// Numbas version: exam_results_page_options
{"name": "Are these graphs isomorphic?", "extensions": ["permutations", "graph-theory"], "custom_part_types": [{"source": {"pk": 2, "author": {"name": "Christian Lawson-Perfect", "pk": 7}, "edit_page": "/part_type/2/edit"}, "name": "List of numbers", "short_name": "list-of-numbers", "description": "<p>The answer is a comma-separated list of numbers.</p>\n<p>The list is marked correct if each number occurs the same number of times as in the expected answer, and no extra numbers are present.</p>\n<p>You can optionally treat the answer as a set, so the number of occurrences doesn't matter, only whether each number is included or not.</p>", "help_url": "", "input_widget": "string", "input_options": {"correctAnswer": "join(\n  if(settings[\"correctAnswerFractions\"],\n    map(let([a,b],rational_approximation(x), string(a/b)),x,settings[\"correctAnswer\"])\n  ,\n    settings[\"correctAnswer\"]\n  ),\n  settings[\"separator\"] + \" \"\n)", "hint": {"static": false, "value": "if(settings[\"show_input_hint\"],\n  \"Enter a list of numbers separated by <code>{settings['separator']}</code>.\",\n  \"\"\n)"}, "allowEmpty": {"static": true, "value": true}}, "can_be_gap": true, "can_be_step": true, "marking_script": "bits:\nlet(b,filter(x<>\"\",x,split(studentAnswer,settings[\"separator\"])),\n    if(isSet,list(set(b)),b)\n)\n\nexpected_numbers:\nlet(l,settings[\"correctAnswer\"] as \"list\",\n    if(isSet,list(set(l)),l)\n)\n\nvalid_numbers:\nif(all(map(not isnan(x),x,interpreted_answer)),\n    true,\n    let(index,filter(isnan(interpreted_answer[x]),x,0..len(interpreted_answer)-1)[0], wrong, bits[index],\n      warn(wrong+\" is not a valid number\");\n      fail(wrong+\" is not a valid number.\")\n    )\n  )\n\nis_sorted:\nassert(sort(interpreted_answer)=interpreted_answer,\n    multiply_credit(0.5,\"Not in order\")\n  )\n\nincluded:\nmap(\n      let(\n        num_student,len(filter(x=y,y,interpreted_answer)),\n        num_expected,len(filter(x=y,y,expected_numbers)),\n        switch(\n          num_student=num_expected,\n            true,\n          num_student<num_expected,\n            incorrect(\n              if(isSet,\"Your answer does not contain \"+x+\" but should.\",\n                \"Your answer does not contain \"+x+\" enough times.\"\n              )\n            );false\n          ,\n            incorrect(\n              if(isSet,\"Your answer contains \"+x+\" but should not.\",\n                \"Your answer contains \"+x+\" too many times.\"\n              )\n            );false\n        )\n      )\n      ,\n      x,\n      sort(list(set(expected_numbers)))\n  )\n\nall_included:\nall(included)\n\nno_extras:\nif(all(map(x in expected_numbers, x, interpreted_answer)),\n    true\n  ,\n    incorrect(\"Your answer contains \"+extra_numbers[0]+\" but should not.\");\n    false\n  )\n\ninterpreted_answer:\nif(lower(studentAnswer) in [\"empty\",\"\u2205\"],[],\n  map(\n    if(settings[\"allowFractions\"],parsenumber_or_fraction(x,notationStyles), parsenumber(x,notationStyles))\n    ,x\n    ,bits\n  )\n)\n\nmark:\nif(studentanswer=\"\",fail(\"You have not entered an answer\"),false);\napply(valid_numbers);\napply(included);\napply(no_extras);\ncorrectif(all_included and no_extras)\n\nnotationStyles:\n[\"en\"]\n\nisSet:\nsettings[\"isSet\"]\n\nextra_numbers:\nfilter(not (x in expected_numbers),x,interpreted_answer)", "marking_notes": [{"name": "bits", "description": "<p>The separate items in the student's answer</p>", "definition": "let(b,filter(x<>\"\",x,split(studentAnswer,settings[\"separator\"])),\n    if(isSet,list(set(b)),b)\n)"}, {"name": "expected_numbers", "description": "", "definition": "let(l,settings[\"correctAnswer\"] as \"list\",\n    if(isSet,list(set(l)),l)\n)"}, {"name": "valid_numbers", "description": "<p>Is every number in the student's list valid?