// Numbas version: exam_results_page_options
{"name": "Group theory", "metadata": {"description": "<p>Questions used in a university course titled \"Group theory\".</p>\n<p>The questions deal with permutations written in cycle or two-line notation.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "duration": 0, "percentPass": 0, "showQuestionGroupNames": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", "", "", ""], "questions": [{"name": "Conjugate of permutations", "extensions": ["permutations"], "custom_part_types": [{"source": {"pk": 68, "author": {"name": "Christian Lawson-Perfect", "pk": 7}, "edit_page": "/part_type/68/edit"}, "name": "Write a permutation in cycle notation", "short_name": "write-a-permutation-in-cycle-notation", "description": "<p>The student must write a permutation in cycle notation. You can optionally require the cycles to be disjoint and/or all of length 2.</p>", "help_url": "", "input_widget": "string", "input_options": {"correctAnswer": "let(ans,\n  if(settings[\"must_be_transpositions\"],\n    as_transpositions(settings[\"correct_answer\"]),\n    show(settings[\"correct_answer\"])\n  ),\n  if(ans=\"\",settings[\"identity_name\"],ans)\n)", "hint": {"static": false, "value": "if(settings[\"show_hint\"],\n  if(settings[\"must_be_disjoint\"],\n    if(settings[\"must_be_transpositions\"],\n      \"Write a permutation in cycle notation as a product of disjoint transpositions.\",\n      \"Write a permutation in cycle notation as a product of disjoint cycles.\"\n    ),\n    if(settings[\"must_be_transpositions\"],\n      \"Write a permutation in cycle notation as a product of transpositions.\",\n      \"Write a permutation in cycle notation.\"\n    )\n  ),\n  \"\"\n)"}, "allowEmpty": {"static": true, "value": false}}, "can_be_gap": true, "can_be_step": true, "marking_script": "mark:\napply(student_permutation);\napply(is_correct);\napply(is_disjoint);\napply(is_transpositions)\n\ninterpreted_answer:\nstudent_permutation\n\nis_correct:\ncorrectif(student_permutation=settings[\"correct_answer\"])\n\nis_disjoint:\nif(is_disjoint(studentAnswer),\n  true\n,\n  assert(not settings[\"must_be_disjoint\"],\n    set_credit(0,\"You have not written your answer as a product of disjoint cycles.\"),\n  );\n  false\n)\n\nstudent_permutation:\ntry(\n  if(studentAnswer=settings[\"identity_name\"],\n    permutation(\"\"), \n    permutation(studentAnswer)\n  ),\n  message,\n    warn(\"Your answer is not valid cycle notation: \"+message);\n    fail(\"Your answer is not valid cycle notation: \"+message)\n)\n\nis_transpositions:\nif(is_transpositions(studentAnswer),\n  true\n,\n  assert(not settings[\"must_be_transpositions\"],\n    set_credit(0,\"You have not written your answer as a product of transpositions.\")\n  );\n  false\n)", "marking_notes": [{"name": "mark", "description": "This is the main marking note. It should award credit and provide feedback based on the student's answer.", "definition": "apply(student_permutation);\napply(is_correct);\napply(is_disjoint);\napply(is_transpositions)"}, {"name": "interpreted_answer", "description": "A value representing the student's answer to this part.", "definition": "student_permutation"}, {"name": "is_correct", "description": "", "definition": "correctif(student_permutation=settings[\"correct_answer\"])"}, {"name": "is_disjoint", "description": "", "definition": "if(is_disjoint(studentAnswer),\n  true\n,\n  assert(not settings[\"must_be_disjoint\"],\n    set_credit(0,\"You have not written your answer as a product of disjoint cycles.\"),\n  );\n  false\n)"}, {"name": "student_permutation", "description": "", "definition": "try(\n  if(studentAnswer=settings[\"identity_name\"],\n    permutation(\"\"), \n    permutation(studentAnswer)\n  ),\n  message,\n    warn(\"Your answer is not valid cycle notation: \"+message);\n    fail(\"Your answer is not valid cycle notation: \"+message)\n)"}, {"name": "is_transpositions", "description": "", "definition": "if(is_transpositions(studentAnswer),\n  true\n,\n  assert(not settings[\"must_be_transpositions\"],\n    set_credit(0,\"You have not written your answer as a product of transpositions.\")\n  );\n  false\n)"}], "settings": [{"name": "correct_answer", "label": "Correct answer", "help_url": "", "hint": "", "input_type": "code", "default_value": "", "evaluate": true}, {"name": "show_hint", "label": "Show answer input hint?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": true}, {"name": "must_be_disjoint", "label": "Must the cycles be disjoint?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": false}, {"name": "must_be_transpositions", "label": "Must the cycles be transpositions?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": false}, {"name": "identity_name", "label": "Name of the identity permutation", "help_url": "", "hint": "The student should write this string for the identity permutation.", "input_type": "string", "default_value": "e", "subvars": true}], "public_availability": "always", "published": true, "extensions": ["permutations"]}], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "tags": ["permutations"], "metadata": {"description": "", "licence": "None specified"}, "statement": "", "advice": "<p>You could find $\\tau \\sigma \\tau^{-1}$ by first finding $\\tau^{-1}$ and then performing the composition by hand, but there's a quicker way: $\\tau \\sigma \\tau^{-1}$ has the same cycle structure as $\\sigma$, but each $n$ is replaced by $\\tau(n)$.</p>\n<h5>Proof</h5>\n<p>Assume, without loss of generality, that $\\sigma$ is a single cycle $(b_1, \\dots, b_n)$, and let $\\pi = \\tau \\sigma \\tau^{-1}$. Look at what happens to $\\tau(b_i)$ when acted on by $\\sigma$:</p>\n<p>\\[ \\tau(b_i) \\overset{\\tau^{-1}}\\mapsto b_i \\overset{\\sigma}\\mapsto b_{i+1} \\overset{\\tau}\\mapsto \\tau(b_{i+1}) \\]</p>\n<p>So, for any $1 \\lt i \\leq n$, $\\tau(b_i) \\overset{\\pi}\\mapsto \\tau(b_{i+1})$.</p>\n<p>For any $x \\notin \\{b_1,\\dots,b_n\\}$, $\\sigma(x) = x$, so $\\pi(x) = x$.</p>\n<p>Hence, $\\pi$ is the cycle $(\\tau(b_1), \\dots, \\tau(b_n))$.