{"id":176,"date":"2019-06-26T22:54:40","date_gmt":"2019-06-26T22:54:40","guid":{"rendered":"http:\/\/funfacts.104.42.120.246.xip.io\/?page_id=176"},"modified":"2020-01-03T21:37:30","modified_gmt":"2020-01-03T21:37:30","slug":"sum-of-cubes","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/funfacts\/sum-of-cubes\/","title":{"rendered":"Sum of Cubes"},"content":{"rendered":"\n

The sum of cubes is just the square of the triangular numbers!<\/p>\n\n\n\n

1<\/strong>3<\/sup>\u00a0+\u00a02<\/strong>3<\/sup>\u00a0+ … +\u00a0n<\/strong>3<\/sup>\u00a0= (1<\/strong>\u00a0+\u00a02<\/strong>\u00a0+ … +\u00a0n<\/strong>)2<\/sup>.<\/p>\n\n\n\n

And there is a nice proof by picture, too. Can you figure out how this diagram illustrates the identity?<\/p>\n\n\n\n

A<\/font>BB<\/font>CCC<\/font>
BAA<\/font>BBB<\/font>
BAA<\/font>BBB<\/font>
BCCAAA<\/font>
BCCAAA<\/font>
BCCAAA<\/font><\/p>\n\n\n\n

Presentation\u00a0Suggestions:<\/strong>
Draw this picture and see if your students can figure out why the diagram is a “proof without words”!<\/p>\n\n\n\n

The Math Behind the Fact:<\/strong>
The diagram illustrates the identity for n<\/strong>=3. Note that the square has (1+2+3)2<\/sup> letters in it. But now I also claim that there is 13<\/sup> red letter, 23<\/sup> green letters, and 33<\/sup> blue letters.<\/p>\n\n\n\n

This can be seen by arranging the letters in “layers” of a cube! The red cube has one layer (A). The green cube has two layers (A and B) with 4 letters in each. The blue cube has three layers (A, B, and C) with 9 letters in each.<\/p>\n\n\n\n

This construction easily generalizes for arbitrary\u00a0n<\/strong>. You can follow this with the Fun Fact Sum of Cubes and Beyond.<\/p>\n\n\n\n

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

References:<\/strong>
John H. Conway and Richard K. Guy, The Book of Numbers<\/a>, pp.58.<\/p>\n\n\n\n

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

The sum of cubes is just the square of the triangular numbers! 13\u00a0+\u00a023\u00a0+ … +\u00a0n3\u00a0= (1\u00a0+\u00a02\u00a0+ … +\u00a0n)2. And there...<\/p>\n","protected":false},"author":7,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[3,8,10,27],"class_list":["post-176","page","type-page","status-publish","hentry","tag-easy","tag-geometry","tag-numtheory","tag-proofs-without-words"],"jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/176","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=176"}],"version-history":[{"count":3,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/176\/revisions"}],"predecessor-version":[{"id":1659,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/176\/revisions\/1659"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/media?parent=176"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/tags?post=176"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}