{"id":248,"date":"2019-06-26T23:11:04","date_gmt":"2019-06-26T23:11:04","guid":{"rendered":"http:\/\/funfacts.104.42.120.246.xip.io\/?page_id=248"},"modified":"2019-11-22T17:11:45","modified_gmt":"2019-11-22T17:11:45","slug":"four-color-theorem","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/funfacts\/four-color-theorem\/","title":{"rendered":"Four Color Theorem"},"content":{"rendered":"\n
\"\"<\/figure><\/div>\n\n\n\n

Are four colors always enough to color any map so that no two countries that share a border (in more than single points) have the same color?<\/p>\n\n\n\n

It is easy to show that you need at least<\/em> four colors, because Figure 1 shows a map with four countries, each of which is touching the other. But is four sufficient for any map?<\/p>\n\n\n\n

Francis Guthrie made this conjecture in 1852, but it remained unproven until 1976, when Wolfgang Haken and Kenneth Appel showed that it was true!<\/p>\n\n\n\n

Also, quite interestingly, this proof required the assistance of a computer to check 1,936 different cases that every other case can be reduced to! To date no one knows a quick short proof of this theorem.<\/p>\n\n\n\n

Presentation Suggestions:<\/strong>
Draw a few pictures to illustrate why the problem is difficult, and why (as some might ask) it is not valid to try to “check by example”.<\/p>\n\n\n\n

The\u00a0Math\u00a0Behind\u00a0the\u00a0Fact:<\/strong>
By the way, showing that five colors is sufficient is relatively easy, and was proved in 1890. The ideas involved in this and the four color theorem come from\u00a0graph theory: each map can be represented by a graph in which each country is a node, and two nodes are connected by an edge if they share a common border. The four color theorem is true for maps on a plane or a sphere. The answer is different for geographic maps on a\u00a0torus; it turns out that 7 colors is necessary and sufficient then…<\/p>\n\n\n\n

How to Cite this Page:<\/strong> 
Su, Francis E., et al. “Four Color Theorem.” Math Fun Facts<\/em>. <http:\/\/www.math.hmc.edu\/funfacts>.<\/p>\n\n\n\n

Fun Fact suggested by: <\/strong>
Lesley Ward <\/p>\n","protected":false},"excerpt":{"rendered":"

Are four colors always enough to color any map so that no two countries that share a border (in more...<\/p>\n","protected":false},"author":7,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[109,9,3,88,12],"class_list":["post-248","page","type-page","status-publish","hentry","tag-coloring-problems","tag-combinatorics","tag-easy","tag-graph-theory","tag-other"],"jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/248","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=248"}],"version-history":[{"count":3,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/248\/revisions"}],"predecessor-version":[{"id":1445,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/248\/revisions\/1445"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/media?parent=248"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/tags?post=248"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}