{"id":1529,"date":"2017-12-20T22:35:28","date_gmt":"2017-12-20T22:35:28","guid":{"rendered":"http:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/?post_type=chapter&#038;p=1529"},"modified":"2017-12-31T22:14:23","modified_gmt":"2017-12-31T22:14:23","slug":"painted-cubes","status":"publish","type":"chapter","link":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/chapter\/painted-cubes\/","title":{"raw":"Painted Cubes","rendered":"Painted Cubes"},"content":{"raw":"<p class=\"p1\">You can build up squares from smaller squares:<\/p>\r\n\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: center\"><img src=\"http:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/12\/1x1sq.png\" alt=\"\" width=\"91\" height=\"87\" class=\"alignnone wp-image-2573\" \/><\/td>\r\n<td style=\"text-align: center\"><img src=\"http:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/12\/2x2sq-300x294.png\" alt=\"\" width=\"157\" height=\"154\" class=\"alignnone wp-image-2574\" \/><\/td>\r\n<td style=\"text-align: center\"><img src=\"http:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/12\/3x3q-300x300.png\" alt=\"\" width=\"276\" height=\"277\" class=\"alignnone wp-image-2575\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center\">1\u00a0\u00d7 1 square<\/td>\r\n<td style=\"text-align: center\">2\u00a0\u00d7 2 square<\/td>\r\n<td style=\"text-align: center\">3\u00a0\u00d7 3 square<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p class=\"p2\">In a similar way, you can build up cubes from smaller cubes:<\/p>\r\n\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: center\"><a title=\"Attribution must be given to Robert Webb's Stella software as the creator of this image along with a link to the website: http:\/\/www.software3d.com\/Stella.php. [Attribution], via Wikimedia Commons\" href=\"https:\/\/commons.wikimedia.org\/wiki\/File%3AFace_colored_cube.png\"><img width=\"98\" alt=\"Face colored cube\" src=\"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/0\/02\/Face_colored_cube.png\/512px-Face_colored_cube.png\" height=\"99\" class=\"\" \/><\/a><\/td>\r\n<td style=\"text-align: center\"><a title=\"By Mike Gonzalez (TheCoffee) (Work by Mike Gonzalez (TheCoffee)) [CC BY-SA 3.0 (https:\/\/creativecommons.org\/licenses\/by-sa\/3.0) or GFDL (http:\/\/www.gnu.org\/copyleft\/fdl.html)], via Wikimedia Commons\" href=\"https:\/\/commons.wikimedia.org\/wiki\/File%3APocket_cube_solved.jpg\"><img width=\"202\" alt=\"Pocket cube solved\" src=\"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/9\/93\/Pocket_cube_solved.jpg\" height=\"202\" class=\"aligncenter\" \/><\/a><\/td>\r\n<td style=\"text-align: center\"><a title=\"By Mike Gonzalez (TheCoffee) (Work by Mike Gonzalez (TheCoffee)) [CC BY-SA 3.0 (https:\/\/creativecommons.org\/licenses\/by-sa\/3.0) or GFDL (http:\/\/www.gnu.org\/copyleft\/fdl.html)], via Wikimedia Commons\" href=\"https:\/\/commons.wikimedia.org\/wiki\/File%3ARubiks_cube_scrambled.jpg\"><img width=\"300\" alt=\"Rubiks cube scrambled\" src=\"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/e\/e5\/Rubiks_cube_scrambled.jpg\" height=\"301\" class=\"\" \/><\/a><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center\">1\u00a0\u00d7 1\u00a0\u00d7 1 cube[footnote]Image by\u00a0<span class=\"licensetpl_attr\">Robert Webb's\u00a0<a href=\"https:\/\/en.wikipedia.org\/wiki\/Stella_(software)\" class=\"extiw\" title=\"en:Stella (software)\">Stella software<\/a>:\u00a0<a href=\"http:\/\/www.software3d.com\/Stella.php\">http:\/\/www.software3d.com\/Stella.php<\/a>, via Wikimedia Commons.<\/span>[\/footnote]<\/td>\r\n<td style=\"text-align: center\">2\u00a0\u00d7 2\u00a0\u00d7 2 cube[footnote]Image by Mike Gonzalez (TheCoffee) (Work by Mike Gonzalez (TheCoffee)) [<a href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/3.0\">CC BY-SA 3.0<\/a>], via Wikimedia Commons.