</p>", "definition": "if(all(map(not isnan(x),x,interpreted_answer)),\n    true,\n    let(index,filter(isnan(interpreted_answer[x]),x,0..len(interpreted_answer)-1)[0], wrong, bits[index],\n      warn(wrong+\" is not a valid number\");\n      fail(wrong+\" is not a valid number.\")\n    )\n  )"}, {"name": "is_sorted", "description": "<p>Are the student's answers in ascending order?</p>", "definition": "assert(sort(interpreted_answer)=interpreted_answer,\n    multiply_credit(0.5,\"Not in order\")\n  )"}, {"name": "included", "description": "<p>Is each number in the expected answer present in the student's list the correct number of times?</p>", "definition": "map(\n      let(\n        num_student,len(filter(x=y,y,interpreted_answer)),\n        num_expected,len(filter(x=y,y,expected_numbers)),\n        switch(\n          num_student=num_expected,\n            true,\n          num_student<num_expected,\n            incorrect(\n              if(isSet,\"Your answer does not contain \"+x+\" but should.\",\n                \"Your answer does not contain \"+x+\" enough times.\"\n              )\n            );false\n          ,\n            incorrect(\n              if(isSet,\"Your answer contains \"+x+\" but should not.\",\n                \"Your answer contains \"+x+\" too many times.\"\n              )\n            );false\n        )\n      )\n      ,\n      x,\n      sort(list(set(expected_numbers)))\n  )"}, {"name": "all_included", "description": "<p>Has every number been included the right number of times?</p>", "definition": "all(included)"}, {"name": "no_extras", "description": "<p>True if the student's list doesn't contain any numbers that aren't in the expected answer.</p>", "definition": "if(all(map(x in expected_numbers, x, interpreted_answer)),\n    true\n  ,\n    incorrect(\"Your answer contains \"+extra_numbers[0]+\" but should not.\");\n    false\n  )"}, {"name": "interpreted_answer", "description": "A value representing the student's answer to this part.", "definition": "if(lower(studentAnswer) in [\"empty\",\"\u2205\"],[],\n  map(\n    if(settings[\"allowFractions\"],parsenumber_or_fraction(x,notationStyles), parsenumber(x,notationStyles))\n    ,x\n    ,bits\n  )\n)"}, {"name": "mark", "description": "This is the main marking note. It should award credit and provide feedback based on the student's answer.", "definition": "if(studentanswer=\"\",fail(\"You have not entered an answer\"),false);\napply(valid_numbers);\napply(included);\napply(no_extras);\ncorrectif(all_included and no_extras)"}, {"name": "notationStyles", "description": "", "definition": "[\"en\"]"}, {"name": "isSet", "description": "<p>Should the answer be considered as a set, so the number of times an element occurs doesn't matter?</p>", "definition": "settings[\"isSet\"]"}, {"name": "extra_numbers", "description": "<p>Numbers included in the student's answer that are not in the expected list.</p>", "definition": "filter(not (x in expected_numbers),x,interpreted_answer)"}], "settings": [{"name": "correctAnswer", "label": "Correct answer", "help_url": "", "hint": "The list of numbers that the student should enter. The order does not matter.", "input_type": "code", "default_value": "", "evaluate": true}, {"name": "allowFractions", "label": "Allow the student to enter fractions?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": false}, {"name": "correctAnswerFractions", "label": "Display the correct answers as fractions?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": false}, {"name": "isSet", "label": "Is the answer a set?", "help_url": "", "hint": "If ticked, the number of times an element occurs doesn't matter, only whether it's included at all.", "input_type": "checkbox", "default_value": false}, {"name": "show_input_hint", "label": "Show the input hint?