</p>\n<h4>a)</h4>\n<p>$\\var{conjugate1}$</p>\n<h4>b)</h4>\n<p>$\\var{conjugate2}$</p>\n", "rulesets": {}, "variables": {"sigma1": {"name": "sigma1", "group": "Ungrouped variables", "definition": "perm(randomcycle(7))", "description": "", "templateType": "anything"}, "conjugate1": {"name": "conjugate1", "group": "Ungrouped variables", "definition": "tau1*sigma1*inverse(tau1)", "description": "", "templateType": "anything"}, "tau1": {"name": "tau1", "group": "Ungrouped variables", "definition": "perm(deal(random(5..11)))", "description": "", "templateType": "anything"}, "tau2": {"name": "tau2", "group": "Ungrouped variables", "definition": "perm(deal(random(5..11)))", "description": "", "templateType": "anything"}, "conjugate2": {"name": "conjugate2", "group": "Ungrouped variables", "definition": "tau2*sigma2*inverse(tau2)", "description": "", "templateType": "anything"}, "sigma2": {"name": "sigma2", "group": "Ungrouped variables", "definition": "perm(deal(random(5..11)))", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["tau1", "tau2", "sigma1", "sigma2", "conjugate1", "conjugate2"], "variable_groups": [], "functions": {"randompartition": {"parameters": [["n", "number"]], "type": "list", "language": "javascript", "definition": "var l = [];\nwhile(n) {\n  var t = Numbas.math.randomint(n)+1;\n  l.push(t);\n  n -= t;\n}\nreturn l;"}, "tracemap": {"parameters": [["n", "number"], ["perms", "list"]], "type": "string", "language": "javascript", "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;"}, "sum": {"parameters": [["l", "list"]], "type": "number", "language": "javascript", "definition": "var t = 0;\nfor(var i=0;i<l.length;i++) {\n  t = Numbas.math.add(t,l[i]);\n}\nreturn t;"}, "permwithorder": {"parameters": [["n", "number"]], "type": "list", "language": "javascript", "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;"}, "randomcycle": {"parameters": [["n", "number"]], "type": "list", "language": "javascript", "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;"}, "primepowerfactors": {"parameters": [["n", "number"]], "type": "list", "language": "javascript", "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;"}, "showlist": {"parameters": [["l", "list"]], "type": "string", "language": "javascript", "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;"}, "orderpossible": {"parameters": [["n", "number"], ["s_n", "number"]], "type": "boolean", "language": "javascript", "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;"}, "permfrompartition": {"parameters": [["cycle_sizes", "list"]], "type": "list", "language": "javascript", "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;"}, "showsum": {"parameters": [["l", "list"]], "type": "string", "language": "javascript", "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;"}, "show_factorisation": {"parameters": [["n", "number"]], "type": "string", "language": "javascript", "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;"}}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Where $\\sigma = \\var{sigma1}$ and $\\tau = \\var{tau1}$, write down the permutation $\\tau \\sigma \\tau^{-1}$.</p>\n<p>$\\tau \\sigma \\tau^{-1}$ = [[0]]</p>", "gaps": [{"type": "write-a-permutation-in-cycle-notation", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correct_answer": "conjugate1", "must_be_disjoint": false, "must_be_transpositions": false}}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Where $\\sigma = \\var{sigma2}$ and $\\tau = \\var{tau2}$, write down the permutation $\\tau \\sigma \\tau^{-1}$.</p>\n<p>$\\tau \\sigma \\tau^{-1}$ = [[0]]\n</p>", "gaps": [{"type": "write-a-permutation-in-cycle-notation", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correct_answer": "conjugate2", "must_be_disjoint": false, "must_be_transpositions": false}}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Permutations of a given order", "extensions": ["permutations"], "custom_part_types": [{"source": {"pk": 68, "author": {"name": "Christian Lawson-Perfect", "pk": 7}, "edit_page": "/part_type/68/edit"}, "name": "Write a permutation in cycle notation", "short_name": "write-a-permutation-in-cycle-notation", "description": "<p>The student must write a permutation in cycle notation. You can optionally require the cycles to be disjoint and/or all of length 2.</p>", "help_url": "", "input_widget": "string", "input_options": {"correctAnswer": "let(ans,\n  if(settings[\"must_be_transpositions\"],\n    as_transpositions(settings[\"correct_answer\"]),\n    show(settings[\"correct_answer\"])\n  ),\n  if(ans=\"\",settings[\"identity_name\"],ans)\n)", "hint": {"static": false, "value": "if(settings[\"show_hint\"],\n  if(settings[\"must_be_disjoint\"],\n    if(settings[\"must_be_transpositions\"],\n      \"Write a permutation in cycle notation as a product of disjoint transpositions.\",\n      \"Write a permutation in cycle notation as a product of disjoint cycles.\"\n    ),\n    if(settings[\"must_be_transpositions\"],\n      \"Write a permutation in cycle notation as a product of transpositions.\",\n      \"Write a permutation in cycle notation.\"\n    )\n  ),\n  \"\"\n)"}, "allowEmpty": {"static": true, "value": false}}, "can_be_gap": true, "can_be_step": true, "marking_script": "mark:\napply(student_permutation);\napply(is_correct);\napply(is_disjoint);\napply(is_transpositions)\n\ninterpreted_answer:\nstudent_permutation\n\nis_correct:\ncorrectif(student_permutation=settings[\"correct_answer\"])\n\nis_disjoint:\nif(is_disjoint(studentAnswer),\n  true\n,\n  assert(not settings[\"must_be_disjoint\"],\n    set_credit(0,\"You have not written your answer as a product of disjoint cycles.\"),\n  );\n  false\n)\n\nstudent_permutation:\ntry(\n  if(studentAnswer=settings[\"identity_name\"],\n    permutation(\"\"), \n    permutation(studentAnswer)\n  ),\n  message,\n    warn(\"Your answer is not valid cycle notation: \"+message);\n    fail(\"Your answer is not valid cycle notation: \"+message)\n)\n\nis_transpositions:\nif(is_transpositions(studentAnswer),\n  true\n,\n  assert(not settings[\"must_be_transpositions\"],\n    set_credit(0,\"You have not written your answer as a product of transpositions.\")\n  );\n  false\n)", "marking_notes": [{"name": "mark", "description": "This is the main marking note. It should award credit and provide feedback based on the student's answer.", "definition": "apply(student_permutation);\napply(is_correct);\napply(is_disjoint);\napply(is_transpositions)"}, {"name": "interpreted_answer", "description": "A value representing the student's answer to this part.", "definition": "student_permutation"}, {"name": "is_correct", "description": "", "definition": "correctif(student_permutation=settings[\"correct_answer\"])"}, {"name": "is_disjoint", "description": "", "definition": "if(is_disjoint(studentAnswer),\n  true\n,\n  assert(not settings[\"must_be_disjoint\"],\n    set_credit(0,\"You have not written your answer as a product of disjoint cycles.