{"id":148,"date":"2020-01-22T15:52:22","date_gmt":"2020-01-22T15:52:22","guid":{"rendered":"http:\/\/mathfun4kids.com\/mlog\/?p=148"},"modified":"2020-10-12T15:55:00","modified_gmt":"2020-10-12T15:55:00","slug":"mathcounts-exercises-15","status":"publish","type":"post","link":"https:\/\/mathfun4kids.com\/mlog\/?p=148","title":{"rendered":"MATHCOUNTS Exercises &#8211; 15"},"content":{"rendered":"\n<p>How many distinct unit cubes are there with two faces painted red, two faces\npainted green and two faces painted blue? Two unit cubes are considered distinct\nif one unit cube cannot be obtained by rotating the other. <em>SOURCE: 2014 MATHCOUNTS Chapter Target Round.<\/em> <a onclick='toggle_visibility(\"mathcounts-exercises-15-solution\");'>Click here for the solution.<\/a><\/p>\n<div id='mathcounts-exercises-15-solution' style='display:none'>\n<p><strong>Solution<\/strong> First, there is only one way to paint the cube with all opposite faces painted in the same color, as shown below, with the front face colored in red:<\/p>\n<a href=\"wp-content\/uploads\/2020\/10\/Screen-Shot-2020-01-21-at-11.25.39-1-PM.png\"><img width=\"128\" src=\"wp-content\/uploads\/2020\/10\/Screen-Shot-2020-01-21-at-11.25.39-1-PM.png\" alt=\"\" class=\"aligncenter wp-image-149\"\/><\/a>\n<!-- Click <a onclick='toggle_visibility(\"mathcounts-exercises-15-cube-1\");'>here<\/a> to view  animated turns of the cube with opposite faces in same colors.<\/p>\n<div id='mathcounts-exercises-15-cube-1' style='display:none'>\n<table border=0 align=center width=\"100%\">\n<tr>\n<td width=\"100%\" align=\"center\">\n<p class=\"aligncenter\"><iframe align=\"center\" style=\"margin: auto; width: 272px; height: 360px; overflow: hidden;\" src=\"\/cube\/cube-1.htm\" width=\"272\" height=\"360\" scrolling=\"no\">iframes not supported<\/iframe><\/p>\n<td>\n<\/tr>\n<\/table>\n<\/div>\n-->\n<p>Second, there are 3 ways to paint the cube with one pair of opposite faces in the same color, while the other two pairs of opposite faces in different colors, as shown below, with the front face in the same color of the back face:<\/p>\n<a href=\"wp-content\/uploads\/2020\/10\/Screen-Shot-2019-12-27-at-11.59.34-1-PM.png\"><img loading=\"lazy\" src=\"wp-content\/uploads\/2020\/10\/Screen-Shot-2019-12-27-at-11.59.34-1-PM.png\" alt=\"\" title=\"Screen Shot 2019-12-27 at 11.59.34-1 PM\" width=\"363\" height=\"124\" class=\"aligncenter size-full wp-image-150\" srcset=\"https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2020\/10\/Screen-Shot-2019-12-27-at-11.59.34-1-PM.png 726w, https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2020\/10\/Screen-Shot-2019-12-27-at-11.59.34-1-PM-300x102.png 300w\" sizes=\"(max-width: 363px) 100vw, 363px\" \/><\/a>\n<p>Finally, there are 2 ways to paint the cube without any pair of opposite faces in the same color, as shown below, with the front face colored in red:<\/p>\n<a href=\"wp-content\/uploads\/2020\/10\/Screen-Shot-2020-01-21-at-11.26.32-1-PM.png\"><img loading=\"lazy\" src=\"wp-content\/uploads\/2020\/10\/Screen-Shot-2020-01-21-at-11.26.32-1-PM.png\" alt=\"\" title=\"Screen Shot 2020-01-21 at 11.26.32-1 PM\" width=\"248\" height=\"124\" class=\"aligncenter size-full wp-image-151\" srcset=\"https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2020\/10\/Screen-Shot-2020-01-21-at-11.26.32-1-PM.png 496w, https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2020\/10\/Screen-Shot-2020-01-21-at-11.26.32-1-PM-300x150.png 300w\" sizes=\"(max-width: 248px) 100vw, 248px\" \/><\/a>\n<!--\nClick <a onclick='toggle_visibility(\"mathcounts-exercises-15-cube-5-6\");'>here<\/a> to view animated turns of two different cubes with faces in same colors are always adjacent.<\/p>\n<div id='mathcounts-exercises-15-cube-5-6' style='display:none'>\n<table border=0>\n<tr>\n<td>\n<p class=\"aligncenter\"><iframe style=\"margin: auto; width: 272px; height: 360px; overflow: hidden;\" src=\"\/cube\/cube-5.htm\" width=\"272\" height=\"360\" scrolling=\"no\">iframes not supported<\/iframe><\/p>\n<td>\n<td>\n<p class=\"aligncenter\"><iframe style=\"margin: auto; width: 272px; height: 360px; overflow: hidden;\" src=\"\/cube\/cube-6.htm\" width=\"272\" height=\"360\" scrolling=\"no\">iframes not supported<\/iframe><\/p>\n<\/td>\n<\/tr>\n<\/table>\n<\/div>\n-->\n<p>Therefore, the final answer to the question is 6.\n<\/p><\/div>\n","protected":false},"excerpt":{"rendered":"<p>How many distinct unit cubes are there with two faces painted red, two faces painted green and two faces painted blue? Two unit cubes are considered distinct if one unit cube cannot be obtained by rotating the other. SOURCE: 2014 &hellip; <a href=\"https:\/\/mathfun4kids.com\/mlog\/?p=148\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false},"categories":[10,9,11],"tags":[],"_links":{"self":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/148"}],"collection":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=148"}],"version-history":[{"count":3,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/148\/revisions"}],"predecessor-version":[{"id":154,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/148\/revisions\/154"}],"wp:attachment":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=148"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=148"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=148"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}