{"id":23,"date":"2010-03-06T21:26:45","date_gmt":"2010-03-06T13:26:45","guid":{"rendered":"http:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/?p=23"},"modified":"2024-02-16T21:14:29","modified_gmt":"2024-02-16T13:14:29","slug":"a-little-conundrum","status":"publish","type":"post","link":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/2010\/03\/a-little-conundrum\/","title":{"rendered":"A Little Conundrum"},"content":{"rendered":"<p>Consider the sentence :<\/p>\n<p style=\"text-align: center;\"><span style=\"color: #0000ff;\"><em><strong>&#8220;Thsi sentence hsa three errors.&#8221;<\/strong><\/em><\/span><\/p>\n<p><span style=\"color: #0000ff;\"><span style=\"color: #000000;\">The puzzle is to find the errors. Two of them are easy to find :<br \/>\n<\/span><\/span><\/p>\n<ul>\n<li><span style=\"color: #0000ff;\"><span style=\"color: #000000;\">&#8220;Thsi&#8221; should read &#8220;This&#8221;,<\/span><\/span><\/li>\n<li><span style=\"color: #0000ff;\"><span style=\"color: #000000;\">&#8220;hsa&#8221; should read &#8220;has&#8221;,<\/span><\/span><\/li>\n<\/ul>\n<p><span style=\"color: #0000ff;\"><span style=\"color: #000000;\"><br \/>\nbut where&#8217;s the third error?<\/span><\/span><\/p>\n<p><span style=\"color: #0000ff;\"><span style=\"color: #000000;\"><!--more--> See if you can find it! Search and search&#8230; eventually it will dawn on you that<\/span><\/span><\/p>\n<ul>\n<li><span style=\"color: #0000ff;\"><span style=\"color: #000000;\">the sentence, in fact, has only two errors, not three.<\/span><\/span><\/li>\n<\/ul>\n<p><span style=\"color: #0000ff;\"><span style=\"color: #000000;\"><br \/>\nSo that&#8217;s the third error! Found it!<\/span><\/span><\/p>\n<p><span style=\"color: #0000ff;\"><span style=\"color: #000000;\">But wait&#8230;. that means the sentence really does have three errors. So we were wrong about the third error.<\/span><\/span><\/p>\n<p><span style=\"color: #0000ff;\"><span style=\"color: #000000;\">I guess it has only two errors after all.<\/span><\/span><\/p>\n<p><span style=\"color: #0000ff;\"><span style=\"color: #000000;\">Only two errors! But the sentence says it has three! Aha! There&#8217;s the third error!<\/span><\/span><\/p>\n<p><span style=\"color: #0000ff;\"><span style=\"color: #000000;\">Oh. That means it only has two errors. Hmm&#8230;.<\/span><\/span><\/p>\n<p><span style=\"color: #0000ff;\"><span style=\"color: #000000;\">Now if this is confusing you, let&#8217;s cut to the chase, and ask the basic question &#8211; <\/span><\/span><\/p>\n<ul>\n<li><span style=\"color: #0000ff;\"><span style=\"color: #000000;\">Is the sentence in error about the number of errors it contains? Does it contain two? or three?<\/span><\/span><\/li>\n<\/ul>\n<p><span style=\"color: #0000ff;\"><span style=\"color: #000000;\"><br \/>\nI showed my son this little conundrum this afternoon. He recognized it quickly as a variant of similar paradoxes I&#8217;ve shown him in the past &#8211; all related to the <a href=\"http:\/\/en.wikipedia.org\/wiki\/Epimenides_paradox\" target=\"_blank\" rel=\"noopener\">Epimenides Paradox<\/a> &#8220;<em>This sentence is not true<\/em>&#8220;.<br \/>\n<\/span><\/span><\/p>\n<p><span style=\"color: #0000ff;\"><span style=\"color: #000000;\">Although this seems, in ways, like a silly logical trick, it keeps turning up in different disguises as you dig deeper into mathematics. In fact, it changed the course of mathematical thinking in the 1930&#8217;s, when a young mathematician named Kurt G\u00f6del constructed the mathematical equivalent of the sentence &#8220;<em>This statement can&#8217;t be proven<\/em>&#8220;<\/span><\/span><\/p>\n<p><span style=\"color: #0000ff;\"><span style=\"color: #000000;\">This meant that the statement had to be true &#8211; or mathematics riddled with deep contradictions. <\/span><\/span><\/p>\n<ul>\n<li><span style=\"color: #0000ff;\"><span style=\"color: #000000;\">If the statement could be proven, it would be wrong. The proof would also be an example proving the opposite of the statement, so that there would be mathematically provable statements whose opposites could also be proven. This would be the equivalent of someone finding a proof that 2+2 was 5.<\/span><\/span><\/li>\n<li><span style=\"color: #0000ff;\"><span style=\"color: #000000;\">If the statement could not be proven, however, that would mean it was true &#8211; but unprovable. This would mean that there are mathematical statements that could never be proven one way or the other. <\/span><\/span><\/li>\n<\/ul>\n<p><span style=\"color: #0000ff;\"><span style=\"color: #000000;\"><br \/>\nThis, and similar ideas, had a big impact on how mathematician thought about their topic over the following decades, and affected other fields also, including computer sciences and the design of modern computers.<br \/>\n<\/span><\/span><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Consider the sentence : &#8220;Thsi sentence hsa three errors.&#8221; The puzzle is to find the errors. Two of them are easy to find : &#8220;Thsi&#8221; should read &#8220;This&#8221;, &#8220;hsa&#8221; should read &#8220;has&#8221;, but where&#8217;s the third error?<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[21,18,19,17,20],"_links":{"self":[{"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/posts\/23"}],"collection":[{"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/comments?post=23"}],"version-history":[{"count":7,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/posts\/23\/revisions"}],"predecessor-version":[{"id":1434,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/posts\/23\/revisions\/1434"}],"wp:attachment":[{"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/media?parent=23"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/categories?post=23"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/tags?post=23"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}