// Numbas version: finer_feedback_settings
{"name": "19. Factors", "metadata": {"description": "<p>Overview of what a factor is.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "duration": 0, "percentPass": 0, "showQuestionGroupNames": false, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", "", "", ""], "variable_overrides": [[], [], [], [], []], "questions": [{"name": "19.a Which numbers have a factor of 5?", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Mash Sheffield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4679/"}, {"name": "Andrew Neate", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21832/"}], "tags": [], "metadata": {"description": "<p>Simple question on times tables tests for 5s.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Which of the following numbers have a factor of 5?</p>", "advice": "<p>If a number has a factor of 5 then it must end in either a 5 or a 0.</p>\n<p><a href=\"https://www.sheffield.ac.uk/mash/maths-resources/maths-all#Factors\"><span style=\"font-weight: 400;\">Use this link to find some resources which will help you revise this topic.</span></a></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"Nums": {"name": "Nums", "group": "Ungrouped variables", "definition": "string(random(1..50)) for: x of:1..10", "description": "", "templateType": "anything", "can_override": false}, "MarkList": {"name": "MarkList", "group": "Ungrouped variables", "definition": "if(5|dec(x),1,0) for: x of:Nums", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "sum(MarkList)>1", "maxRuns": 100}, "ungrouped_variables": ["Nums", "MarkList"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "m_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": "1", "shuffleChoices": true, "displayType": "checkbox", "displayColumns": 0, "minAnswers": 0, "maxAnswers": 0, "warningType": "none", "showCellAnswerState": true, "markingMethod": "all-or-nothing", "choices": "Nums", "matrix": "MarkList"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "19.b Which numbers have a factor of 3?", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Mash Sheffield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4679/"}, {"name": "Andrew Neate", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21832/"}], "tags": [], "metadata": {"description": "<p>Simple question on times tables tests for 3s.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Which of the following numbers have a factor of 3?</p>", "advice": "<p>If a number has a factor of 3 then so does the sum of the digits. This can help make by making a number smaller, for instance 12 has 1+2=3 as the sum of its digits and hence 12 has a factor of 3. Similarly 96 has sum of digits 9+6=15 and 1+5=6 which has a factor of 3 so 96 has a factor of 3.</p>\n<p><a href=\"https://www.sheffield.ac.uk/mash/maths-resources/maths-all#Factors\"><span style=\"font-weight: 400;\">Use this link to find some resources which will help you revise this topic.</span></a></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"Nums": {"name": "Nums", "group": "Ungrouped variables", "definition": "string(random(1..50)) for: x of:1..10", "description": "", "templateType": "anything", "can_override": false}, "MarkList": {"name": "MarkList", "group": "Ungrouped variables", "definition": "if(3|dec(x),1,0) for: x of:Nums", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "sum(MarkList)>1", "maxRuns": 100}, "ungrouped_variables": ["Nums", "MarkList"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "m_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": "1", "shuffleChoices": true, "displayType": "checkbox", "displayColumns": 0, "minAnswers": 0, "maxAnswers": 0, "warningType": "none", "showCellAnswerState": true, "markingMethod": "all-or-nothing", "choices": "Nums", "matrix": "MarkList"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "19.c Find factors of a number", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Rachel Staddon", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/901/"}, {"name": "Mash Sheffield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4679/"}, {"name": "Andrew Neate", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21832/"}], "tags": [], "metadata": {"description": "<p>Multiples, factors, lowest common multiples and highest common factors.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Choose all of&nbsp;the factors of $\\var{product}$.</p>", "advice": "<p>The factors of a number are the numbers which divide it. In this case we need to check which of the above numbers divide $\\var{product}$ which is just a case of testing them each individually.</p>\n<p><a href=\"https://www.sheffield.ac.uk/mash/maths-resources/maths-all#Factors\"><span style=\"font-weight: 400;\">Use this link to find some resources which will help you revise this topic.