{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Rubik's 3x3x3 Cube ##"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "#RC3\n",
    "S48=SymmetricGroup(48)\n",
    "R=S48(\"(25,27,32,30)(26,29,31,28)(3,38,43,19)(5,36,45,21)(8,33,48,24)\")\n",
    "L=S48(\"(9,11,16,14)(10,13,15,12)(1,17,41,40)(4,20,44,37)(6,22,46,35)\")\n",
    "U=S48(\"(1,3,8,6)(2,5,7,4)(9,33,25,17)(10,34,26,18)(11,35,27,19)\")\n",
    "D=S48(\"(41,43,48,46)(42,45,47,44)(14,22,30,38)(15,23,31,39)(16,24,32,40)\")\n",
    "F=S48(\"(17,19,24,22)(18,21,23,20)(6,25,43,16)(7,28,42,13)(8,30,41,11)\")\n",
    "B=S48(\"(33,35,40,38)(34,37,39,36)(3,9,46,32)(2,12,47,29)(1,14,48,27)\")\n",
    "RC3=S48.subgroup([R,L,U,D,F,B])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "S48(\"(7,18)\") in RC3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "S48(\"(7,18)(5,26)\") in RC3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Sage has a built in cube solver:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\"F2 R2 B' F' D' F D B R2 F' R' F' R\""
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rubik=CubeGroup();\n",
    "state = rubik(\"(7,18)(5,26)\")  # current state of the cube\n",
    "rubik.solve(state)             # calls the solve algorithm"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Rubik's 2x2x2 Cube ##"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "#RC2\n",
    "S24=SymmetricGroup(24)\n",
    "R=S24(\"(13,14,16,15)(10,2,19,22)(12,4,17,24)\") \n",
    "L=S24(\"(5,6,8,7)(3,11,23,18)(1,9,21,20)\")\n",
    "U=S24(\"(1,2,4,3)(9,5,17,13)(10,6,18,14)\")\n",
    "D=S24(\"(21,22,24,23)(11,15,19,7)(12,16,20,8)\") \n",
    "F=S24(\"(9,10,12,11),(3,13,22,8),(4,15,21,6)\")\n",
    "B=S24(\"(17,18,20,19),(1,7,24,14),(2,5,23,16)\")\n",
    "RC2=S24.subgroup([R,D,F])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3674160"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "RC2.order()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "H=S24.subgroup([R,D,F,U,L,B])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "88179840"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H.order()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "24"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H.order()/RC2.order()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Oval Track Puzzle ###"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "#OT\n",
    "S20=SymmetricGroup(20)\n",
    "R=S20(\"(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)\")\n",
    "T=S20(\"(1,4)(2,3)\")\n",
    "OT=S20.subgroup([R,T])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "OT.order()==factorial(20)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Hungarian Rings Puzzle ###"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "#HR\n",
    "S38=SymmetricGroup(38)\n",
    "L=S38(\"(1,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2)\")\n",
    "R=S38(\"(1,38,37,36,35,6,34,33,32,31,30,29,28,27,26,25,24,23,22,21)\")\n",
    "HR=S38.subgroup([L,R])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
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   "display_name": "SageMath 7.3",
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   "codemirror_mode": {
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