{"id":527,"date":"2024-12-11T08:00:00","date_gmt":"2024-12-11T07:00:00","guid":{"rendered":"https:\/\/control-advent.com\/?p=527"},"modified":"2025-11-28T19:04:54","modified_gmt":"2025-11-28T18:04:54","slug":"11-endless-magic-the-curious-cycle-of-cookies-and-elves","status":"publish","type":"post","link":"https:\/\/control-advent.com\/?p=527","title":{"rendered":"11 &#8211; Endless Magic: The Curious Cycle of Cookies and Elves"},"content":{"rendered":"\n<p><strong>Author of this question:<\/strong> Steffi Knorn, <em>Fachgebiet <em>Regelungstechnik<\/em>, <\/em>Technische Universit\u00e4t Berlin, Germany<\/p>\n\n\n\n<p>In the bustling elves&#8217; workshop, cookies are more than just a treat \u2014 they\u2019re the magical fuel that powers the elves\u2019 tireless efforts. But these aren&#8217;t ordinary cookies; they mysteriously multiply, as if the workshop itself conspires to keep them flowing. Yet there&#8217;s a twist: the more cookies there are, the more elves appear, drawn by the sweet aroma and enchanted by the magic. And the more elves, the more cookies they devour, spinning the workshop into an endless cycle of sugary abundance. It&#8217;s a delicate system, one that Santa watches over with a twinkle in his eye and a strategic mind to keep it balanced. Keeping the magic in an equilibrium isn\u2019t as simple as it seems!<\/p>\n\n\n\n<p>After observing the workshop for a while, Santa noticed how the cookies and elves behave, and the factors seem to be:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Cookies multiply: Each night, as the elves bid the workshop farewell, the cookies left behind undergo a magical transformation. For every single cookie resting on the trays, ten new ones appear by morning, multiplying the total by a factor of 11. It&#8217;s as if the cookies can\u2019t resist the call to grow, ensuring the elves wake up to an overflowing bounty of sugary delights to fuel their day.<\/li>\n\n\n\n<li>The elves expect to eat cookies: Indeed the elves can\u2019t help but chatter about the number of cookies left at the end of each day &#8211; it\u2019s a topic as hot as fresh-baked gingerbread. By morning, each elf arrives with high hopes, and each plans to eat exactly as many cookies as there were left on the tray the previous evening. (Needless to say, if possible, elves follow through with their plans.)<\/li>\n\n\n\n<li>Elves dislike crowded workspaces: Elves may be hardworking, but they have their limits \u2014 especially when it comes to crowded situations. If the cookie supply runs empty and the sweet incentive disappears, the workshop suddenly feels a lot less appealing. In fact, when not considering the allurement to the workshop due to cookies only half the elves who showed up the previous day will bother coming back, the others will prefer a quieter day off.<\/li>\n\n\n\n<li>Elves spread the word very fast: Fortunately, elves are passionate about cookies, and good news travels fast among them. If there\u2019s a hefty stash of cookies left at the end of the day, excitement bubbles over, and word spreads like frosting on a gingerbread house. By morning, the workshop sees a surge of eager elves, with the number of newcomers equal to 0.1 times the number of cookies left the night before, multiplied by the elves who worked the previous day. In other words, more cookies and more elves to have spread the word mean more elves next day, drawn in by the irresistible promise of sugary delight!<\/li>\n<\/ul>\n\n\n\n<p>The constant fluctuation of cookies and elves in the workshop is driving Santa crazy and making it difficult for him to plan ahead. Tired of this chaos, he decides that he wants to ensure the same number of elves show up in the workshop every day. What should he do to achieve this?