[\/footnote]<\/td>\r\n<td style=\"text-align: center\">3\u00a0\u00d7 3\u00a0\u00d7 3 cube[footnote]Image by Mike Gonzalez (TheCoffee) (Work by Mike Gonzalez (TheCoffee)) [CC BY-SA 3.0], via Wikimedia Commons.[\/footnote]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"textbox examples\">\r\n<h3 itemprop=\"educationalUse\">Think \/ Pair \/ Share<\/h3>\r\n<p class=\"p1\">We call a 1\u00a0\u00d7 1\u00a0\u00d7 1 cube\u00a0a <b>unit cube<\/b>.<\/p>\r\n\r\n<ul>\r\n \t<li>How many unit cubes are in a\u00a02\u00a0\u00d7 2\u00a0\u00d7 2\u00a0cube?<\/li>\r\n \t<li>How many unit cubes are in a\u00a03\u00a0\u00d7 3\u00a0\u00d7 3 cube?<\/li>\r\n \t<li>How many unit cubes are in a\u00a0<em>n<\/em>\u00a0\u00d7 <em>n<\/em>\u00a0\u00d7 <em>n<\/em> cube?<\/li>\r\n<\/ul>\r\nExplain your answers.\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3 itemprop=\"educationalUse\">Problem 10<\/h3>\r\n<p class=\"p1\">Imagine you build a\u00a03\u00a0\u00d7 3\u00a0\u00d7 3 cube\u00a0from 27 small white unit cubes. Then you take your cube and dip it into a bucket of bright blue paint. After the cube dries, you take it apart, separating the small unit cubes.<\/p>\r\n\r\n<ol>\r\n \t<li>After you take the cube apart, some of the unit cubes are still all white (no blue paint). How many? How do you know you are right?<\/li>\r\n \t<li>After you take the cube apart, some of the unit cubes have blue paint on just one face. How many? How do you know you are right?<\/li>\r\n \t<li>After you take the cube apart, some of the unit cubes have blue paint on two faces. How many? How do you know you are right?<\/li>\r\n \t<li>After you take the cube apart, some of the unit cubes have blue paint on three faces. How many? How do you know you are right?<\/li>\r\n \t<li>After you take the cube apart, do any of the unit cubes have blue paint on more than three faces? How many? How do you know you are right?<\/li>\r\n<\/ol>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3 itemprop=\"educationalUse\">Problem 11<\/h3>\r\n<p class=\"p1\">Generalize your work on Problem 10. What if you started with a 2 \u00d7 2 \u00d7 2 cube? Answer the same questions. What about a 4 \u00d7 4 \u00d7 4 cube? How about an\u00a0<em>n<\/em>\u00a0\u00d7 <em>n<\/em>\u00a0\u00d7 <em>n<\/em> cube? Be sure to justify what you say.<\/p>\r\n\r\n<\/div>","rendered":"<p class=\"p1\">You can build up squares from smaller squares:<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: center\"><img loading=\"lazy\" decoding=\"async\" src=\"\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/12\/1x1sq.png\" alt=\"\" width=\"91\" height=\"87\" class=\"alignnone wp-image-2573\" srcset=\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/12\/1x1sq.png 182w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/12\/1x1sq-65x62.png 65w\" sizes=\"auto, (max-width: 91px) 100vw, 91px\" \/><\/td>\n<td style=\"text-align: center\"><img loading=\"lazy\" decoding=\"async\" src=\"\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/12\/2x2sq-300x294.png\" alt=\"\" width=\"157\" height=\"154\" class=\"alignnone wp-image-2574\" srcset=\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/12\/2x2sq-300x294.png 300w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/12\/2x2sq-65x64.png 65w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/12\/2x2sq-225x221.png 225w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/12\/2x2sq.png 314w\" sizes=\"auto, (max-width: 157px) 100vw, 157px\" \/><\/td>\n<td style=\"text-align: center\"><img loading=\"lazy\" decoding=\"async\" src=\"\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/12\/3x3q-300x300.png\" alt=\"\" width=\"276\" height=\"277\" class=\"alignnone wp-image-2575\" srcset=\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/12\/3x3q-300x300.png 300w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/12\/3x3q-150x150.png 