{"id":1019,"date":"2009-04-14T16:45:13","date_gmt":"2009-04-14T15:45:13","guid":{"rendered":"http:\/\/www.walkingrandomly.com\/?p=1019"},"modified":"2010-02-01T11:48:47","modified_gmt":"2010-02-01T10:48:47","slug":"pandigital-fraction-sums","status":"publish","type":"post","link":"https:\/\/walkingrandomly.com\/?p=1019","title":{"rendered":"Pandigital fraction sums"},"content":{"rendered":"

Over at Wild about Math, Sol introduced pandigital fraction sums<\/a> of the form<\/p>\n

\" \light 50 \frac{1}{2} + 49 \frac{38}{76} =100\"<\/p>\n

I call these sums pandigital because the left hand side contains every digit from 0-9 once and once only.\u00a0 Sol challenged his readers to come up with more equations of this type and so I had a crack using Mathematica.<\/p>\n

In order to keep the number of solutions down (and to keep them as close as possible to Sol’s original examples), I only considered sums with four terms – 2 of which were integers and 2 of which were fractions. In other words sums of the form<\/p>\n

\" \light a+b+\frac{c}{d}+\frac{e}{f}=100\"<\/p>\n

My next simplification was to only consider numerators and denominators of up to 2 digits each so fractions such as 123\/456 were immediately out.\u00a0 As you might expect, I also didn’t count trivial re-arrangements of the terms as separate solutions.<\/p>\n

Finally, I didn’t insist that the fractions were written in their lowest terms so, for example, 38\/76 is allowed even though it could be rewritten as 1\/2.\u00a0 Since one of Sol’s original examples contained such a fraction, I didn’t want to remove them.\u00a0 This also meant that I allowed fractions that could be rewritten as integers such as 48\/16.<\/p>\n

Even with all of these restrictions in place I came up with 580 solutions<\/a>!\u00a0 One of which is<\/p>\n

\" \light 9+87+\frac{16}{5}+\frac{32}{40}=100\"<\/p>\n

Interestingly, (unless I have made a mistake) it seems that if you insist on having fractions in their lowest terms then there are no solutions at all!<\/p>\n

Does anyone out there have anything to add to this little problem?\u00a0 Have I messed up and missed a solution or is one of my solutions incorrect?\u00a0 Have I missed something obvious?\u00a0 Are there more interesting variations?\u00a0 Anything?<\/p>\n","protected":false},"excerpt":{"rendered":"

Over at Wild about Math, Sol introduced pandigital fraction sums of the form I call these sums pandigital because the left hand side contains every digit from 0-9 once and once only.\u00a0 Sol challenged his readers to come up with more equations of this type and so I had a crack using Mathematica. In order […]<\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[6],"tags":[],"class_list":["post-1019","post","type-post","status-publish","format-standard","hentry","category-general-math"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p3swhs-gr","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=\/wp\/v2\/posts\/1019","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1019"}],"version-history":[{"count":24,"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=\/wp\/v2\/posts\/1019\/revisions"}],"predecessor-version":[{"id":1042,"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=\/wp\/v2\/posts\/1019\/revisions\/1042"}],"wp:attachment":[{"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1019"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1019"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1019"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}