<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"583\" src=\"https:\/\/control-advent.com\/wp-content\/uploads\/2024\/11\/11-Dec-2024-1024x583.jpg\" alt=\"\" class=\"wp-image-424\" srcset=\"https:\/\/control-advent.com\/wp-content\/uploads\/2024\/11\/11-Dec-2024-1024x583.jpg 1024w, https:\/\/control-advent.com\/wp-content\/uploads\/2024\/11\/11-Dec-2024-300x171.jpg 300w, https:\/\/control-advent.com\/wp-content\/uploads\/2024\/11\/11-Dec-2024-768x437.jpg 768w, https:\/\/control-advent.com\/wp-content\/uploads\/2024\/11\/11-Dec-2024-24x14.jpg 24w, https:\/\/control-advent.com\/wp-content\/uploads\/2024\/11\/11-Dec-2024-36x20.jpg 36w, https:\/\/control-advent.com\/wp-content\/uploads\/2024\/11\/11-Dec-2024-48x27.jpg 48w, https:\/\/control-advent.com\/wp-content\/uploads\/2024\/11\/11-Dec-2024.jpg 1232w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n<\/div>\n\n\n<p>a. \ud83c\udf85\ud83e\udddd\u200d\u2642\ufe0f\ud83c\udf6a Santa should make sure that there are ten elves at work each day and five cookies left in the evening, then the mount of elves and cookies is not going to change from day to day.<\/p>\n\n\n\n<p>b. \ud83e\udddd\u200d\u2642\ufe0f\ud83d\udd1f\u2714\ufe0f Regardless of the amount of cookies, Santa has to make sure that ten elves are in the workshop every day.<\/p>\n\n\n\n<p>c. \ud83e\udddd\u200d\u2642\ufe0f5\ufe0f\u20e3\u2714\ufe0f Regardless of the amount of cookies, Santa has to make sure that five elves are in the workshop every day.<\/p>\n\n\n\n<p>d. \ud83d\udeab\ud83e\udddd\u200d\u2642\ufe0f\ud83d\udcc9 No matter how many cookies there are, the only option is to keep the elves out of the workshop entirely. While this would result in zero productivity, at least the number of elves would remain constant!<\/p>\n\n\n\n<p>e. \ud83c\udf6a\ud83d\udd1f\ud83c\udf19 The key is to make sure that one night only ten cookies are left in the workshop. Then, the rest will sort itself out, regardless of how many elves were in the workshop that day.<\/p>\n\n\n\n<p>f. \ud83c\udf6a5\ufe0f\u20e3\ud83c\udf19 The key is to make sure that one night only five cookies are left in the workshop. Then, the rest will sort itself out, regardless of how many elves were in the workshop that day.<\/p>\n\n\n\n<p>g. \ud83c\udf85\ud83c\udf6a\u274c Santa should make sure there are no cookies. Only if he makes sure that he eats all cookies during the night, so that none are left the next morning, only then the number of elves will be constant afterwards.<\/p>\n\n\n<div class='bootstrap-yop yop-poll-mc'>\n\t\t\t\t\t\t\t<div class=\"basic-yop-poll-container\" style=\"background-color:#b70004; border:0px; border-style:solid; border-color:#b70004; border-radius:0px; padding:10px 10px;\" data-id=\"16\" data-temp=\"basic\" data-skin=\"minimal\" data-cscheme=\"black\" data-cap=\"0\" data-access=\"wordpress\" data-tid=\"\" data-uid=\"4ad8387a3931087a0dadf427795801b6\" data-pid=\"527\" data-resdet=\"percentages\" data-show-results-to=\"\" data-show-results-moment=\"never\" data-show-results-only=\"false\" data-show-message=\"true\" data-show-results-as=\"bar\" data-sort-results-by=\"as-defined\" data-sort-results-rule=\"asc\"data-is-ended=\"0\" data-gdpr=\"no\" data-gdpr-sol=\"consent\" data-css=\".basic-yop-poll-container[data-uid] .basic-vote { text-align: center; }\" data-counter=\"0\" data-load-with=\"1\" data-notification-section=\"top\"><div class=\"row\"><div class=\"col-md-12\"><div class=\"basic-inner\"><div class=\"basic-message hide\" style=\"border-left: 10px solid #008000; padding: 0px 10px;\" data-error=\"#ff0000\" data-success=\"#008000\"><p class=\"basic-message-text\" style=\"color:#000000; font-size:14px; font-weight:normal;\"><\/p><\/div><div class=\"basic-overlay hide\"><div class=\"basic-vote-options\"><\/div><div class=\"basic-preloader\"><div class=\"basic-windows8\"><div class=\"basic-wBall basic-wBall_1\"><div class=\"basic-wInnerBall\"><\/div><\/div><div class=\"basic-wBall basic-wBall_2\"><div class=\"basic-wInnerBall\"><\/div><\/div><div class=\"basic-wBall basic-wBall_3\"><div class=\"basic-wInnerBall\"><\/div><\/div><div class=\"basic-wBall basic-wBall_4\"><div class=\"basic-wInnerBall\"><\/div><\/div><div class=\"basic-wBall basic-wBall_5\"><div class=\"basic-wInnerBall\"><\/div><\/div><\/div><\/div><\/div><form