\"),\n  );\n  false\n)"}, {"name": "student_permutation", "description": "", "definition": "try(\n  if(studentAnswer=settings[\"identity_name\"],\n    permutation(\"\"), \n    permutation(studentAnswer)\n  ),\n  message,\n    warn(\"Your answer is not valid cycle notation: \"+message);\n    fail(\"Your answer is not valid cycle notation: \"+message)\n)"}, {"name": "is_transpositions", "description": "", "definition": "if(is_transpositions(studentAnswer),\n  true\n,\n  assert(not settings[\"must_be_transpositions\"],\n    set_credit(0,\"You have not written your answer as a product of transpositions.\")\n  );\n  false\n)"}], "settings": [{"name": "correct_answer", "label": "Correct answer", "help_url": "", "hint": "", "input_type": "code", "default_value": "", "evaluate": true}, {"name": "show_hint", "label": "Show answer input hint?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": true}, {"name": "must_be_disjoint", "label": "Must the cycles be disjoint?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": false}, {"name": "must_be_transpositions", "label": "Must the cycles be transpositions?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": false}, {"name": "identity_name", "label": "Name of the identity permutation", "help_url": "", "hint": "The student should write this string for the identity permutation.", "input_type": "string", "default_value": "e", "subvars": true}], "public_availability": "always", "published": true, "extensions": ["permutations"]}], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "tags": ["permutations"], "metadata": {"description": "", "licence": "None specified"}, "statement": "", "advice": "<h4>\n  <span style=\"font-size: 1em;\">a)</span>\n</h4>\n<p>Note that $\\var{s_order} = \\var{latex(show_factorisation(s_order))}$. The order of a permutation is the lowest common multiple of the lengths of its disjoint cycles, so it suffices to write a permutation with {pluralise(len(factorise(s_order)),'a cycle of length','cycles of lengths')} {showlist(primepowerfactors(s_order))}.</p>\n<p>One such permutation is $\\var{permutation(permwithorder(s_order))}$.</p>\n<h4>b)</h4>\n<p>There exists an element of $S_\\var{s_n}$ of order $x$ if there exist integers $(a_1,\\dots,a_n)$ whose lowest common multiple is $x$ and whose sum is less than or equal to $\\var{s_n}$. You can establish whether this is the case by looking at the prime factorisation of $x$ - the LCM of the prime power factors of $x$ is $x$, so you just need to check that they sum to at most $\\var{s_n}$.</p>\n", "rulesets": {}, "variables": {"example_element": {"name": "example_element", "group": "Ungrouped variables", "definition": "let(\n  p,perm(permwithorder(s_order)),\n  d,s_n-size(p),\n  perm(map(\n      if(j<=d,j,d+p[j-d])-1,j,1..s_n))\n)", "description": "", "templateType": "anything"}, "candidate_possible": {"name": "candidate_possible", "group": "Ungrouped variables", "definition": "map(orderpossible(x,s_n),x,candidate_orders)", "description": "", "templateType": "anything"}, "max_order": {"name": "max_order", "group": "Ungrouped variables", "definition": "floor(e^(s_n/e))", "description": "", "templateType": "anything"}, "s_order": {"name": "s_order", "group": "Ungrouped variables", "definition": "lcm(partition)", "description": "", "templateType": "anything"}, "candidate_orders": {"name": "candidate_orders", "group": "Ungrouped variables", "definition": "distinct(possible_orders+other_orders)[0..6]", "description": "", "templateType": "anything"}, "s_n": {"name": "s_n", "group": "Ungrouped variables", "definition": "random(9..20)", "description": "", "templateType": "anything"}, "possible_orders": {"name": "possible_orders", "group": "Ungrouped variables", "definition": "map(lcm(x),x,possible_partitions) except s_order", "description": "", "templateType": "anything"}, "possible_partitions": {"name": "possible_partitions", "group": "Ungrouped variables", "definition": "repeat(randompartition(s_n),3)", "description": "", "templateType": "anything"}, "partition": {"name": "partition", "group": "Ungrouped variables", "definition": "randompartition(s_n)", "description": "", "templateType": "anything"}, "other_orders": {"name": "other_orders", "group": "Ungrouped variables", "definition": "map(x+1,x,deal(max_order)) except possible_orders", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": "100"}, "ungrouped_variables": ["possible_orders", "possible_partitions", "partition", "candidate_possible", "other_orders", "s_order", "candidate_orders", "max_order", "s_n", "example_element"], "variable_groups": [], "functions": {"randompartition": {"parameters": [["n", "number"]], "type": "list", "language": "javascript", "definition": "var l = [];\nwhile(n) {\n  var t = Numbas.math.randomint(n)+1;\n  l.push(t);\n  n -= t;\n}\nreturn l;"}, "tracemap": {"parameters": [["n", "number"], ["perms", "list"]], "type": "string", "language": "javascript", "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;"}, "show_factorisation": {"parameters": [["n", "number"]], "type": "string", "language": "javascript", "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;"}, "permfrompartition": {"parameters": [["cycle_sizes", "list"]], "type": "list", "language": "javascript", "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;"}, "permwithorder": {"parameters": [["n", "number"]], "type": "list", "language": "javascript", "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;"}, "randomcycle": {"parameters": [["n", "number"]], "type": "list", "language": "javascript", "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;"}, "primepowerfactors": {"parameters": [["n", "number"]], "type": "list", "language": "javascript", "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;"}, "showlist": {"parameters": [["l", "list"]], "type": "string", "language": "javascript", "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;"}, "orderpossible": {"parameters": [["n", "number"], ["s_n", "number"]], "type": "boolean", "language": "javascript", "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;"}, "sum": {"parameters": [["n", "list"]], "type": "number", "language": "javascript", "definition": "var t = 0;\nfor(var i=0;i<l.length;i++) {\n  t = Numbas.math.add(t,l[i]);\n}\nreturn t;"}, "showsum": {"parameters": [["l", "list"]], "type": "string", "language": "javascript", "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;"}}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Write down an element of $S_{\\var{s_n}}$ of order $\\var{s_order}$.