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": true}, {"name": "separator", "label": "Separator", "help_url": "", "hint": "The substring that should separate items in the student's list", "input_type": "string", "default_value": ",", "subvars": false}], "public_availability": "always", "published": true, "extensions": []}, {"source": {"pk": 197, "author": {"name": "Christian Lawson-Perfect", "pk": 7}, "edit_page": "/part_type/197/edit"}, "name": "Write a permutation in two-line form", "short_name": "write-a-permutation-in-two-line-form", "description": "<p>The student must write out a permutation (an element of $S_n$) in two-line form: the top line always contains the numbers 1 to $n$, and on the bottom line they must write the numbers that they are mapped to.</p>", "help_url": "", "input_widget": "matrix", "input_options": {"correctAnswer": "matrix(map(settings[\"correctAnswer\"][i],i,1..settings[\"size\"]))", "hint": {"static": true, "value": ""}, "allowResize": {"static": true, "value": false}, "numRows": {"static": true, "value": 1}, "numColumns": {"static": false, "value": "settings[\"size\"]"}, "parseCells": {"static": true, "value": true}, "allowedNotationStyles": {"static": true, "value": ["plain"]}, "allowFractions": {"static": true, "value": false}, "showBrackets": {"static": true, "value": true}, "rowHeaders": {"static": false, "value": "[]"}, "columnHeaders": {"static": false, "value": "map(string(n),n,1..settings[\"size\"])"}}, "can_be_gap": true, "can_be_step": true, "marking_script": "mark:\napply(valid_permutation);\ncorrectif(student_permutation=settings[\"correctAnswer\"])\n\nstudent_numbers:\nlist(studentAnswer[0])\n\nstudent_permutation:\npermutation(map(x-1,x,student_numbers))\n\nvalid_permutation:\n  assert(all(map(isint(x) and x>=1 and x<=settings[\"size\"], x, student_numbers)),\n    fail(\"Your answer is not a valid permutation: each entry must be one of the numbers 1 \u2026 {settings['size']}.\")\n  );\n  assert(len(set(student_numbers))=settings[\"size\"],\n    fail(\"Your answer is not a valid permutation: some entries occur more than once.\")\n  )\n\ninterpreted_answer:\nstudent_permutation", "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(valid_permutation);\ncorrectif(student_permutation=settings[\"correctAnswer\"])"}, {"name": "student_numbers", "description": "<p>The numbers that the student typed.</p>", "definition": "list(studentAnswer[0])"}, {"name": "student_permutation", "description": "", "definition": "permutation(map(x-1,x,student_numbers))"}, {"name": "valid_permutation", "description": "", "definition": "  assert(all(map(isint(x) and x>=1 and x<=settings[\"size\"], x, student_numbers)),\n    fail(\"Your answer is not a valid permutation: each entry must be one of the numbers 1 \u2026 {settings['size']}.\")\n  );\n  assert(len(set(student_numbers))=settings[\"size\"],\n    fail(\"Your answer is not a valid permutation: some entries occur more than once.\")\n  )"}, {"name": "interpreted_answer", "description": "A value representing the student's answer to this part.", "definition": "student_permutation"}], "settings": [{"name": "correctAnswer", "label": "Correct answer", "help_url": "", "hint": "", "input_type": "code", "default_value": "", "evaluate": true}, {"name": "size", "label": "Size", "help_url": "", "hint": "", "input_type": "code", "default_value": "", "evaluate": true}], "public_availability": "always", "published": true, "extensions": []}, {"source": {"pk": 1, "author": {"name": "Christian Lawson-Perfect", "pk": 7}, "edit_page": "/part_type/1/edit"}, "name": "Yes/no", "short_name": "yes-no", "description": "<p>The student is shown two radio choices: \"Yes\" and \"No\". One of them is correct.