class=\"basic-form\"><input type=\"hidden\" name=\"_token\" value=\"eb63eb0959\" autocomplete=\"off\"><div class=\"basic-elements\"><div class=\"basic-element basic-question basic-question-text-horizontal\" data-id=\"16\" data-uid=\"ab1c02253ac87cbe1ef2c9b51de3f4b8\" data-type=\"question\" data-question-type=\"text\" data-allow-multiple=\"no\" data-min=\"1\" data-max=\"1\" data-display=\"horizontal\" data-colnum=\"\" data-display-others=\"no\" data-others-color=\"\" data-others=\"\"><div role=\"heading\" aria-level=\"5\" class=\"basic-question-title\" style=\"color:#ffffff; font-size:14px; font-weight:normal; text-align:center;\">Choose an option:<\/div><ul class=\"basic-answers basic-h-answers\"><li class=\"basic-answer col-xs-6 col-sm-6 cols-7\" style=\"padding:0px 0px;\" data-id=\"100\" data-type=\"text\" data-vn=\"72\" data-color=\"#000000\" data-make-link=\"no\" data-link=\"\"><div class=\"basic-answer-content basic-text-horizontal\"><label for=\"answer[100]\" class=\"basic-answer-label\"><input type=\"radio\" id=\"answer[100]\" name=\"answer[16]\" value=\"100\" autocomplete=\"off\"><span class=\"basic-text\" style=\"color: #ffffff; font-size: 14px; font-weight: normal;\" tabindex=\"0\">a<\/span><\/label><\/div><\/li><li class=\"basic-answer col-xs-6 col-sm-6 cols-7\" style=\"padding:0px 0px;\" data-id=\"101\" data-type=\"text\" data-vn=\"12\" data-color=\"#000000\" data-make-link=\"no\" data-link=\"\"><div class=\"basic-answer-content basic-text-horizontal\"><label for=\"answer[101]\" class=\"basic-answer-label\"><input type=\"radio\" id=\"answer[101]\" name=\"answer[16]\" value=\"101\" autocomplete=\"off\"><span class=\"basic-text\" style=\"color: #ffffff; font-size: 14px; font-weight: normal;\" tabindex=\"0\">b<\/span><\/label><\/div><\/li><li class=\"basic-answer col-xs-6 col-sm-6 cols-7\" style=\"padding:0px 0px;\" data-id=\"102\" data-type=\"text\" data-vn=\"4\" data-color=\"#000000\" data-make-link=\"no\" data-link=\"\"><div class=\"basic-answer-content basic-text-horizontal\"><label for=\"answer[102]\" class=\"basic-answer-label\"><input type=\"radio\" id=\"answer[102]\" name=\"answer[16]\" value=\"102\" autocomplete=\"off\"><span class=\"basic-text\" style=\"color: #ffffff; font-size: 14px; font-weight: normal;\" tabindex=\"0\">c<\/span><\/label><\/div><\/li><li class=\"basic-answer col-xs-6 col-sm-6 cols-7\" style=\"padding:0px 0px;\" data-id=\"133\" data-type=\"text\" data-vn=\"17\" data-color=\"#000000\" data-make-link=\"no\" data-link=\"\"><div class=\"basic-answer-content basic-text-horizontal\"><label for=\"answer[133]\" class=\"basic-answer-label\"><input type=\"radio\" id=\"answer[133]\" name=\"answer[16]\" value=\"133\" autocomplete=\"off\"><span class=\"basic-text\" style=\"color: #ffffff; font-size: 14px; font-weight: normal;\" tabindex=\"0\">d<\/span><\/label><\/div><\/li><li class=\"basic-answer col-xs-6 col-sm-6 cols-7\" style=\"padding:0px 0px;\" data-id=\"134\" data-type=\"text\" data-vn=\"19\" data-color=\"#000000\" data-make-link=\"no\" data-link=\"\"><div class=\"basic-answer-content basic-text-horizontal\"><label for=\"answer[134]\" class=\"basic-answer-label\"><input type=\"radio\" id=\"answer[134]\" name=\"answer[16]\" value=\"134\" autocomplete=\"off\"><span class=\"basic-text\" style=\"color: #ffffff; font-size: 14px; font-weight: normal;\" tabindex=\"0\">e<\/span><\/label><\/div><\/li><li class=\"basic-answer col-xs-6 col-sm-6 cols-7\" style=\"padding:0px 0px;\" data-id=\"135\" data-type=\"text\" data-vn=\"26\" data-color=\"#000000\" data-make-link=\"no\" data-link=\"\"><div class=\"basic-answer-content basic-text-horizontal\"><label for=\"answer[135]\" class=\"basic-answer-label\"><input type=\"radio\" id=\"answer[135]\" name=\"answer[16]\" value=\"135\" autocomplete=\"off\"><span class=\"basic-text\" style=\"color: #ffffff; font-size: 14px; font-weight: normal;\" tabindex=\"0\">f<\/span><\/label><\/div><\/li><li class=\"basic-answer col-xs-6 col-sm-6 cols-7\" style=\"padding:0px 