</span></a></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"ca2": {"name": "ca2", "group": "Ungrouped variables", "definition": "random(answers except ca1)", "description": "", "templateType": "anything", "can_override": false}, "ca1": {"name": "ca1", "group": "Ungrouped variables", "definition": "random(answers)", "description": "", "templateType": "anything", "can_override": false}, "hcf": {"name": "hcf", "group": "Ungrouped variables", "definition": "gcd(k,l)", "description": "", "templateType": "anything", "can_override": false}, "lcm4": {"name": "lcm4", "group": "Ungrouped variables", "definition": "lcm(p,q)", "description": "", "templateType": "anything", "can_override": false}, "lcm2": {"name": "lcm2", "group": "Ungrouped variables", "definition": "lcm(n,m)", "description": "", "templateType": "anything", "can_override": false}, "product": {"name": "product", "group": "Ungrouped variables", "definition": "a*b*c", "description": "", "templateType": "anything", "can_override": false}, "answers": {"name": "answers", "group": "Ungrouped variables", "definition": "[a,b,c,a*b,a*c,b*c,1,product]", "description": "", "templateType": "anything", "can_override": false}, "hcf3": {"name": "hcf3", "group": "Ungrouped variables", "definition": "gcd(p,q)", "description": "", "templateType": "anything", "can_override": false}, "hcf2": {"name": "hcf2", "group": "Ungrouped variables", "definition": "gcd(n,m)+1", "description": "", "templateType": "anything", "can_override": false}, "lcm": {"name": "lcm", "group": "Ungrouped variables", "definition": "lcm(k,l)", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1,2,3)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(5,7)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(2,3)", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(1..12)", "description": "", "templateType": "anything", "can_override": false}, "g": {"name": "g", "group": "Ungrouped variables", "definition": "random(2..10 except b)", "description": "", "templateType": "anything", "can_override": false}, "f": {"name": "f", "group": "Ungrouped variables", "definition": "random(2..10)", "description": "", "templateType": "anything", "can_override": false}, "h": {"name": "h", "group": "Ungrouped variables", "definition": "random(1..100 except [d*1,d*2,d*3,d*4,d*5,d*6,d*7,d*8,d*9,d*10,f*1,f*2,f*3,f*4,f*5,f*6,f*7,f*8,f*9,f*10,g*1,g*2,g*3,g*4,g*5,g*6,g*7,g*8,g*9,g*10])", "description": "", "templateType": "anything", "can_override": false}, "k": {"name": "k", "group": "Ungrouped variables", "definition": "random(4..12)", "description": "", "templateType": "anything", "can_override": false}, "j": {"name": "j", "group": "Ungrouped variables", "definition": "random(1..50 except [d*1,d*2,d*3,d*4,d*5,d*6,d*7,d*8,d*9,d*10,f*1,f*2,f*3,f*4,f*5,f*6,f*7,f*8,f*9,f*10,g*1,g*2,g*3,g*4,g*5,g*6,g*7,g*8,g*9,g*10,h])", "description": "", "templateType": "anything", "can_override": false}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "random(2..15 except [k,l,lcm])", "description": "", "templateType": "anything", "can_override": false}, "l": {"name": "l", "group": "Ungrouped variables", "definition": "random(4..15 except k)", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(6..15 except [k,l,lcm,m])", "description": "", "templateType": "anything", "can_override": false}, "q": {"name": "q", "group": "Ungrouped variables", "definition": "random(4..36 except p)", "description": "", "templateType": "anything", "can_override": false}, "p": {"name": "p", "group": "Ungrouped variables", "definition": "random(6..24)", "description": "", "templateType": "anything", "can_override": false}, "wa2": {"name": "wa2", "group": "Ungrouped variables", "definition": "random(1..product except [a,b,c,a*b,a*c,b*c,1,product,wa1])", "description": "", "templateType": "anything", "can_override": false}, "wa1": {"name": "wa1", "group": "Ungrouped variables", "definition": "random(1..product except answers)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["hcf2", "lcm2", "a", "b", "c", "product", "answers", "wa1", "wa2", "ca1", "ca2", "d", "f", "g", "h", "j", "k", "l", "lcm", "m", "n", "hcf", "p", "q", "hcf3", "lcm4"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "m_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": true, "displayType": "checkbox", "displayColumns": 0, "minAnswers": 0, "maxAnswers": "0", "warningType": "warn", "showCellAnswerState": true, "markingMethod": "sum ticked cells", "choices": ["<p>$\\var{ca1}$</p>", "<p>$\\var{ca2}$</p>", "<p>$\\var{wa1}$</p>", "<p>$\\var{wa2}$</p>"], "matrix": ["1", "1", 0, 0], "distractors": ["", "", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "19.d Which numbers have a factor of 9?", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Mash Sheffield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4679/"}, {"name": "Andrew Neate", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21832/"}], "tags": [], "metadata": {"description": "<p>factors of 9</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Which of the following numbers have a factor of 9?</p>", "advice": "<p>A number has a factor of 9 if it has two factors of 3. First check which numbers have a factor of 3. Divide each number with a factor of 3 by 3 and see if they still have a factor of 3.</p>\n<p>Remember If a number has a factor of 3 then so does the sum of the digits. This can help make by making a number smaller, for instance 12 has 1+2=3 as the sum of its digits and hence 12 has a factor of 3. Similarly 96 has sum of digits 9+6=15 and 1+5=6 which has a factor of 3 so 96 has a factor of 3.</p>\n<p><a href=\"https://www.sheffield.ac.uk/mash/maths-resources/maths-all#Factors\"><span style=\"font-weight: 400;\">Use this link to find some resources which will help you revise this topic.</span></a></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"Nums": {"name": "Nums", "group": "Ungrouped variables", "definition": "string(random(1..50)) for: x of:1..10", "description": "", "templateType": "anything", "can_override": false}, "MarkList": {"name": "MarkList", "group": "Ungrouped variables", "definition": "if(9|dec(x),1,0) for: x of:Nums", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "sum(MarkList)>1", "maxRuns": 100}, "ungrouped_variables": ["Nums", "MarkList"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "m_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": "1", "shuffleChoices": true, "displayType": "checkbox", "displayColumns": 0, "minAnswers": 0, "maxAnswers": 0, "warningType": "none", "showCellAnswerState": true, "markingMethod": "all-or-nothing", "choices": "Nums", "matrix": "MarkList"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "19.e. Finding all factors of a number", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Mash Sheffield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4679/"}, {"name": "Andrew Neate", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21832/"}], "tags": [], "metadata": {"description": "<p>Prime factor decomposition starting steps.