{"id":1052,"date":"2009-04-17T11:48:02","date_gmt":"2009-04-17T10:48:02","guid":{"rendered":"http:\/\/www.walkingrandomly.com\/?p=1052"},"modified":"2019-01-25T04:29:52","modified_gmt":"2019-01-25T03:29:52","slug":"problem-of-the-week-6-cannonballs","status":"publish","type":"post","link":"https:\/\/walkingrandomly.com\/?p=1052","title":{"rendered":"Problem of the week #6 &#8211; Cannonballs"},"content":{"rendered":"<p>Imagine that you were the Captain of a sailing ship a few hundred years ago and, in order to protect yourself from pirates, you had a few cannons.\u00a0 Cannons need cannonballs and it is well known that the best way to stack cannon balls is to arrange them as a <a href=\"http:\/\/en.wikipedia.org\/wiki\/Square_pyramid\">square pyamid<\/a> as in the image below.<\/p>\n<p style=\"text-align: center;\"><img decoding=\"async\" class=\"aligncenter\" src=\"\/images\/pow\/cannonballs.png\" alt=\"Stack of Cannonballs\" \/><\/p>\n<p>So, in this example you have 16 balls in the bottom layer, 9 balls in the next layer, then 4 and, finally, one at the top giving a total of 16+9+4+1 = 30 balls and we say that 30 is the 4th <a href=\"http:\/\/en.wikipedia.org\/wiki\/Square_pyramidal_number\">square pyramidal number<\/a>.\u00a0 The first few such numbers are<\/p>\n<ul>\n<li>1<\/li>\n<li>5 (4+1)<\/li>\n<li>14 (9+4+1)<\/li>\n<li>30 (16+9+4+1)<\/li>\n<li>55 (25+16+9+4+1)<\/li>\n<\/ul>\n<p>Now, there is a well known problem called the <a href=\"http:\/\/mathworld.wolfram.com\/CannonballProblem.html\">Cannonball problem (Spolier alert: This link contains the solution)<\/a> which asks &#8216;What is the smallest square number that is also square pyramidal number?&#8217; but the traditional cannonball problem has been stated and solved by many people and so it <strong>isn&#8217;t<\/strong> my problem of the week.<\/p>\n<p>My problem is as follows &#8216;<strong>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>&#8216;<\/p>\n<p>As always, there are no prizes I&#8217;m afraid (but if you are a company who would like to sponsor prizes for future POTWs then let me know).\u00a0 I imagine that the solution to this will require a diagram so it might be best to put your solution on a pdf file, web page or some other visual media rather than using the comments section.\u00a0 Finding my email adress is yet another (easy) <a href=\"https:\/\/www.walkingrandomly.com\/?p=53\">puzzle to solve<\/a>.<\/p>\n<p>Have fun.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Imagine that you were the Captain of a sailing ship a few hundred years ago and, in order to protect yourself from pirates, you had a few cannons.\u00a0 Cannons need cannonballs and it is well known that the best way to stack cannon balls is to arrange them as a square pyamid as in the [&hellip;]<\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"closed","ping_status":"closed","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-1052","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-gY","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=\/wp\/v2\/posts\/1052","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=1052"}],"version-history":[{"count":7,"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=\/wp\/v2\/posts\/1052\/revisions"}],"predecessor-version":[{"id":1074,"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=\/wp\/v2\/posts\/1052\/revisions\/1074"}],"wp:attachment":[{"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1052"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1052"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1052"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}