0px;\" data-id=\"136\" data-type=\"text\" data-vn=\"11\" data-color=\"#000000\" data-make-link=\"no\" data-link=\"\"><div class=\"basic-answer-content basic-text-horizontal\"><label for=\"answer[136]\" class=\"basic-answer-label\"><input type=\"radio\" id=\"answer[136]\" name=\"answer[16]\" value=\"136\" autocomplete=\"off\"><span class=\"basic-text\" style=\"color: #ffffff; font-size: 14px; font-weight: normal;\" tabindex=\"0\">g<\/span><\/label><\/div><\/li><\/ul><\/div><div class=\"clearfix\"><\/div><\/div><div class=\"basic-vote\"><a href=\"#\" class=\"button basic-vote-button\" role=\"button\" style=\"background:#ffffff; border:1px; border-style: solid; border-color:#000000; border-radius:0px; padding:5px 10px; color:#000000; font-size:14px; font-weight:normal;\">&gt;&gt;&gt;<\/a><\/div><\/form><div class=\"yop-poll-modal\"><div class=\"yop-poll-modal-content\"><div class=\"yop-poll-modal-header\"><span class=\"yop-poll-modal-close\">&times;<\/span><\/div><div class=\"yop-poll-modal-body\"><div class=\"yop-poll-form-group\"><label class=\"yop-poll-user-or-email-label\">Username or Email Address<\/label><input type=\"text\" class=\"yop-poll-user-or-email-input\" autocomplete=\"off\"><\/div><div class=\"yop-poll-form-group\"><label class=\"yop-poll-password-label\">Password<\/label><input type=\"password\" class=\"yop-poll-password-input\" autocomplete=\"off\"><\/div><div class=\"yop-poll-form-group submit\"><button class=\"button button-primary button-large yop-poll-login-button\">Log In<\/button><\/div><div class=\"yop-poll-form-group yop-poll-section-login-options\"><p><a href=\"https:\/\/control-advent.com\/wp-login.php?action=register\" target=\"_blank\">No account? Register here<\/a><\/p><p><a href=\"https:\/\/control-advent.com\/wp-login.php?action=lostpassword\" target=\"_blank\">Forgot password<\/a><\/p><\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/div>\n\t\t\t\t\t\t<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Author of this question: Steffi Knorn, Fachgebiet Regelungstechnik, Technische Universit\u00e4t Berlin, Germany In the bustling elves&#8217; workshop, cookies are more than just a treat \u2014 they\u2019re the magical fuel that powers the elves\u2019 tireless efforts. But these aren&#8217;t ordinary cookies; &hellip; <span class=\"more-link\"><a href=\"https:\/\/control-advent.com\/?p=527\" class=\"readmore\">Continue reading<span class=\"screen-reader-text\">11 &#8211; Endless Magic: The Curious Cycle of Cookies and Elves<\/span><\/a><\/span><\/p>\n","protected":false},"author":4,"featured_media":424,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[22],"tags":[99],"class_list":["post-527","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-english","tag-99"],"_links":{"self":[{"href":"https:\/\/control-advent.com\/index.php?rest_route=\/wp\/v2\/posts\/527","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/control-advent.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/control-advent.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/control-advent.com\/index.php?rest_route=\/wp\/v2\/users\/4"}],"replies":[{"embeddable":true,"href":"https:\/\/control-advent.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=527"}],"version-history":[{"count":2,"href":"https:\/\/control-advent.com\/index.php?rest_route=\/wp\/v2\/posts\/527\/revisions"}],"predecessor-version":[{"id":1041,"href":"https:\/\/control-advent.com\/index.php?rest_route=\/wp\/v2\/posts\/527\/revisions\/1041"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/control-advent.com\/index.php?rest_route=\/wp\/v2\/media\/424"}],"wp:attachment":[{"href":"https:\/\/control-advent.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=527"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/control-advent.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=527"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/control-advent.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=527"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}