150w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/12\/3x3q-65x65.png 65w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/12\/3x3q-225x226.png 225w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/12\/3x3q-350x352.png 350w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/12\/3x3q.png 452w\" sizes=\"auto, (max-width: 276px) 100vw, 276px\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center\">1\u00a0\u00d7 1 square<\/td>\n<td style=\"text-align: center\">2\u00a0\u00d7 2 square<\/td>\n<td style=\"text-align: center\">3\u00a0\u00d7 3 square<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p class=\"p2\">In a similar way, you can build up cubes from smaller cubes:<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: center\"><a title=\"Attribution must be given to Robert Webb's Stella software as the creator of this image along with a link to the website: http:\/\/www.software3d.com\/Stella.php. [Attribution], via Wikimedia Commons\" href=\"https:\/\/commons.wikimedia.org\/wiki\/File%3AFace_colored_cube.png\"><img loading=\"lazy\" decoding=\"async\" width=\"98\" alt=\"Face colored cube\" src=\"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/0\/02\/Face_colored_cube.png\/512px-Face_colored_cube.png\" height=\"99\" class=\"\" \/><\/a><\/td>\n<td style=\"text-align: center\"><a title=\"By Mike Gonzalez (TheCoffee) (Work by Mike Gonzalez (TheCoffee)) [CC BY-SA 3.0 (https:\/\/creativecommons.org\/licenses\/by-sa\/3.0) or GFDL (http:\/\/www.gnu.org\/copyleft\/fdl.html)], via Wikimedia Commons\" href=\"https:\/\/commons.wikimedia.org\/wiki\/File%3APocket_cube_solved.jpg\"><img loading=\"lazy\" decoding=\"async\" width=\"202\" alt=\"Pocket cube solved\" src=\"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/9\/93\/Pocket_cube_solved.jpg\" height=\"202\" class=\"aligncenter\" \/><\/a><\/td>\n<td style=\"text-align: center\"><a title=\"By Mike Gonzalez (TheCoffee) (Work by Mike Gonzalez (TheCoffee)) [CC BY-SA 3.0 (https:\/\/creativecommons.org\/licenses\/by-sa\/3.0) or GFDL (http:\/\/www.gnu.org\/copyleft\/fdl.html)], via Wikimedia Commons\" href=\"https:\/\/commons.wikimedia.org\/wiki\/File%3ARubiks_cube_scrambled.jpg\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" alt=\"Rubiks cube scrambled\" src=\"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/e\/e5\/Rubiks_cube_scrambled.jpg\" height=\"301\" class=\"\" \/><\/a><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center\">1\u00a0\u00d7 1\u00a0\u00d7 1 cube<a class=\"footnote\" title=\"Image by\u00a0Robert Webb's\u00a0Stella software:\u00a0http:\/\/www.software3d.com\/Stella.php, via Wikimedia Commons.\" id=\"return-footnote-1529-1\" href=\"#footnote-1529-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a><\/td>\n<td style=\"text-align: center\">2\u00a0\u00d7 2\u00a0\u00d7 2 cube<a class=\"footnote\" title=\"Image by Mike Gonzalez (TheCoffee) (Work by Mike Gonzalez (TheCoffee)) [CC BY-SA 3.0], via Wikimedia Commons.\" id=\"return-footnote-1529-2\" href=\"#footnote-1529-2\" aria-label=\"Footnote 2\"><sup class=\"footnote\">[2]<\/sup><\/a><\/td>\n<td style=\"text-align: center\">3\u00a0\u00d7 3\u00a0\u00d7 3 cube<a class=\"footnote\" title=\"Image by Mike Gonzalez (TheCoffee) (Work by Mike Gonzalez (TheCoffee)) [CC BY-SA 3.0], via Wikimedia Commons.\" id=\"return-footnote-1529-3\" href=\"#footnote-1529-3\" aria-label=\"Footnote 3\"><sup class=\"footnote\">[3]<\/sup><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox examples\">\n<h3 itemprop=\"educationalUse\">Think \/ Pair \/ Share<\/h3>\n<p class=\"p1\">We call a 1\u00a0\u00d7 1\u00a0\u00d7 1 cube\u00a0a <b>unit cube<\/b>.<\/p>\n<ul>\n<li>How many unit cubes are in a\u00a02\u00a0\u00d7 2\u00a0\u00d7 2\u00a0cube?<\/li>\n<li>How many unit cubes are in a\u00a03\u00a0\u00d7 3\u00a0\u00d7 3 cube?