</p>", "help_url": "", "input_widget": "radios", "input_options": {"correctAnswer": "if(eval(settings[\"correct_answer_expr\"]), 0, 1)", "hint": {"static": true, "value": ""}, "choices": {"static": true, "value": ["Yes", "No"]}}, "can_be_gap": true, "can_be_step": true, "marking_script": "mark:\nif(studentanswer=correct_answer,\n  correct(),\n  incorrect()\n)\n\ninterpreted_answer:\nstudentAnswer=0\n\ncorrect_answer:\nif(eval(settings[\"correct_answer_expr\"]),0,1)", "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": "if(studentanswer=correct_answer,\n  correct(),\n  incorrect()\n)"}, {"name": "interpreted_answer", "description": "A value representing the student's answer to this part.", "definition": "studentAnswer=0"}, {"name": "correct_answer", "description": "", "definition": "if(eval(settings[\"correct_answer_expr\"]),0,1)"}], "settings": [{"name": "correct_answer_expr", "label": "Is the answer \"Yes\"?", "help_url": "", "hint": "An expression which evaluates to <code>true</code> or <code>false</code>.", "input_type": "mathematical_expression", "default_value": "true", "subvars": false}], "public_availability": "always", "published": true, "extensions": []}], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Are these graphs isomorphic?", "tags": [], "metadata": {"description": "<p>The student is shown two labelled graphs. They are asked:</p>\n<ul>\n<li>Number of vertices in each</li>\n<li>Number of edges in each</li>\n<li>Degree sequences for each</li>\n<li>Is there an isomorphism between them? If so, write one.</li>\n</ul>\n<p>The number of vertices is always equal, so this is a gimme.</p>\n<p>If the edges or degree sequences are different, the student is expected to realise that there cannot be an isomorphism.</p>\n<p>If these values are the same, then there will be an isomorphism (else the question is a bit too tricky).</p>\n<p>Numbas expects a particular isomorphism, but there may be more than one, all of which would be accepted.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Consider the following labelled graphs, and their adjacency matrices:</p>\n<table style=\"width: 600px; height: 36px;\">\n<tbody>\n<tr style=\"height: 18px;\">\n<td style=\"text-align: center; width: 285.85px; height: 18px;\">$\\Gamma_1$</td>\n<td style=\"text-align: center; width: 281.15px; height: 18px;\">$\\Gamma_2$</td>\n</tr>\n<tr style=\"height: 18px;\">\n<td style=\"text-align: center; width: 285.85px; height: 18px;\">{draw(g1)}</td>\n<td style=\"text-align: center; width: 281.15px; height: 18px;\">{draw(g2)}</td>\n</tr>\n<tr>\n<td style=\"text-align: center; width: 285.85px;\">$\\var{adjacency_matrix(g1)}$</td>\n<td style=\"text-align: center; width: 281.15px;\">$\\var{adjacency_matrix(g2)}$</td>\n</tr>\n<tr>\n<td style=\"text-align: center; width: 285.85px;\"></td>\n<td style=\"text-align: center; width: 281.15px;\"></td>\n</tr>\n</tbody>\n</table>\n<p></p>", "advice": "<p>The <strong>vertices</strong> are the labelled \"dots\". In each graph there are {n+1} vertices.</p>\n<p>The <strong>edges</strong> are the lines connecting vertices. In $\\Gamma_1$ there are {len(edges(g1))} edges, and in $\\Gamma_2$ there are {len(edges(g2))}.</p>\n<p>The <strong>degree </strong>of a vertex is the number of edges incident to it.</p>\n<p>The <strong>degree sequence </strong>is a list of the degrees of all of the vertices of a graph, usually ordered from smallest to largest. The degree sequence of $\\Gamma_1$ is {join(sort(vertex_degrees(g1)),\", \")}.</p>\n<p>Similarly, the degree sequence of $\\Gamma_2$ is {join(sort(vertex_degrees(g2)),\", \")}. This is because, for example, {len(filter(i=2, i, vertex_degrees(g2)))} of the vertices in $\\Gamma_2$ have degree 2, {len(filter(i=3, i, vertex_degrees(g2)))} of the vertices in $\\Gamma_2$ have degree 3, and so on.