</p>\n<p>[[0]]</p>", "gaps": [{"type": "write-a-permutation-in-cycle-notation", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "student_size: size(student_permutation)\n\nstudent_order: order(student_permutation)\n\nis_right_size:\n  assert(student_size=s_n,\n    incorrect(\"Your permutation belongs to $S_{\"+student_size+\"}$, not $S_{\"+s_n+\"}$.\");\n    end()\n  )\n\nhas_right_order:\n  assert(student_order=s_order,\n    incorrect(\"Your permutation is an element of $S_{\"+s_n+\"}$ but it has order \"+student_order+\", not \"+s_order);\n    end()\n  )\n\nmark:\n  apply(student_permutation);\n  apply(is_right_size);\n  apply(has_right_order);\n  correct()", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correct_answer": "example_element", "must_be_disjoint": false, "must_be_transpositions": false}}], "sortAnswers": false}, {"type": "m_n_x", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Which of the following are possible orders of elements of $S_{\\var{s_n}}$?</p>", "minMarks": "0", "maxMarks": "0", "minAnswers": "0", "maxAnswers": "0", "shuffleChoices": false, "shuffleAnswers": false, "displayType": "radiogroup", "warningType": "none", "showCellAnswerState": true, "choices": ["$\\var{candidate_orders[0]}$", "$\\var{candidate_orders[1]}$", "$\\var{candidate_orders[2]}$", "$\\var{candidate_orders[3]}$", "$\\var{candidate_orders[4]}$", "$\\var{candidate_orders[5]}$"], "matrix": "let(mark,1/2, map(if(orderpossible(x,s_n),[mark,-mark],[-mark,mark]),x,candidate_orders))", "layout": {"type": "all", "expression": ""}, "answers": ["Possible", "Not possible"]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Permutations: two-line form", "extensions": ["permutations"], "custom_part_types": [{"source": {"pk": 68, "author": {"name": "Christian Lawson-Perfect", "pk": 7}, "edit_page": "/part_type/68/edit"}, "name": "Write a permutation in cycle notation", "short_name": "write-a-permutation-in-cycle-notation", "description": "<p>The student must write a permutation in cycle notation. You can optionally require the cycles to be disjoint and/or all of length 2.</p>", "help_url": "", "input_widget": "string", "input_options": {"correctAnswer": "let(ans,\n  if(settings[\"must_be_transpositions\"],\n    as_transpositions(settings[\"correct_answer\"]),\n    show(settings[\"correct_answer\"])\n  ),\n  if(ans=\"\",settings[\"identity_name\"],ans)\n)", "hint": {"static": false, "value": "if(settings[\"show_hint\"],\n  if(settings[\"must_be_disjoint\"],\n    if(settings[\"must_be_transpositions\"],\n      \"Write a permutation in cycle notation as a product of disjoint transpositions.\",\n      \"Write a permutation in cycle notation as a product of disjoint cycles.\"\n    ),\n    if(settings[\"must_be_transpositions\"],\n      \"Write a permutation in cycle notation as a product of transpositions.\",\n      \"Write a permutation in cycle notation.\"\n    )\n  ),\n  \"\"\n)"}, "allowEmpty": {"static": true, "value": false}}, "can_be_gap": true, "can_be_step": true, "marking_script": "mark:\napply(student_permutation);\napply(is_correct);\napply(is_disjoint);\napply(is_transpositions)\n\ninterpreted_answer:\nstudent_permutation\n\nis_correct:\ncorrectif(student_permutation=settings[\"correct_answer\"])\n\nis_disjoint:\nif(is_disjoint(studentAnswer),\n  true\n,\n  assert(not settings[\"must_be_disjoint\"],\n    set_credit(0,\"You have not written your answer as a product of disjoint cycles.\"),\n  );\n  false\n)\n\nstudent_permutation:\ntry(\n  if(studentAnswer=settings[\"identity_name\"],\n    permutation(\"\"), \n    permutation(studentAnswer)\n  ),\n  message,\n    warn(\"Your answer is not valid cycle notation: \"+message);\n    fail(\"Your answer is not valid cycle notation: \"+message)\n)\n\nis_transpositions:\nif(is_transpositions(studentAnswer),\n  true\n,\n  assert(not settings[\"must_be_transpositions\"],\n    set_credit(0,\"You have not written your answer as a product of transpositions.\")\n  );\n  false\n)", "marking_notes": [{"name": "mark", "description": "This is the main marking note. It should award credit and provide feedback based on the student's answer.", "definition": "apply(student_permutation);\napply(is_correct);\napply(is_disjoint);\napply(is_transpositions)"}, {"name": "interpreted_answer", "description": "A value representing the student's answer to this part.", "definition": "student_permutation"}, {"name": "is_correct", "description": "", "definition": "correctif(student_permutation=settings[\"correct_answer\"])"}, {"name": "is_disjoint", "description": "", "definition": "if(is_disjoint(studentAnswer),\n  true\n,\n  assert(not settings[\"must_be_disjoint\"],\n    set_credit(0,\"You have not written your answer as a product of disjoint cycles.\"),\n  );\n  false\n)"}, {"name": "student_permutation", "description": "", "definition": "try(\n  if(studentAnswer=settings[\"identity_name\"],\n    permutation(\"\"), \n    permutation(studentAnswer)\n  ),\n  message,\n    warn(\"Your answer is not valid cycle notation: \"+message);\n    fail(\"Your answer is not valid cycle notation: \"+message)\n)"}, {"name": "is_transpositions", "description": "", "definition": "if(is_transpositions(studentAnswer),\n  true\n,\n  assert(not settings[\"must_be_transpositions\"],\n    set_credit(0,\"You have not written your answer as a product of transpositions.\")\n  );\n  false\n)"}], "settings": [{"name": "correct_answer", "label": "Correct answer", "help_url": "", "hint": "", "input_type": "code", "default_value": "", "evaluate": true}, {"name": "show_hint", "label": "Show answer input hint?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": true}, {"name": "must_be_disjoint", "label": "Must the cycles be disjoint?