{"id":192,"date":"2019-06-26T22:57:52","date_gmt":"2019-06-26T22:57:52","guid":{"rendered":"http:\/\/funfacts.104.42.120.246.xip.io\/?page_id=192"},"modified":"2020-01-03T22:43:46","modified_gmt":"2020-01-03T22:43:46","slug":"toggling-light-switches","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/funfacts\/toggling-light-switches\/","title":{"rendered":"Toggling Light Switches"},"content":{"rendered":"\n<div class=\"wp-block-image\"><figure class=\"alignright\"><img loading=\"lazy\" decoding=\"async\" width=\"144\" height=\"186\" data-attachment-id=\"1682\" data-permalink=\"https:\/\/math.hmc.edu\/funfacts\/toggling-light-switches\/10001-4-5-1-2\/\" data-orig-file=\"https:\/\/math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2020\/01\/10001.4-5.1.gif\" data-orig-size=\"144,186\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"10001.4-5.1\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2020\/01\/10001.4-5.1.gif\" src=\"https:\/\/math.hmc.edu\/funfacts\/wp-content\/uploads\/sites\/4\/2020\/01\/10001.4-5.1.gif\" alt=\"\" class=\"wp-image-1682\"\/><\/figure><\/div>\n\n\n\n<p>Imagine 100 light bulbs with light switches numbered 1 through 100, all in a row, all off. Suppose you do the following: toggle all switches that are multiples of 1, then toggle all switches that are multiples of 2, then toggle all switches that are multiples of 3, etc.<\/p>\n\n\n\n<p>By the time you are finished (and have toggled multiples of 100, which is just the last switch), which light bulbs are on and which are off?<\/p>\n\n\n\n<p>Fun fact: The light bulbs which are on are the ones numbered 1, 4, 9, 16, &#8230; all the squares!<\/p>\n\n\n\n<p><strong>Presentation\u00a0Suggestions:<\/strong><br>Draw a suggestive picture and work out whether the first few light bulbs are on or off. Let them see or conjecture a\u00a0pattern, then have them (as a fun homework) see and figure out why it is true! Maybe a light bulb will go on when they do this!<\/p>\n\n\n\n<p><strong>The\u00a0Math\u00a0Behind\u00a0the\u00a0Fact:<\/strong><br>The\u00a0<em>n<\/em>-th light bulb is toggled once for every factor of\u00a0<em>n<\/em>. Squares are the only numbers with an odd number of\u00a0factors, which can be seen because every factor J of a number, has a co-factor K for which JK=n. This pairs up all the factors of n, unless J=K, which only occurs when n is a square.<\/p>\n\n\n\n<p><strong>How to Cite this Page:<\/strong>&nbsp;<br>Su, Francis E., et al. &#8220;Toggling Light Switches.&#8221;&nbsp;<em>Math Fun Facts<\/em>. &lt;http:\/\/www.math.hmc.edu\/funfacts&gt;.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Imagine 100 light bulbs with light switches numbered 1 through 100, all in a row, all off. Suppose you do&#46;&#46;&#46;<\/p>\n","protected":false},"author":7,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[9,3,10],"class_list":["post-192","page","type-page","status-publish","hentry","tag-combinatorics","tag-easy","tag-numtheory"],"jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/192","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/users\/7"}],"replies":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/comments?post=192"}],"version-history":[{"count":3,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/192\/revisions"}],"predecessor-version":[{"id":1683,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/192\/revisions\/1683"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/media?parent=192"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/tags?post=192"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}