{"id":113,"date":"2017-11-27T15:39:05","date_gmt":"2017-11-27T21:39:05","guid":{"rendered":"http:\/\/blogs.acu.edu\/mu_sigma\/?page_id=113"},"modified":"2017-11-27T15:39:05","modified_gmt":"2017-11-27T21:39:05","slug":"problem-of-the-week-problem-2-solution-october-30-2017","status":"publish","type":"page","link":"https:\/\/blogs.acu.edu\/mu_sigma\/problem-of-the-week-problem-2-solution-october-30-2017\/","title":{"rendered":"Problem of the Week &#8211; Problem 2 Solution &#8211; October 30, 2017"},"content":{"rendered":"<p><span style=\"font-size: large\">Solution to <b>Cards of the Round Table<\/b><\/span><\/p>\n<p><span style=\"font-size: small\">Correct solutions were submitted by: \u00a0Rui Bi<\/span><\/p>\n<p>First consider the case when the number of cards is a power of two. For example, if <i>n<\/i> is 8, then the first time around the table you remove 2, 4, 6, and 8. At this time you are at card number 1, and the rest of the problem can be considered as an original problem but with <i>n<\/i> = 4. We deduce that, in general, if the number of cards is a power of two, then the final remaining card is number 1.<\/p>\n<p>For the general problem, go around the table removing cards until the first time the number of cards remaining is a power of two. From the preceding analysis, we know that the card where we are standing will be the final remaining card.<\/p>\n<p>To actually find the card number, we first find the integer <i>k<\/i> such that 2<i><sup>k<\/sup><\/i> is no more than <i>n<\/i> but <i>n<\/i> is less than 2<sup><i>k<\/i>+1<\/sup>. This number <i>k<\/i> is the greatest integer less than or equal to <i>log<\/i><sub>2<\/sub> <i>n<\/i>, denoted by <b>floor<\/b>(<i>log<\/i><sub><span style=\"font-size: xx-small\">2<\/span><\/sub> <i>n<\/i>). Thus <i>n<\/i> &#8211; 2<sup><b>floor<\/b>(<i>log<\/i><sub><span style=\"font-size: xx-small\">2<\/span><\/sub> <i>n<\/i>)<\/sup>is the number of cards that have been removed. We have removed cards with even numbers, so we are at the card with the odd number<\/p>\n<p align=\"CENTER\">2(<i>n<\/i> &#8211; 2<sup><b>floor<\/b>(<i>log<\/i><sub><span style=\"font-size: xx-small\">2<\/span><\/sub> <i>n<\/i>)<\/sup> ) + 1.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Solution to Cards of the Round Table Correct solutions were submitted by: \u00a0Rui Bi First consider the case when the number of cards is a power of two. For example, if n is 8, then the first time around the table you remove 2, 4, 6, and 8. At this time you are at card [&hellip;]<\/p>\n","protected":false},"author":130,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_et_pb_use_builder":"","_et_pb_old_content":"","_et_gb_content_width":"","footnotes":""},"class_list":["post-113","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/pages\/113","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/users\/130"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/comments?post=113"}],"version-history":[{"count":1,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/pages\/113\/revisions"}],"predecessor-version":[{"id":114,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/pages\/113\/revisions\/114"}],"wp:attachment":[{"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/media?parent=113"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}