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": false}, {"name": "must_be_transpositions", "label": "Must the cycles be transpositions?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": false}, {"name": "identity_name", "label": "Name of the identity permutation", "help_url": "", "hint": "The student should write this string for the identity permutation.", "input_type": "string", "default_value": "e", "subvars": true}], "public_availability": "always", "published": true, "extensions": ["permutations"]}], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "tags": ["permutations"], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>\\[ \\pi = \\var{twoline(ppi)} \\]</p>", "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": {}, "variables": {"ppi_num_even_cycles": {"name": "ppi_num_even_cycles", "group": "Ungrouped variables", "definition": "sum(map(1-mod(x,2),x,ppi_cycle_lengths))", "description": "", "templateType": "anything"}, "ppi_cycle_lengths": {"name": "ppi_cycle_lengths", "group": "Ungrouped variables", "definition": "map(len(x),x,nontrivial_cycles(ppi))", "description": "", "templateType": "anything"}, "ppi": {"name": "ppi", "group": "Ungrouped variables", "definition": "perm(deal(random(6..11)))", "description": "", "templateType": "anything"}, "ppi_num_cycles": {"name": "ppi_num_cycles", "group": "Ungrouped variables", "definition": "len(nontrivial_cycles(ppi))", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["ppi_num_even_cycles", "ppi_num_cycles", "ppi", "ppi_cycle_lengths"], "variable_groups": [], "functions": {"randompartition": {"parameters": [["n", "number"]], "type": "list", "language": "javascript", "definition": "var l = [];\nwhile(n) {\n  var t = Numbas.math.randomint(n)+1;\n  l.push(t);\n  n -= t;\n}\nreturn l;"}, "tracemap": {"parameters": [["n", "number"], ["perms", "list"]], "type": "string", "language": "javascript", "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;"}, "sum": {"parameters": [["l", "list"]], "type": "number", "language": "javascript", "definition": "var t = 0;\nfor(var i=0;i<l.length;i++) {\n  t = Numbas.math.add(t,l[i]);\n}\nreturn t;"}, "permfrompartition": {"parameters": [["cycle_sizes", "list"]], "type": "list", "language": "javascript", "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;"}, "permwithorder": {"parameters": [["n", "number"]], "type": "list", "language": "javascript", "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;"}, "randomcycle": {"parameters": [["n", "number"]], "type": "list", "language": "javascript", "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;"}, "primepowerfactors": {"parameters": [["n", "number"]], "type": "list", "language": "javascript", "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;"}, "showlist": {"parameters": [["l", "list"]], "type": "string", "language": "javascript", "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;"}, "orderpossible": {"parameters": [["n", "number"], ["s_n", "number"]], "type": "boolean", "language": "javascript", "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;"}, "show_factorisation": {"parameters": [["n", "number"]], "type": "string", "language": "javascript", "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;"}, "showsum": {"parameters": [["l", "list"]], "type": "string", "language": "javascript", "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;"}}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "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>", "gaps": [{"type": "write-a-permutation-in-cycle-notation", "useCustomName": true, "customName": "$\\pi$", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [{"variables": [{"name": "ppi_num_even_cycles", "value": "1"}, {"name": "ppi_num_cycles", "value": "1"}, {"name": "ppi", "value": "perm(\"(1,6,7,5,4,2)\")"}, {"name": "ppi_cycle_lengths", "value": "[ 6 ]"}], "answer": {"empty": false, "valid": true, "value": "(1,6,7,5,4,2)"}, "notes": [{"name": "mark", "expected": {"value": "nothing", "messages": ["Your answer is correct.\n\nYou were awarded <strong>1</strong> mark."], "warnings": [], "error": "", "valid": true, "credit": 1}}]}], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correct_answer": "ppi", "must_be_disjoint": true, "must_be_transpositions": false}}, {"type": "write-a-permutation-in-cycle-notation", "useCustomName": true, "customName": "$\\pi^{-1}$", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correct_answer": "inverse(ppi)", "must_be_disjoint": true, "must_be_transpositions": false}}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>What is the order of $\\pi$? [[0]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "order(ppi)", "maxValue": "order(ppi)", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\pi$ is [[0]]</p>", "gaps": [{"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["<p>an even permutation</p>", "<p>an odd permutation</p>"], "matrix": "if(even(ppi),[1,0],[0,1])"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Write a permutation as a product of disjoint cycles", "extensions": ["permutations"], "custom_part_types": [{"source": {"pk": 68, "author": {"name": "Christian Lawson-Perfect", "pk": 7}, "edit_page": "/part_type/68/edit"}, "name": "Write a permutation in cycle notation", "short_name": "write-a-permutation-in-cycle-notation", "description": "<p>The student must write a permutation in cycle notation. You can optionally require the cycles to be disjoint and/or all of length 2.</p>", "help_url": "", "input_widget": "string", "input_options": {"correctAnswer": "let(ans,\n  if(settings[\"must_be_transpositions\"],\n    as_transpositions(settings[\"correct_answer\"]),\n    show(settings[\"correct_answer\"])\n  ),\n  if(ans=\"\",settings[\"identity_name\"],ans)\n)", "hint": {"static": false, "value": "if(settings[\"show_hint\"],\n  if(settings[\"must_be_disjoint\"],\n    if(settings[\"must_be_transpositions\"],\n      \"Write a permutation in cycle notation as a product of disjoint transpositions.\",\n      \"Write a permutation in cycle notation as a product of disjoint cycles.\"\n    ),\n    if(settings[\"must_be_transpositions\"],\n      \"Write a permutation in cycle notation as a product of transpositions.\",\n      \"Write a permutation in cycle notation.