{"id":6392,"date":"2011-05-08T20:18:52","date_gmt":"2011-05-09T01:18:52","guid":{"rendered":"http:\/\/blog.kenperlin.com\/?p=6392"},"modified":"2011-05-08T20:25:02","modified_gmt":"2011-05-09T01:25:02","slug":"more-on-suggester-shapes","status":"publish","type":"post","link":"http:\/\/blog.kenperlin.com\/?p=6392","title":{"rendered":"More on suggester shapes"},"content":{"rendered":"<p>I loved the comments on yesterday&#8217;s post.  Anton&#8217;s solution to the puzzle was very creative.  My only caveat is that it might be hard for someone looking at that shape to realize that it is suggesting a puzzle about squares, since the shape isn&#8217;t built from squares, but rather from shapes with a width:height ratio of 4:3.<\/p>\n<p>My own solution is based on the observation that you can always fold up a corner of a shape made up of squares, thereby removing a square while preserving the length of the perimeter:<\/p>\n<p><center><br \/>\n<img decoding=\"async\" src=\"http:\/\/blog.kenperlin.com\/wp-content\/uploads\/2011\/05\/squares.jpg\"><br \/>\n<\/center><\/p>\n<p>Applying this idea iteratively to the puzzle of finding a shape with the same perimeter as a square, but half the area of that square, I found the following solution:<\/p>\n<p><center><br \/>\n<img decoding=\"async\" src=\"http:\/\/blog.kenperlin.com\/wp-content\/uploads\/2011\/05\/shape1.jpg\"><br \/>\n<\/center><\/p>\n<p>Although to make things more fun (and prettier) in posing the puzzle, I would present the shape rotated 45<sup>o<\/sup>, so it would look more like this:<\/p>\n<p><center><br \/>\n<img decoding=\"async\" src=\"http:\/\/blog.kenperlin.com\/wp-content\/uploads\/2011\/05\/shape2.jpg\"><br \/>\n<\/center><\/p>\n<p>The puzzle Alec discussed in his comment &#8212; to find a shape with the same perimeter as a circle, but half the area of that circle &#8212; has a solution very different from the one he proposes, reminiscent of my recent <a href=http:\/\/blog.kenperlin.com\/?p=6158 target=1>Yin Yang post<\/a>.  Actually, it has an infinite number of solutions, of which these are the first two in a series:<\/p>\n<p><center><br \/>\n<img decoding=\"async\" src=\"http:\/\/blog.kenperlin.com\/wp-content\/uploads\/2011\/05\/circle11.jpg\"><\/p>\n<p><img decoding=\"async\" src=\"http:\/\/blog.kenperlin.com\/wp-content\/uploads\/2011\/05\/circle2.jpg\"><br \/>\n<\/center><\/p>\n","protected":false},"excerpt":{"rendered":"<p>I loved the comments on yesterday&#8217;s post. Anton&#8217;s solution to the puzzle was very creative. My only caveat is that it might be hard for someone looking at that shape to realize that it is suggesting a puzzle about squares, since the shape isn&#8217;t built from squares, but rather from shapes with a width:height ratio &hellip; <a href=\"http:\/\/blog.kenperlin.com\/?p=6392\" class=\"more-link\">Continue reading<span class=\"screen-reader-text\"> &#8220;More on suggester shapes&#8221;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[1],"tags":[],"_links":{"self":[{"href":"http:\/\/blog.kenperlin.com\/index.php?rest_route=\/wp\/v2\/posts\/6392"}],"collection":[{"href":"http:\/\/blog.kenperlin.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/blog.kenperlin.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/blog.kenperlin.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/blog.kenperlin.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=6392"}],"version-history":[{"count":10,"href":"http:\/\/blog.kenperlin.com\/index.php?rest_route=\/wp\/v2\/posts\/6392\/revisions"}],"predecessor-version":[{"id":6408,"href":"http:\/\/blog.kenperlin.com\/index.php?rest_route=\/wp\/v2\/posts\/6392\/revisions\/6408"}],"wp:attachment":[{"href":"http:\/\/blog.kenperlin.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6392"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/blog.kenperlin.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6392"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/blog.kenperlin.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6392"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}