// Numbas version: finer_feedback_settings
{"name": "Write a permutation as a product of disjoint cycles", "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": "Write a permutation as a product of disjoint cycles", "tags": ["permutations"], "type": "question", "advice": "<p>Each set of brackets represents a <em>cycle</em>, with each number being mapped to the one on its right, and the last number in the sequence mapped to the first.</p>\n<p>Follow each number through the cycles, starting with the rightmost cycle and reading leftwards. For example, here's how $1$ is mapped to $\\var{prod[1]}$:</p>\n<p>\\[ \\var{latex(tracemap(1,perms))} \\]</p>\n<p>Then repeat this for $\\var{prod[1]}$, until you get back to $1$. This is a single cycle. If any number is not included in this cycle, repeat this process starting at that number to get another, disjoint, cycle. Some numbers are mapped to themselves - these are trivial cycles and you don't need to write them down. Once you've covered every number, you should have the following:</p>\n<p>\\[ \\var{prod} \\]</p>\n", "rulesets": {}, "parts": [{"prompt": "<p>Write as a product of disjoint cycles</p>\n<p>\\[\\var{perms[0]} \\var{perms[1]} \\var{perms[2]} \\var{perms[3]} \\]</p>", "marks": 1, "showCorrectAnswer": true, "displayAnswer": "{show(prod)}", "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  return;\n}\nvar unwrapValue = Numbas.jme.unwrapValue;\nvar prod = unwrapValue(question.scope.variables.prod);\nif(p.equal(prod)) {\n  this.setCredit(1,\"Correct!\");\n} else {\n  this.setCredit(0,\"Wrong\");\n}"}}, "answer": "", "type": "patternmatch"}], "statement": "", "variable_groups": [], "progress": "in-progress", "preamble": {"css": "", "js": ""}, "variables": {"perms": {"definition": "repeat(perm(randomcycle(n)),4)", "templateType": "anything", "group": "Ungrouped variables", "name": "perms", "description": ""}, "prod": {"definition": "compose(perms[0],perms[1],perms[2],perms[3])", "templateType": "anything", "group": "Ungrouped variables", "name": "prod", "description": ""}, "n": {"definition": "random(4..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "n", "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/"}]}