\"\n    )\n  ),\n  \"\"\n)"}, "allowEmpty": {"static": true, "value": false}}, "can_be_gap": true, "can_be_step": true, "marking_script": "mark:\napply(student_permutation);\napply(is_correct);\napply(is_disjoint);\napply(is_transpositions)\n\ninterpreted_answer:\nstudent_permutation\n\nis_correct:\ncorrectif(student_permutation=settings[\"correct_answer\"])\n\nis_disjoint:\nif(is_disjoint(studentAnswer),\n  true\n,\n  assert(not settings[\"must_be_disjoint\"],\n    set_credit(0,\"You have not written your answer as a product of disjoint cycles.\"),\n  );\n  false\n)\n\nstudent_permutation:\ntry(\n  if(studentAnswer=settings[\"identity_name\"],\n    permutation(\"\"), \n    permutation(studentAnswer)\n  ),\n  message,\n    warn(\"Your answer is not valid cycle notation: \"+message);\n    fail(\"Your answer is not valid cycle notation: \"+message)\n)\n\nis_transpositions:\nif(is_transpositions(studentAnswer),\n  true\n,\n  assert(not settings[\"must_be_transpositions\"],\n    set_credit(0,\"You have not written your answer as a product of transpositions.\")\n  );\n  false\n)", "marking_notes": [{"name": "mark", "description": "This is the main marking note. It should award credit and provide feedback based on the student's answer.", "definition": "apply(student_permutation);\napply(is_correct);\napply(is_disjoint);\napply(is_transpositions)"}, {"name": "interpreted_answer", "description": "A value representing the student's answer to this part.", "definition": "student_permutation"}, {"name": "is_correct", "description": "", "definition": "correctif(student_permutation=settings[\"correct_answer\"])"}, {"name": "is_disjoint", "description": "", "definition": "if(is_disjoint(studentAnswer),\n  true\n,\n  assert(not settings[\"must_be_disjoint\"],\n    set_credit(0,\"You have not written your answer as a product of disjoint cycles.\"),\n  );\n  false\n)"}, {"name": "student_permutation", "description": "", "definition": "try(\n  if(studentAnswer=settings[\"identity_name\"],\n    permutation(\"\"), \n    permutation(studentAnswer)\n  ),\n  message,\n    warn(\"Your answer is not valid cycle notation: \"+message);\n    fail(\"Your answer is not valid cycle notation: \"+message)\n)"}, {"name": "is_transpositions", "description": "", "definition": "if(is_transpositions(studentAnswer),\n  true\n,\n  assert(not settings[\"must_be_transpositions\"],\n    set_credit(0,\"You have not written your answer as a product of transpositions.\")\n  );\n  false\n)"}], "settings": [{"name": "correct_answer", "label": "Correct answer", "help_url": "", "hint": "", "input_type": "code", "default_value": "", "evaluate": true}, {"name": "show_hint", "label": "Show answer input hint?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": true}, {"name": "must_be_disjoint", "label": "Must the cycles be disjoint?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": false}, {"name": "must_be_transpositions", "label": "Must the cycles be transpositions?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": false}, {"name": "identity_name", "label": "Name of the identity permutation", "help_url": "", "hint": "The student should write this string for the identity permutation.", "input_type": "string", "default_value": "e", "subvars": true}], "public_availability": "always", "published": true, "extensions": ["permutations"]}], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "tags": ["checked2015", "groups", "permutations"], "metadata": {"description": "", "licence": "None specified"}, "statement": "", "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": {}, "variables": {"prod": {"name": "prod", "group": "Ungrouped variables", "definition": "compose(perms[0],perms[1],perms[2],perms[3])", "description": "", "templateType": "anything"}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(4..9)", "description": "", "templateType": "anything"}, "perms": {"name": "perms", "group": "Ungrouped variables", "definition": "repeat(perm(randomcycle(n)),4)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["perms", "prod", "n"], "variable_groups": [], "functions": {"randompartition": {"parameters": [["n", "number"]], "type": "list", "language": "javascript", "definition": "var l = [];\nwhile(n) {\n  var t = Numbas.math.randomint(n)+1;\n  l.push(t);\n  n -= t;\n}\nreturn l;"}, "tracemap": {"parameters": [["n", "number"], ["perms", "list"]], "type": "string", "language": "javascript", "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;"}, "show_factorisation": {"parameters": [["n", "number"]], "type": "string", "language": "javascript", "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;"}, "permfrompartition": {"parameters": [["cycle_sizes", "list"]], "type": "list", "language": "javascript", "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;"}, "permwithorder": {"parameters": [["n", "number"]], "type": "list", "language": "javascript", "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;"}, "randomcycle": {"parameters": [["n", "number"]], "type": "list", "language": "javascript", "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;"}, "primepowerfactors": {"parameters": [["n", "number"]], "type": "list", "language": "javascript", "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;"}, "showlist": {"parameters": [["l", "list"]], "type": "string", "language": "javascript", "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;"}, "orderpossible": {"parameters": [["n", "number"], ["s_n", "number"]], "type": "boolean", "language": "javascript", "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;"}, "sum": {"parameters": [["l", "list"]], "type": "number", "language": "javascript", "definition": "var t = 0;\nfor(var i=0;i<l.length;i++) {\n  t = Numbas.math.add(t,l[i]);\n}\nreturn t;"}, "showsum": {"parameters": [["l", "list"]], "type": "string", "language": "javascript", "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;"}}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "write-a-permutation-in-cycle-notation", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "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>", "settings": {"correct_answer": "prod", "must_be_disjoint": true, "must_be_transpositions": false}}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Write a permutation as a product of transpositions", "extensions": ["permutations"], "custom_part_types": [{"source": {"pk": 68, "author": {"name": "Christian Lawson-Perfect", "pk": 7}, "edit_page": "/part_type/68/edit"}, "name": "Write a permutation in cycle notation", "short_name": "write-a-permutation-in-cycle-notation", "description": "<p>The student must write a permutation in cycle notation. You can optionally require the cycles to be disjoint and/or all of length 2.