// Numbas version: exam_results_page_options
{"name": "Counting 2: a deck of cards", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"statement": "<p>When using the pigeonhole principle, you will often need to decide what are the pigeons and what are the containers.</p>\n<p>In a standard deck of 52 cards,&nbsp;each card is one of 4 different suits (of which there are 2 red and 2 black suits) and one of 13 different values.&nbsp;</p>\n<p>Suppose you are dealt a hand of $\\var{n}$ cards from a standard deck.&nbsp;</p>", "variables": {"n": {"definition": "random(14..40)", "description": "", "name": "n", "group": "Ungrouped variables", "templateType": "anything"}}, "preamble": {"js": "", "css": ""}, "ungrouped_variables": ["n"], "tags": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "rulesets": {}, "variable_groups": [], "functions": {}, "extensions": [], "metadata": {"description": "<p>Applicaiton of the pigeonhole principle</p>", "licence": "Creative Commons Attribution-ShareAlike 4.0 International"}, "parts": [{"stepsPenalty": 0, "variableReplacements": [], "failureRate": 1, "type": "jme", "valuegenerators": [], "customMarkingAlgorithm": "", "showPreview": true, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "vsetRangePoints": 5, "extendBaseMarkingAlgorithm": true, "checkVariableNames": false, "prompt": "<p>The hand is guaranteed to contain at least how many cards sharing the same colour?</p>", "useCustomName": false, "unitTests": [], "customName": "", "vsetRange": [0, 1], "checkingType": "absdiff", "answer": "{ceil(n/2)}", "steps": [{"prompt": "<p>Let the cards be the pigeons, and let the two colours be the containers. By the pigeonhole principle there is one colour which has at least $\\left\\lceil \\dfrac{\\var{n}}{2}\\right\\rceil$ cards.</p>", "variableReplacements": [], "useCustomName": false, "unitTests": [], "type": "information", "customName": "", "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "marks": 0, "extendBaseMarkingAlgorithm": true, "scripts": {}}], "marks": 1, "checkingAccuracy": 0.001, "scripts": {}}, {"stepsPenalty": 0, "variableReplacements": [], "failureRate": 1, "type": "jme", "valuegenerators": [], "customMarkingAlgorithm": "", "showPreview": true, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "vsetRangePoints": 5, "extendBaseMarkingAlgorithm": true, "checkVariableNames": false, "prompt": "<p>The hand is guaranteed to contain at least how many cards&nbsp;of the same suit?</p>", "useCustomName": false, "unitTests": [], "customName": "", "vsetRange": [0, 1], "checkingType": "absdiff", "answer": "{ceil(n/4)}", "steps": [{"prompt": "<p>Let the cards be the pigeons, and let the four suits be the containers. By the pigeonhole principle there is one (perhaps more, but <em>at least</em> one) suit which has at least $\\left\\lceil \\dfrac{\\var{n}}{4}\\right\\rceil$ cards.</p>", "variableReplacements": [], "useCustomName": false, "unitTests": [], "type": "information", "customName": "", "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "marks": 0, "extendBaseMarkingAlgorithm": true, "scripts": {}}], "marks": 1, "checkingAccuracy": 0.001, "scripts": {}}, {"stepsPenalty": 0, "variableReplacements": [], "failureRate": 1, "type": "jme", "valuegenerators": [], "customMarkingAlgorithm": "", "showPreview": true, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "vsetRangePoints": 5, "extendBaseMarkingAlgorithm": true, "checkVariableNames": false, "prompt": "<p>The hand is guaranteed to contain at least how many cards sharing the same value?</p>", "useCustomName": false, "unitTests": [], "customName": "", "vsetRange": [0, 1], "checkingType": "absdiff", "answer": "{ceil(n/13)}", "steps": [{"prompt": "<p>Let the cards be the pigeons, and let the thirteen values be the containers. By the pigeonhole principle there is one (perhaps more, but <em>at least</em> one) value which has at least $\\left\\lceil \\dfrac{\\var{n}}{13}\\right\\rceil$ cards.</p>", "variableReplacements": [], "useCustomName": false, "unitTests": [], "type": "information", "customName": "", "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "marks": 0, "extendBaseMarkingAlgorithm": true, "scripts": {}}], "marks": 1, "checkingAccuracy": 0.001, "scripts": {}}], "name": "Counting 2: a deck of cards", "advice": "<p>a. $\\left\\lceil \\dfrac{\\var{n}}{2} \\right\\rceil = \\var{ceil(n/2)}$.</p>\n<p>b. $\\left\\lceil \\dfrac{\\var{n}}{4} \\right\\rceil = \\var{ceil(n/4)}$.</p>\n<p>c. $\\left\\lceil \\dfrac{\\var{n}}{13} \\right\\rceil = \\var{ceil(n/13)}$.</p>", "contributors": [{"name": "Daniel Mansfield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/743/"}, {"name": "Sean Gardiner", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2443/"}]}]}], "contributors": [{"name": "Daniel Mansfield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/743/"}, {"name": "Sean Gardiner", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2443/"}]}