{"id":977,"date":"2014-07-31T23:48:50","date_gmt":"2014-08-01T03:48:50","guid":{"rendered":"http:\/\/www.curtisbright.com\/bln\/?p=977"},"modified":"2014-07-31T23:48:50","modified_gmt":"2014-08-01T03:48:50","slug":"an-extended-hat-puzzle","status":"publish","type":"post","link":"http:\/\/localhost\/blog\/index.php\/2014\/07\/31\/an-extended-hat-puzzle\/","title":{"rendered":"An extended hat puzzle"},"content":{"rendered":"<p>Shortly after hearing about the <a href=\"http:\/\/www.curtisbright.com\/bln\/2014\/06\/30\/a-hat-puzzle\/\">hat puzzle<\/a> I wrote about last month I came across an interesting extension of the problem, which replaces the 100 wizards with an infinite number of wizards:<\/p>\n<p style=\"padding-left: 30px;\"><em>A countably infinite number of wizards are each given a red or blue hat with 50% probability. <em>Each wizard can see everyone\u2019s hat except their own. The wizards have to guess the colour of their hat without communicating in any way, but will be allowed to devise a strategy to coordinate their guesses beforehand. How can they ensure that only a finite number of them guess incorrectly? You may assume the axiom of choice.<\/em><\/em><\/p>\n<p>This seems paradoxical since somehow knowing about other wizard&#8217;s hats&#8212;which are chosen independently from a wizard&#8217;s own hat&#8212;allows each wizard to conclude that they will almost surely guess their hat colour correctly.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Shortly after hearing about the hat puzzle I wrote about last month I came across an interesting extension of the problem, which replaces the 100 wizards with an infinite number of wizards: A countably infinite number of wizards are each &hellip; <a href=\"http:\/\/localhost\/blog\/index.php\/2014\/07\/31\/an-extended-hat-puzzle\/\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4,7],"tags":[],"_links":{"self":[{"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/posts\/977"}],"collection":[{"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/comments?post=977"}],"version-history":[{"count":0,"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/posts\/977\/revisions"}],"wp:attachment":[{"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/media?parent=977"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/categories?post=977"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/tags?post=977"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}