</p>", "help_url": "", "input_widget": "string", "input_options": {"correctAnswer": "let(ans,\n  if(settings[\"must_be_transpositions\"],\n    as_transpositions(settings[\"correct_answer\"]),\n    show(settings[\"correct_answer\"])\n  ),\n  if(ans=\"\",settings[\"identity_name\"],ans)\n)", "hint": {"static": false, "value": "if(settings[\"show_hint\"],\n  if(settings[\"must_be_disjoint\"],\n    if(settings[\"must_be_transpositions\"],\n      \"Write a permutation in cycle notation as a product of disjoint transpositions.\",\n      \"Write a permutation in cycle notation as a product of disjoint cycles.\"\n    ),\n    if(settings[\"must_be_transpositions\"],\n      \"Write a permutation in cycle notation as a product of transpositions.\",\n      \"Write a permutation in cycle notation.\"\n    )\n  ),\n  \"\"\n)"}, "allowEmpty": {"static": true, "value": false}}, "can_be_gap": true, "can_be_step": true, "marking_script": "mark:\napply(student_permutation);\napply(is_correct);\napply(is_disjoint);\napply(is_transpositions)\n\ninterpreted_answer:\nstudent_permutation\n\nis_correct:\ncorrectif(student_permutation=settings[\"correct_answer\"])\n\nis_disjoint:\nif(is_disjoint(studentAnswer),\n  true\n,\n  assert(not settings[\"must_be_disjoint\"],\n    set_credit(0,\"You have not written your answer as a product of disjoint cycles.\"),\n  );\n  false\n)\n\nstudent_permutation:\ntry(\n  if(studentAnswer=settings[\"identity_name\"],\n    permutation(\"\"), \n    permutation(studentAnswer)\n  ),\n  message,\n    warn(\"Your answer is not valid cycle notation: \"+message);\n    fail(\"Your answer is not valid cycle notation: \"+message)\n)\n\nis_transpositions:\nif(is_transpositions(studentAnswer),\n  true\n,\n  assert(not settings[\"must_be_transpositions\"],\n    set_credit(0,\"You have not written your answer as a product of transpositions.\")\n  );\n  false\n)", "marking_notes": [{"name": "mark", "description": "This is the main marking note. It should award credit and provide feedback based on the student's answer.", "definition": "apply(student_permutation);\napply(is_correct);\napply(is_disjoint);\napply(is_transpositions)"}, {"name": "interpreted_answer", "description": "A value representing the student's answer to this part.", "definition": "student_permutation"}, {"name": "is_correct", "description": "", "definition": "correctif(student_permutation=settings[\"correct_answer\"])"}, {"name": "is_disjoint", "description": "", "definition": "if(is_disjoint(studentAnswer),\n  true\n,\n  assert(not settings[\"must_be_disjoint\"],\n    set_credit(0,\"You have not written your answer as a product of disjoint cycles.\"),\n  );\n  false\n)"}, {"name": "student_permutation", "description": "", "definition": "try(\n  if(studentAnswer=settings[\"identity_name\"],\n    permutation(\"\"), \n    permutation(studentAnswer)\n  ),\n  message,\n    warn(\"Your answer is not valid cycle notation: \"+message);\n    fail(\"Your answer is not valid cycle notation: \"+message)\n)"}, {"name": "is_transpositions", "description": "", "definition": "if(is_transpositions(studentAnswer),\n  true\n,\n  assert(not settings[\"must_be_transpositions\"],\n    set_credit(0,\"You have not written your answer as a product of transpositions.\")\n  );\n  false\n)"}], "settings": [{"name": "correct_answer", "label": "Correct answer", "help_url": "", "hint": "", "input_type": "code", "default_value": "", "evaluate": true}, {"name": "show_hint", "label": "Show answer input hint?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": true}, {"name": "must_be_disjoint", "label": "Must the cycles be disjoint?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": false}, {"name": "must_be_transpositions", "label": "Must the cycles be transpositions?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": false}, {"name": "identity_name", "label": "Name of the identity permutation", "help_url": "", "hint": "The student should write this string for the identity permutation.", "input_type": "string", "default_value": "e", "subvars": true}], "public_availability": "always", "published": true, "extensions": ["permutations"]}], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "tags": ["permutations"], "metadata": {"description": "", "licence": "None specified"}, "statement": "", "advice": "<h4>a)</h4>\n<p>\\[ \\var{as_transpositions(pi1)} \\]</p>\n<h4>b)</h4>\n<p>\\[ \\var{as_transpositions(pi2)} \\]</p>", "rulesets": {}, "variables": {"pi1": {"name": "pi1", "group": "Ungrouped variables", "definition": "perm(randomcycle(random(5..9)))", "description": "", "templateType": "anything"}, "pi2": {"name": "pi2", "group": "Ungrouped variables", "definition": "perm(deal(random(11..17)))", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["pi1", "pi2"], "variable_groups": [], "functions": {"randompartition": {"parameters": [["n", "number"]], "type": "list", "language": "javascript", "definition": "var l = [];\nwhile(n) {\n  var t = Numbas.math.randomint(n)+1;\n  l.push(t);\n  n -= t;\n}\nreturn l;"}, "tracemap": {"parameters": [["n", "number"], ["perms", "list"]], "type": "string", "language": "javascript", "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;"}, "sum": {"parameters": [["l", "list"]], "type": "number", "language": "javascript", "definition": "var t = 0;\nfor(var i=0;i<l.length;i++) {\n  t = Numbas.math.add(t,l[i]);\n}\nreturn t;"}, "permfrompartition": {"parameters": [["cycle_sizes", "list"]], "type": "list", "language": "javascript", "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;"}, "permwithorder": {"parameters": [["n", "number"]], "type": "list", "language": "javascript", "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;"}, "randomcycle": {"parameters": [["n", "number"]], "type": "list", "language": "javascript", "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;"}, "primepowerfactors": {"parameters": [["n", "number"]], "type": "list", "language": "javascript", "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;"}, "showlist": {"parameters": [["l", "list"]], "type": "string", "language": "javascript", "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;"}, "orderpossible": {"parameters": [["n", "number"], ["s_n", "number"]], "type": "boolean", "language": "javascript", "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;"}, "show_factorisation": {"parameters": [["n", "number"]], "type": "string", "language": "javascript", "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;"}, "showsum": {"parameters": [["l", "list"]], "type": "string", "language": "javascript", "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;"}}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Write the permutation $\\pi_1 = \\var{pi1}$ as a product of transpositions.