 19, 30]\n",
" [32, 4, 5, 11, 25, 24, 12, 13, 3, 1, 15, 21, 28, 8, 17, 19, 30, 6, 10, 26, 23, 2, 14, 22, 27, 9, 16, 20, 29, 7, 18, 31]\n",
" [32, 17, 19, 30, 6, 3, 13, 12, 4, 5, 11, 25, 24, 1, 8, 28, 21, 15, 10, 26, 23, 2, 14, 22, 27, 9, 16, 20, 29, 7, 18, 31]\n",
" [32, 17, 19, 30, 6, 3, 13, 12, 24, 25, 11, 5, 4, 21, 28, 8, 1, 15, 10, 26, 23, 2, 14, 22, 27, 9, 16, 20, 29, 7, 18, 31]\n"
]
}
],
"source": [
"import time\n",
"\n",
"results = []\n",
"for n in range(15,33):\n",
" g = make_graph(n)\n",
" start = time.monotonic()\n",
" paths = list(hamiltonian_paths(g))\n",
" dur = time.monotonic() - start\n",
" print((n, len(paths), dur))\n",
" for p in paths: print(\" \", p)"
]
}
],
"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": "3.9.1"
}
},
"nbformat": 4,
"nbformat_minor": 5
}
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