// Numbas version: finer_feedback_settings
{"name": "Permutations: two-line form", "extensions": ["permutations"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {"orderpossible": {"definition": "var factors = Numbas.math.factorise(n);\nvar t = 0;\nfor(var i=0;i<factors.length;i++) {\n  if(factors[i]) {\n    t += Math.pow(Numbas.math.primes[i],factors[i]);\n  }\n}\nreturn t<=s_n;", "type": "boolean", "language": "javascript", "parameters": [["n", "number"], ["s_n", "number"]]}, "randomcycle": {"definition": "var off = Numbas.math.randomint(n);\nvar d = Numbas.math.deal(n);\nwhile(d[off]==off) {\n  d = Numbas.math.deal(n);\n}\nvar out = [];\nfor(var i = 0;i<n;i++) {\n  out[i]=i;\n}\nvar old = off;\nvar i = off;\nwhile((i=d[i])!=off) {\n  out[old]=i;\n  old = i;\n}\nout[old] = i;\nreturn out;", "type": "list", "language": "javascript", "parameters": [["n", "number"]]}, "show_factorisation": {"definition": "var factors = Numbas.math.factorise(n);\nvar out = '';\nfor(var i=0;i<factors.length;i++) {\n  if(factors[i]) {\n    if(out.length) {\n      out += ' \\\\cdot ';\n    }\n    out += Numbas.math.primes[i];\n    if(factors[i]>1) {\n      out += '^{'+factors[i]+'}';\n    }\n  }\n}\nreturn out;", "type": "string", "language": "javascript", "parameters": [["n", "number"]]}, "sum": {"definition": "var t = 0;\nfor(var i=0;i<l.length;i++) {\n  t = Numbas.math.add(t,l[i]);\n}\nreturn t;", "type": "number", "language": "javascript", "parameters": [["l", "list"]]}, "permwithorder": {"definition": "var factors = Numbas.math.factorise(n);\nvar off = 0;\nvar out = [];\nfor(var i=0;i<factors.length;i++) {\n  if(factors[i]>0) {\n    var cycle_length = Math.pow(Numbas.math.primes[i],factors[i]);\n    for(var j=0;j<cycle_length-1;j++) {\n      out[off+j]=off+j+1;\n    }\n    out[off+cycle_length-1]=off;\n    off = off+cycle_length;\n  }\n}\nreturn out;", "type": "list", "language": "javascript", "parameters": [["n", "number"]]}, "primepowerfactors": {"definition": "var factors = Numbas.math.factorise(n);\nvar out = [];\nfor(var i=0;i<factors.length;i++) {\n  if(factors[i]) {\n    out.push(Math.pow(Numbas.math.primes[i],factors[i]));\n  }\n}\nreturn out;", "type": "list", "language": "javascript", "parameters": [["n", "number"]]}, "showlist": {"definition": "var out = '';\nfor(var i=0;i<l.length;i++) {\n  if(i==l.length-1 && l.length>1) {\n    out += ' and ';\n  } else if(i>0) {\n    out += ', ';\n  }\n  out += '$'+l[i]+'$';\n}\nreturn out;", "type": "string", "language": "javascript", "parameters": [["l", "list"]]}, "permfrompartition": {"definition": "var to = [];\nvar offset = 0;\nfor(var i=0; i<cycle_sizes.length; i++) {\n  var o_offset = offset;\n  for(var j=0;j<cycle_sizes[i]-1;j++) {\n    to[offset] = offset+1;\n    offset += 1;\n  }\n  to[offset] = o_offset;\n  o_offset = offset;\n  offset += 1;\n}\nreturn to;", "type": "list", "language": "javascript", "parameters": [["cycle_sizes", "list"]]}, "showsum": {"definition": "var t = 0;\nvar out = '';\nfor(var i=0;i<l.length;i++) {\n  if(i>0) {\n    out += ' + ';\n  }\n  out += l[i]\n  t += l[i];\n}\nout += ' = '+t;\nreturn out;", "type": "string", "language": "javascript", "parameters": [["l", "list"]]}, "randompartition": {"definition": "var l = [];\nwhile(n) {\n  var t = Numbas.math.randomint(n)+1;\n  l.push(t);\n  n -= t;\n}\nreturn l;", "type": "list", "language": "javascript", "parameters": [["n", "number"]]}, "tracemap": {"definition": "n -= 1;\nvar l = perms.length;\nvar out = ''+(n+1);\nfor(var i=l-1;i>=0;i--) {\n  n = perms[i].map(n);\n  out += ' \\\\to '+(n+1);\n}\nreturn out;", "type": "string", "language": "javascript", "parameters": [["n", "number"], ["perms", "list"]]}}, "name": "Permutations: two-line form", "tags": ["permutations"], "type": "question", "advice": "<p>Each number on the top row is mapped to the number directly beneath it. To find a disjoint cycle, pick a number and trace it through repeated applications of the permutation until you get back to the number you started with. Do this for each number to find every disjoint cycle.</p>\n<p>The order of the permutation $\\pi$ is the lowest common multiple of the orders of its disjoint cycles.</p>\n<p data-bind=\"visible:question.scope.variables.ppi_num_cycles.value>1\">There are $\\var{ppi_num_cycles}$ nontrivial disjoint cycles, of lengths {showlist(ppi_cycle_lengths)}. The lowest common multiple of those numbers is $\\var{lcm(ppi_cycle_lengths)}$, so the order of $\\pi$ is $\\var{lcm(ppi_cycle_lengths)}$.