If you know how to prove things by induction, then here is an amazing fact:<\/p>\n\n\n\n
Theorem. All horses are the same color.<\/p>\n\n\n\n
Proof. We’ll induct on the number of horses. Base case: 1 horse. Clearly with just 1 horse, all horses have the same color.<\/p>\n\n\n\n
Now, for the inductive step: we’ll show that if it is true for any group of N horses, that all have the same color, then it is true for any group of N+1 horses.<\/p>\n\n\n\n
Well, given any set of N+1 horses, if you exclude the last horse, you get a set of N horses. By the inductive step these N horses all have the same color. But by excluding the first<\/em> horse in the pack of N+1 horses, you can conclude that the last N horses also have the same color. Therefore all N+1 horses have the same color. QED.<\/p>\n\n\n\n Hmmn… clearly not all horses have the same color. So what’s wrong with this proof by induction?<\/p>\n\n\n\n Presentation\u00a0Suggestions:<\/strong> The Math Behind the Fact:<\/strong> Well actually, the argument in the inductive step breaks down in going from n=1 to n=2, because the first 1 horse and the last 1 horse have no horses in common, and therefore may not all have the same color.<\/p>\n\n\n\n How to Cite this Page:<\/strong> Fun Fact suggested by:<\/strong> If you know how to prove things by induction, then here is an amazing fact: Theorem. All horses are the...<\/p>\n","protected":false},"author":7,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[57,50,12,52],"class_list":["post-553","page","type-page","status-publish","hentry","tag-advanced","tag-logic","tag-other","tag-puzzle"],"jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/553","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=553"}],"version-history":[{"count":3,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/553\/revisions"}],"predecessor-version":[{"id":1266,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/553\/revisions\/1266"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/media?parent=553"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/tags?post=553"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
This delightful puzzle is an excellent test of student understanding of proofs by\u00a0induction.<\/p>\n\n\n\n
Hint: what could be wrong? You showed the base case. And you showed the inductive step, right?<\/p>\n\n\n\n
Su, Francis E., et al. “All Horses are the Same Color.” Math Fun Facts<\/em>. <http:\/\/www.math.hmc.edu\/funfacts>.<\/p>\n\n\n\n
Arthur Benjamin<\/p>\n","protected":false},"excerpt":{"rendered":"