<\/li>\n<li>How many unit cubes are in a\u00a0<em>n<\/em>\u00a0\u00d7 <em>n<\/em>\u00a0\u00d7 <em>n<\/em> cube?<\/li>\n<\/ul>\n<p>Explain your answers.<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3 itemprop=\"educationalUse\">Problem 10<\/h3>\n<p class=\"p1\">Imagine you build a\u00a03\u00a0\u00d7 3\u00a0\u00d7 3 cube\u00a0from 27 small white unit cubes. Then you take your cube and dip it into a bucket of bright blue paint. After the cube dries, you take it apart, separating the small unit cubes.<\/p>\n<ol>\n<li>After you take the cube apart, some of the unit cubes are still all white (no blue paint). How many? How do you know you are right?<\/li>\n<li>After you take the cube apart, some of the unit cubes have blue paint on just one face. How many? How do you know you are right?<\/li>\n<li>After you take the cube apart, some of the unit cubes have blue paint on two faces. How many? How do you know you are right?<\/li>\n<li>After you take the cube apart, some of the unit cubes have blue paint on three faces. How many? How do you know you are right?<\/li>\n<li>After you take the cube apart, do any of the unit cubes have blue paint on more than three faces? How many? How do you know you are right?<\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3 itemprop=\"educationalUse\">Problem 11<\/h3>\n<p class=\"p1\">Generalize your work on Problem 10. What if you started with a 2 \u00d7 2 \u00d7 2 cube? Answer the same questions. What about a 4 \u00d7 4 \u00d7 4 cube? How about an\u00a0<em>n<\/em>\u00a0\u00d7 <em>n<\/em>\u00a0\u00d7 <em>n<\/em> cube? Be sure to justify what you say.<\/p>\n<\/div>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-1529-1\">Image by\u00a0<span class=\"licensetpl_attr\">Robert Webb's\u00a0<a href=\"https:\/\/en.wikipedia.org\/wiki\/Stella_(software)\" class=\"extiw\" title=\"en:Stella (software)\">Stella software<\/a>:\u00a0<a href=\"http:\/\/www.software3d.com\/Stella.php\">http:\/\/www.software3d.com\/Stella.php<\/a>, via Wikimedia Commons.<\/span> <a href=\"#return-footnote-1529-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><li id=\"footnote-1529-2\">Image by Mike Gonzalez (TheCoffee) (Work by Mike Gonzalez (TheCoffee)) [<a href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/3.0\">CC BY-SA 3.0<\/a>], via Wikimedia Commons. <a href=\"#return-footnote-1529-2\" class=\"return-footnote\" aria-label=\"Return to footnote 2\">&crarr;<\/a><\/li><li id=\"footnote-1529-3\">Image by Mike Gonzalez (TheCoffee) (Work by Mike Gonzalez (TheCoffee)) [CC BY-SA 3.0], via Wikimedia Commons. <a href=\"#return-footnote-1529-3\" class=\"return-footnote\" aria-label=\"Return to footnote 3\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":21,"menu_order":6,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1529","chapter","type-chapter","status-publish","hentry"],"part":1379,"_links":{"self":[{"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/pressbooks\/v2\/chapters\/1529","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/wp\/v2\/users\/21"}],"version-history":[{"count":16,"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/pressbooks\/v2\/chapters\/1529\/revisions"}],"predecessor-version":[{"id":2727,"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/pressbooks\/v2\/chapters\/1529\/revisions\/2727"}],"part":[{"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/pressbooks\/v2\/parts\/1379"}],"metadata":[{"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/pressbooks\/v2\/chapters\/1529\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/wp\/v2\/media?parent=1529"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/pressbooks\/v2\/chapter-type?post=1529"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/wp\/v2\/contributor?post=1529"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/wp\/v2\/license?post=1529"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}