</p>\n<p data-bind=\"visible:question.scope.variables.ppi_num_cycles.value==1\">There is one nontrivial disjoint cycle, of length $\\var{ppi_cycle_lengths[0]}$, so the order of $\\pi$ is $\\var{lcm(ppi_cycle_lengths)}$.</p>\n<p>A permutation is <em>even</em> if it can be written as the product of an even number of transpositions. This is true if and only if, when decomposed into a product of disjoint cycles, there are an even number of cycles of even length. $\\pi$ has $\\var{ppi_num_even_cycles}$ {pluralise(ppi_num_even_cycles,'cycle','cycles')} of even length, so is an <em>{if(even(ppi),'even','odd')}</em> permutation.</p>", "rulesets": {}, "parts": [{"prompt": "<p>Write $\\pi$ and $\\pi^{-1}$ as a product of disjoint cycles.</p>\n<p>$\\pi = $ [[0]]</p>\n<p>$\\pi^{-1} = $ [[1]]</p>", "marks": 0, "gaps": [{"marks": 1, "showCorrectAnswer": true, "displayAnswer": "{show(ppi)}", "scripts": {"validate": {"order": "instead", "script": "var Permutation = Numbas.extensions.permutations.Permutation;\ntry{\n  Permutation.parse(this.studentAnswer);\n} catch(e) {\n  this.giveWarning(\"Invalid notation\");\n  return false;\n}\nreturn true;"}, "mark": {"order": "instead", "script": "var Permutation = Numbas.extensions.permutations.Permutation;\ntry {\n  var p = Permutation.fromString(this.studentAnswer);\n} catch(e) {\n  this.setCredit(0,\"Invalid notation\");\n  return;\n}\nif(!Permutation.is_disjoint(this.studentAnswer)) {\n  this.setCredit(0,\"Your cycles are not disjoint\");\n}\nvar unwrapValue = Numbas.jme.unwrapValue;\nvar prod = unwrapValue(question.scope.variables.ppi);\nif(p.equal(prod)) {\n  this.setCredit(1,\"Correct!\");\n} else {\n  this.setCredit(0,\"Wrong\");\n}"}}, "answer": "", "type": "patternmatch"}, {"marks": 1, "showCorrectAnswer": true, "displayAnswer": "{show(ppi)}", "scripts": {"validate": {"order": "instead", "script": "var Permutation = Numbas.extensions.permutations.Permutation;\ntry{\n  Permutation.parse(this.studentAnswer);\n} catch(e) {\n  this.giveWarning(\"Invalid notation\");\n  return false;\n}\nreturn true;"}, "mark": {"order": "instead", "script": "var Permutation = Numbas.extensions.permutations.Permutation;\ntry {\n  var p = Permutation.fromString(this.studentAnswer);\n} catch(e) {\n  this.setCredit(0,\"Invalid notation\");\n  return;\n}\nif(!Permutation.is_disjoint(this.studentAnswer)) {\n  this.setCredit(0,\"Your cycles are not disjoint\");\n}\nvar unwrapValue = Numbas.jme.unwrapValue;\nvar prod = unwrapValue(question.scope.variables.ppi);\nprod = prod.inverse();\nif(p.equal(prod)) {\n  this.setCredit(1,\"Correct!\");\n} else {\n  this.setCredit(0,\"Wrong\");\n}"}}, "answer": "", "type": "patternmatch"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "<p>What is the order of $\\pi$? [[0]]</p>", "marks": 0, "gaps": [{"marks": 1, "maxValue": "order(ppi)", "minValue": "order(ppi)", "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "<p>$\\pi$ is [[0]]</p>", "marks": 0, "gaps": [{"maxAnswers": 0, "displayColumns": 0, "matrix": "if(even(ppi),[1,0],[0,1])", "minAnswers": 0, "shuffleChoices": false, "choices": ["<p>an even permutation</p>", "<p>an odd permutation</p>"], "displayType": "radiogroup", "maxMarks": 0, "scripts": {}, "distractors": ["", ""], "marks": 0, "showCorrectAnswer": true, "type": "1_n_2", "minMarks": 0}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}], "statement": "<p>\\[ \\pi = \\var{twoline(ppi)} \\]</p>", "variable_groups": [], "progress": "in-progress", "preamble": {"css": "", "js": ""}, "variables": {"ppi_num_even_cycles": {"definition": "sum(map(1-mod(x,2),x,ppi_cycle_lengths))", "templateType": "anything", "group": "Ungrouped variables", "name": "ppi_num_even_cycles", "description": ""}, "ppi_num_cycles": {"definition": "len(nontrivial_cycles(ppi))", "templateType": "anything", "group": "Ungrouped variables", "name": "ppi_num_cycles", "description": ""}, "ppi": {"definition": "perm(deal(random(6..11)))", "templateType": "anything", "group": "Ungrouped variables", "name": "ppi", "description": ""}, "ppi_cycle_lengths": {"definition": "map(len(x),x,nontrivial_cycles(ppi))", "templateType": "anything", "group": "Ungrouped variables", "name": "ppi_cycle_lengths", "description": ""}}, "metadata": {"notes": "", "description": "", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}]}