</p>\n<p>Two graphs $\\Gamma_1, \\Gamma_2$ are isomorphic if there is a bijection from the vertex set of $\\Gamma_1$ to the vertex set of $\\Gamma_2$ that maps a pair of vertices joined by an edge to a pair of vertices joined by an edge, and that maps a pair of vertices not joined by an edge to a pair of vertices not joined by an edge. If there is no such bijection then the graphs are not isomorphic.</p>\n<p><strong>There cannot be a graph isomorphism </strong>between two graphs if they have different numbers of vertices, different numbers of edges, or different degree sequences.</p>\n<p>{advice}</p>", "rulesets": {}, "extensions": ["graph-theory", "permutations"], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"g1": {"name": "g1", "group": "Ungrouped variables", "definition": "set_vertex_labels( graph(map(edge(from,to),[from,to],\n  [[0,1],[0,2]+[0,n-2],[0,n-1]]+\n  flatten(map(map([j,k],k,shuffle(j+1..n-1)[0..random(1,2)]),j,1..n-1))\n)), map(string(x),x,1..n))", "description": "<p>The graph $\\Gamma_1$.</p>\n<p>Vertex 1 will be adjacent to four others.</p>\n<p>The remaining vertices will each be adjacent to either one or two later vertices.</p>\n<p>In this way, no vertex will be adjacent to more than four others.</p>", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(6..8)", "description": "<p>Each graph will contain n vertices.</p>", "templateType": "anything", "can_override": false}, "vertex_perm": {"name": "vertex_perm", "group": "Ungrouped variables", "definition": "perm(shuffle(0..n-1))", "description": "<p>The vertices of graph 2 will be mapped by this permutation to those of graph 1.</p>", "templateType": "anything", "can_override": false}, "diff_degree": {"name": "diff_degree", "group": "Ungrouped variables", "definition": "let(\n  degrees, vertex_degrees(g1),\n  d, max(degrees),\n  v, indices(degrees,d)[0],\n  edges, edges(g1),\n  e, random(filter(not (v in ends(e)),e,edges)),\n  nedges,map(if(e2=e,edge(v,to(e)),e2),e2,edges),\n  graph(nedges)\n)", "description": "<p>Take g1 and find the vertex $v$ with highest degree. Move an edge connecting two other vertices so that one end is on $v$ instead. This changes the degree sequence, so produces a graph not isomorphic to g1.</p>", "templateType": "anything", "can_override": false}, "fewer_edges": {"name": "fewer_edges", "group": "Ungrouped variables", "definition": "let(\n  edges, edges(g1),\n  degrees,vertex_degrees(g1),\n  ok_edges,filter(degrees[from(e)]>1 and degrees[to(e)]>1,e,edges),\n  e,random(ok_edges),\n  graph(filter(e2<>e,e2,edges))\n)", "description": "<p>Remove an edge from g1 to produce a non-isomorphic graph. Only edges connecting vertices with degree at least 2 are considered, so the graph remains connected.</p>", "templateType": "anything", "can_override": false}, "g2": {"name": "g2", "group": "Ungrouped variables", "definition": "set_vertex_labels(\n  auto_layout(permute_vertices(\n    vertex_perm, \n    if(is_isomorphic,\n      graph(edges(g1)),\n      random(diff_degree, fewer_edges)\n    )\n  ))\n  ,\n  map(string(x),x,1..n)\n)", "description": "<p>The graph $\\Gamma_2$. If it should be isomorphic, just take g1, otherwise pick from the non-isomorphic options.</p>\n<p>Permute the vertices according to the permutation vertex_perm, and lay the vertices out again.</p>", "templateType": "anything", "can_override": false}, "advice_none": {"name": "advice_none", "group": "Ungrouped variables", "definition": "\"<p>Therefore there is no isomorphism between $\\\\Gamma_1$ and $\\\\Gamma_2$.</p>\"", "description": "<p>The advice shown if the graphs are not isomorphic.