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Provide all the factors of $\\var{Number}$.</p>", "advice": "<p>For any number $n$ we know 1 and $n$ will always be factors. We will work out the rest of the factors by testing which numbers divide {Number}.</p>\n<p>First try dividing by 2, if this does not work then try 3, then 4 and so on. (In fact, to find the first factor you only need to check the prime numbers 2,3,5,7,...).</p>\n<p>After 1, the first number which divides {Number} is {Factors[1]}. This means that both {Factors[1]} and {Number/Factors[1]} are factors. We can now speed up the process of looking for factors of {Number} by looking for factors of {Number/Factors[1]} instead. Make sure to remember that if $x$ is a factor of {Number/Factors[1]} then both $x$ and $\\var{Factors[1]}\\times x$ are factors of {Number}!</p>\n<p>We only need to look for factors up to $\\sqrt{\\var{Number}}$. Once you have done this you should have the full list of factors {AnswersAsString}.</p>\n<p></p>\n<p><a href=\"https://www.sheffield.ac.uk/mash/maths-resources/maths-all#Factors\"><span style=\"font-weight: 400;\">Use this link to find some resources which will help you revise this topic.</span></a></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"Number": {"name": "Number", "group": "Ungrouped variables", "definition": "random(1..100)", "description": "<p>All numbers less than 119 have 12 or fewer factors so can't increase this range without increasing the number of options</p>", "templateType": "anything", "can_override": false}, "Factors": {"name": "Factors", "group": "Ungrouped variables", "definition": "divisors(Number)", "description": "", "templateType": "anything", "can_override": false}, "DisplayAnswers": {"name": "DisplayAnswers", "group": "Ungrouped variables", "definition": "if(x<abs(Factors),Factors[x],\"\") for: x of:0..11", "description": "", "templateType": "anything", "can_override": false}, "Temp": {"name": "Temp", "group": "Ungrouped variables", "definition": "if(x=1,\"1\",if(x=\"\",\"\",\", \"+string(x))) for: x of:DisplayAnswers", "description": "", "templateType": "anything", "can_override": false}, "AnswersAsString": {"name": "AnswersAsString", "group": "Ungrouped variables", "definition": "Temp[0]+Temp[1]+Temp[2]+Temp[3]+Temp[4]+Temp[5]+Temp[6]+Temp[7]+Temp[8]+Temp[9]+Temp[10]+Temp[11]", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "len(Factors)>2", "maxRuns": 100}, "ungrouped_variables": ["Number", "Factors", "DisplayAnswers", "Temp", "AnswersAsString"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "student_set (The set of numbers that the student gave):\n  set(x for: x of: interpreted_answers where: x<>nothing)\n\nexpected_set (The set of roots of the equation):\n  set(Factors)\n\nnum_correct (The number of values in the student's set which are correct):\n  len(intersection(student_set, expected_set))\n\nnum_answers: len(expected_set)\n\nnum_incorrect: len(student_set)-num_correct\n\nmark:\n  add_credit(\n    num_correct / num_answers, \"You gave {num_correct} of {num_answers} distinct factors.\")\n  sub_credit(\n    num_incorrect / 12, \"You gave {num_incorrect} answers which are not factors\"\n  )", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Leave boxes empty if there is not enough factors.</p>\n<p>[[0]]</p>\n<p>[[1]]</p>\n<p>[[2]]</p>\n<p>[[3]]</p>\n<p>[[4]]</p>\n<p>[[5]]</p>\n<p>[[6]]</p>\n<p>[[7]]</p>\n<p>[[8]]</p>\n<p>[[9]]</p>\n<p>[[10]]</p>\n<p>[[11]]</p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "Gap0", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "student_empty: trim(studentAnswer)=\"\"\n\nmark:  correctif(interpreted_answer in Factors or student_empty)\n//  assert(student_empty, apply(base_mark))\n\ninterpreted_answer:\n  if(student_empty, nothing, base_interpreted_answer)", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "1", "maxValue": "1", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "DisplayAnswers[0]", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "Gap1", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "student_empty: trim(studentAnswer)=\"\"\n\nmark:\n  assert(student_empty, apply(base_mark))\n\ninterpreted_answer:\n  if(student_empty, nothing, base_interpreted_answer)", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "1", "maxValue": "1", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "DisplayAnswers[1]", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "Gap2", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "student_empty: trim(studentAnswer)=\"\"\n\nmark:\n  assert(student_empty, apply(base_mark))\n\ninterpreted_answer:\n  if(student_empty, nothing, base_interpreted_answer)", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "1", "maxValue": "1", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "DisplayAnswers[2]", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "Gap3", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "student_empty: 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