{"id":1435,"date":"2009-06-22T05:27:37","date_gmt":"2009-06-22T04:27:37","guid":{"rendered":"http:\/\/www.walkingrandomly.com\/?p=1435"},"modified":"2010-02-01T11:45:28","modified_gmt":"2010-02-01T10:45:28","slug":"proof-that-math-exams-have-become-easier-over-the-last-169-years","status":"publish","type":"post","link":"https:\/\/walkingrandomly.com\/?p=1435","title":{"rendered":"Proof that math exams have become easier over the last 169 years!"},"content":{"rendered":"

Solution to problem of the week #6<\/strong><\/p>\n

Way back in April<\/a>, I posed the following problem.\u00a0 ‘Consider a square pyramidal pile of identical cannonballs of radius r such that the bottom layer contains 16 cannonballs (such as the pile in the diagram above). Find the volume (in terms of r) of the pyramid that envelops and contains the whole pile’<\/strong><\/p>\n

Since then I have received several answers (check out the comments section of the original post for a few of them) and all but one of them were wrong.\u00a0 In my opinion, this is nothing to be too ashamed about since I couldn’t solve the problem either and I am not about to berate my readers for failing to do something that I couldn’t do myself!<\/p>\n

So if I couldn’t do it then how did I know that all of these answers were incorrect?\u00a0 Well clearly I had cheated and had access to a worked solution.\u00a0 ‘My’ problem was in fact problem number 15 from one the 1840 undergraduate final exams at Cambridge University<\/a> and a worked solution is given in the (now fully digitized, thanks to google) text Solutions for the Cambridge Problems 1840,1841<\/a>. <\/strong><\/p>\n

The solution is<\/p>\n

\"\light v=\frac{2}{3}( 1+3 \sqrt{2} +sqrt{3})^3 r^3\"<\/strong><\/p>\n

You’ll find the official worked solution to this problem on page 20 of ‘Solutions for the Cambridge Problems 180,1841’ (sadly, the diagram is missing) but James Graham-Eagle of the University of Massachusetts Lowell sent me not one but two different solutions in this pdf file<\/a> and they are much<\/strong> easier to follow in my humble opinion.\u00a0 Thanks for that James.<\/p>\n

I’m glad that this problem wasn’t in my final exam!<\/p>\n","protected":false},"excerpt":{"rendered":"

Solution to problem of the week #6 Way back in April, I posed the following problem.\u00a0 ‘Consider a square pyramidal pile of identical cannonballs of radius r such that the bottom layer contains 16 cannonballs (such as the pile in the diagram above). Find the volume (in terms of r) of the pyramid that envelops […]<\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[6,21],"tags":[],"class_list":["post-1435","post","type-post","status-publish","format-standard","hentry","category-general-math","category-problem-of-the-week"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p3swhs-n9","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=\/wp\/v2\/posts\/1435","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1435"}],"version-history":[{"count":12,"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=\/wp\/v2\/posts\/1435\/revisions"}],"predecessor-version":[{"id":2264,"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=\/wp\/v2\/posts\/1435\/revisions\/2264"}],"wp:attachment":[{"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1435"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1435"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1435"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}