</p>", "templateType": "long string", "can_override": false}, "advice_isomorphism": {"name": "advice_isomorphism", "group": "Ungrouped variables", "definition": "\"<p>Since these are all the same for $\\\\Gamma_1$ and $\\\\Gamma_2$, we might be able to find an isomorphism. Indeed, we can map vertex 1 of $\\\\Gamma_1$ to vertex {vertex_perm[1]} of $\\\\Gamma_2$, we can map vertex 2 to vertex {vertex_perm[2]}, vertex 3 to vertex {vertex_perm[3]}, and so on. Therefore a valid isomorphism is</p>\\n<p>\\\\[ \\\\var{twoline(vertex_perm)} \\\\]</p>\\n<p>(Note: there may be more than one isomorphism.)</p>\"", "description": "<p>The advice shown if the graphs are isomorphic.</p>", "templateType": "long string", "can_override": false}, "advice": {"name": "advice", "group": "Ungrouped variables", "definition": "if(is_isomorphic, advice_isomorphism, advice_none)", "description": "<p>The final line of the advice, describing the isomorphism if there is one, or stating that there is none.</p>", "templateType": "anything", "can_override": false}, "is_isomorphic": {"name": "is_isomorphic", "group": "Ungrouped variables", "definition": "random(true,false)", "description": "<p>Should $\\Gamma_1$ and $\\Gamma_2$ be isomorphic?</p>", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["is_isomorphic", "n", "g1", "diff_degree", "fewer_edges", "g2", "vertex_perm", "advice_none", "advice_isomorphism", "advice"], "variable_groups": [], "functions": {}, "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>Compare the properties of $\\Gamma_1$ and $\\Gamma_2$.</p>\n<table style=\"height: 59px;\">\n<tbody>\n<tr style=\"height: 18px;\">\n<td style=\"height: 18px; width: 96.8333px;\"></td>\n<td style=\"height: 18px; width: 185.283px;\">Number of vertices</td>\n<td style=\"height: 18px; width: 171.7px;\">Number of edges</td>\n<td style=\"height: 18px; width: 251.917px;\">Degree sequence</td>\n</tr>\n<tr style=\"height: 23px;\">\n<td style=\"height: 23px; width: 96.8333px;\">$\\Gamma_1$</td>\n<td style=\"height: 23px; width: 185.283px;\">[[0]]</td>\n<td style=\"height: 23px; width: 171.7px;\">[[2]]</td>\n<td style=\"height: 23px; width: 251.917px;\">[[4]]</td>\n</tr>\n<tr style=\"height: 18px;\">\n<td style=\"height: 18px; width: 96.8333px;\">$\\Gamma_2$</td>\n<td style=\"height: 18px; width: 185.283px;\">[[1]]</td>\n<td style=\"height: 18px; width: 171.7px;\">[[3]]</td>\n<td style=\"height: 18px; width: 251.917px;\">[[5]]</td>\n</tr>\n</tbody>\n</table>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "Vertices in $\\Gamma_1$", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "len(vertices(g1))", "maxValue": "len(vertices(g1))", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "Vertices in $\\Gamma_2$", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "len(vertices(g2))", "maxValue": "len(vertices(g2))", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "Edges in $\\Gamma_1$", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "len(edges(g1))", "maxValue": "len(edges(g1))", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "Edges in $\\Gamma_2$", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "len(edges(g2))", "maxValue": "len(edges(g2))", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "list-of-numbers", "useCustomName": true, "customName": "Degree sequence of $\\Gamma_1$", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correctAnswer": "sort(vertex_degrees(g1))", "allowFractions": false, "correctAnswerFractions": false, "isSet": false, "show_input_hint": false}}, {"type": "list-of-numbers", "useCustomName": true, "customName": "Degree sequence of $\\Gamma_2$", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correctAnswer": "sort(vertex_degrees(g2))", "allowFractions": false, "correctAnswerFractions": false, "isSet": false, "show_input_hint": 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>Are the graphs $\\Gamma_1$ and $\\Gamma_2$ 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