</p>\n<p>$\\pi_1 = $ [[0]]\n</p>", "gaps": [{"type": "write-a-permutation-in-cycle-notation", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correct_answer": "pi1", "must_be_disjoint": false, "must_be_transpositions": true}}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Write the permutation $\\pi_2 = \\var{pi2}$ as a product of transpositions.</p>\n<p>$\\pi_2 = $ [[0]]\n</p>", "gaps": [{"type": "write-a-permutation-in-cycle-notation", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correct_answer": "pi2", "must_be_disjoint": false, "must_be_transpositions": true}}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}]}], "navigation": {"allowregen": true, "reverse": true, "browse": true, "allowsteps": true, "showfrontpage": true, "showresultspage": "oncompletion", "navigatemode": "sequence", "onleave": {"action": "none", "message": ""}, "preventleave": true, "startpassword": ""}, "timing": {"allowPause": true, "timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "feedback": {"showactualmark": true, "showtotalmark": true, "showanswerstate": true, "allowrevealanswer": true, "advicethreshold": 0, "intro": "", "reviewshowscore": true, "reviewshowfeedback": true, "reviewshowexpectedanswer": true, "reviewshowadvice": true, "feedbackmessages": []}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": 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You can optionally require the cycles to be disjoint and/or all of length 2.</p>", "help_url": "", "input_widget": "string", "input_options": {"correctAnswer": "let(ans,\n  if(settings[\"must_be_transpositions\"],\n    as_transpositions(settings[\"correct_answer\"]),\n    show(settings[\"correct_answer\"])\n  ),\n  if(ans=\"\",settings[\"identity_name\"],ans)\n)", "hint": {"static": false, "value": "if(settings[\"show_hint\"],\n  if(settings[\"must_be_disjoint\"],\n    if(settings[\"must_be_transpositions\"],\n      \"Write a permutation in cycle notation as a product of disjoint transpositions.\",\n      \"Write a permutation in cycle notation as a product of disjoint cycles.\"\n    ),\n    if(settings[\"must_be_transpositions\"],\n      \"Write a permutation in cycle notation as a product of transpositions.\",\n      \"Write a permutation in cycle notation.\"\n    )\n  ),\n  \"\"\n)"}, "allowEmpty": {"static": true, "value": false}}, "can_be_gap": true, "can_be_step": true, "marking_script": "mark:\napply(student_permutation);\napply(is_correct);\napply(is_disjoint);\napply(is_transpositions)\n\ninterpreted_answer:\nstudent_permutation\n\nis_correct:\ncorrectif(student_permutation=settings[\"correct_answer\"])\n\nis_disjoint:\nif(is_disjoint(studentAnswer),\n  true\n,\n  assert(not settings[\"must_be_disjoint\"],\n    set_credit(0,\"You have not written your answer as a product of disjoint cycles.\"),\n  );\n  false\n)\n\nstudent_permutation:\ntry(\n  if(studentAnswer=settings[\"identity_name\"],\n    permutation(\"\"), \n    permutation(studentAnswer)\n  ),\n  message,\n    warn(\"Your answer is not valid cycle notation: \"+message);\n    fail(\"Your answer is not valid cycle notation: \"+message)\n)\n\nis_transpositions:\nif(is_transpositions(studentAnswer),\n  true\n,\n  assert(not settings[\"must_be_transpositions\"],\n    set_credit(0,\"You have not written your answer as a product of transpositions.\")\n  );\n  false\n)", "marking_notes": [{"name": "mark", "description": "This is the main marking note. It should award credit and provide feedback based on the student's answer.", "definition": "apply(student_permutation);\napply(is_correct);\napply(is_disjoint);\napply(is_transpositions)"}, {"name": "interpreted_answer", "description": "A value representing the student's answer to this part.", "definition": "student_permutation"}, {"name": "is_correct", "description": "", "definition": "correctif(student_permutation=settings[\"correct_answer\"])"}, {"name": "is_disjoint", "description": "", "definition": "if(is_disjoint(studentAnswer),\n  true\n,\n  assert(not settings[\"must_be_disjoint\"],\n    set_credit(0,\"You have not written your answer as a product of disjoint cycles.\"),\n  );\n  false\n)"}, {"name": "student_permutation", "description": "", "definition": "try(\n  if(studentAnswer=settings[\"identity_name\"],\n    permutation(\"\"), \n    permutation(studentAnswer)\n  ),\n  message,\n    warn(\"Your answer is not valid cycle notation: \"+message);\n    fail(\"Your answer is not valid cycle notation: \"+message)\n)"}, {"name": "is_transpositions", "description": "", "definition": "if(is_transpositions(studentAnswer),\n  true\n,\n  assert(not settings[\"must_be_transpositions\"],\n    set_credit(0,\"You have not written your answer as a product of transpositions.\")\n  );\n  false\n)"}], "settings": [{"name": "correct_answer", "label": "Correct answer", "help_url": "", "hint": "", "input_type": "code", "default_value": "", "evaluate": true}, {"name": "show_hint", "label": "Show answer input hint?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": true}, {"name": "must_be_disjoint", "label": "Must the cycles be disjoint?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": false}, {"name": "must_be_transpositions", "label": "Must the cycles be transpositions?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": false}, {"name": "identity_name", "label": "Name of the identity permutation", "help_url": "", "hint": "The student should write this string for the identity permutation.", "input_type": "string", "default_value": "e", "subvars": true}], "public_availability": "always", "published": true, "extensions": ["permutations"]}], "resources": []}