{"id":1098,"date":"2020-05-21T13:30:13","date_gmt":"2020-05-21T20:30:13","guid":{"rendered":"http:\/\/cetinkoc.net\/blog\/?p=1098"},"modified":"2020-08-12T04:19:49","modified_gmt":"2020-08-12T11:19:49","slug":"4-numbers-added-and-multiplied-to-the-same-value","status":"publish","type":"post","link":"https:\/\/cetinkayakoc.net\/blog\/4-numbers-added-and-multiplied-to-the-same-value\/","title":{"rendered":"4 numbers added and multiplied to the same value"},"content":{"rendered":"\n

This puzzle showed up in several places with different anecdotes. The puzzle first specifies a decimal number with two digits after the point, such as n = 7.35. We are supposed to discover 4 different numbers such that n is equal to the sum of and the product of these 4 numbers. This means, given n, we have 2 equations with 4 unknowns:<\/p>\n\n\n\n

a + b + c + d = n
a * b * c * d = n<\/p>\n\n\n\n

The original puzzle gives n as 7.11. <\/p>\n\n\n\n

There is an analytical solution of this problem in the Math Forum<\/a> web site, posted by an author named Dr. Rob. The solution is constructed by finding the factors of the integer 711 and goes through several arguments, and finds the soluton: a = 3.16, b = 1.50, c = 1.25, and d = 1.20. The author also talks about the “uniqueness” of the solution for n = 7.11 which turns out to be true.<\/p>\n\n\n\n

However, for some n values there are more than one solution. I run a Python program, and somewhat exhaustively searched for solutions (my search was not fully exhaustive but included some smart search provisions) for a, b, c, d in the range [0.01,3.99] . I am listing all solutions below that are sorted with respect to n values. <\/p>\n\n\n\n

There are multiple solutions for some n values in my list, but all solutions for that n are different; for example, 7.07 has two different solutions: [0.80, 1.75, 2.02, 2.50] and [1.25, 1.00, 2.02, 2.80]. These pairs of 4 numbers indeed sum and multiply to 7.07<\/p>\n\n\n\n

I also found the same (unique) solution for 7.11: [1.20, 1.25, 1.50, 3.16]<\/p>\n\n\n\n

There are 47 numbers in my list and 36 of them has one solution.
4 of them has two solutions: 6.75, 7.07, 7.92, 8.10
1 of them has three solutions: 7.56<\/p>\n\n\n\n

Here is the List: <\/p>\n\n\n\n

6.44 [1.25, 1.60, 1.75, 1.84]
6.51 [1.25, 1.40, 1.86, 2.00]
6.63 [1.25, 1.25, 1.92, 2.21]
6.75 [1.50, 1.00, 2.25, 2.00]
6.75 [1.20, 1.25, 1.80, 2.50]
6.78 [1.13, 1.25, 2.40, 2.00]
6.84 [1.44, 1.90, 1.00, 2.50]
6.86 [1.40, 1.96, 1.00, 2.50]
7.07 [0.80, 1.75, 2.02, 2.50]
7.07 [1.25, 1.00, 2.02, 2.80]
7.08 [1.18, 1.00, 2.40, 2.50]
7.11 [1.20, 1.25, 1.50, 3.16]
7.13 [1.15, 1.00, 2.48, 2.50]
7.14 [1.02, 1.12, 2.50, 2.50]
7.20 [1.50, 1.50, 1.00, 3.20]
7.25 [0.80, 1.45, 2.50, 2.50]
7.35 [1.05, 1.00, 2.50, 2.80]
7.52 [0.64, 1.88, 2.50, 2.50]
7.56 [0.96, 1.25, 1.75, 3.60]
7.56 [1.12, 1.25, 1.44, 3.75]
7.56 [1.25, 1.25, 1.28, 3.78]
7.62 [1.27, 1.60, 1.00, 3.75]
7.65 [1.20, 1.70, 1.00, 3.75]
7.67 [0.59, 2.08, 2.50, 2.50]
7.70 [1.25, 1.60, 1.00, 3.85]
7.74 [0.80, 1.25, 2.25, 3.44]
7.82 [0.92, 1.00, 2.50, 3.40]
7.86 [0.80, 1.31, 2.00, 3.75]
7.92 [0.75, 1.25, 2.40, 3.52]
7.92 [0.90, 1.00, 2.50, 3.52]
8.01 [0.75, 1.20, 2.50, 3.56]
8.10 [0.75, 1.20, 2.40, 3.75]
8.10 [0.50, 2.40, 2.50, 2.70]
8.16 [0.85, 1.00, 2.56, 3.75]
8.22 [0.48, 2.50, 2.50, 2.74]
8.28 [0.69, 1.25, 2.50, 3.84]
8.40 [0.50, 2.40, 2.00, 3.50]
8.64 [0.50, 1.80, 2.50, 3.84]
8.67 [0.50, 1.70, 2.72, 3.75]
8.82 [0.42, 2.40, 2.50, 3.50]
8.91 [0.45, 2.50, 2.00, 3.96]
8.96 [0.40, 2.50, 2.56, 3.50]
9.00 [0.40, 2.50, 2.50, 3.60]
9.24 [0.64, 1.00, 3.75, 3.85]
9.36 [0.52, 1.25, 3.75, 3.84]
9.48 [0.50, 1.28, 3.75, 3.95]
10.8 [0.25, 3.20, 3.60, 3.75]<\/p>\n","protected":false},"excerpt":{"rendered":"

This puzzle showed up in several places with different anecdotes. The puzzle first specifies a decimal number with two digits<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"nf_dc_page":"","_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[3],"tags":[],"class_list":["post-1098","post","type-post","status-publish","format-standard","hentry","category-cs"],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/cetinkayakoc.net\/blog\/wp-json\/wp\/v2\/posts\/1098","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/cetinkayakoc.net\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/cetinkayakoc.net\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/cetinkayakoc.net\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/cetinkayakoc.net\/blog\/wp-json\/wp\/v2\/comments?post=1098"}],"version-history":[{"count":5,"href":"https:\/\/cetinkayakoc.net\/blog\/wp-json\/wp\/v2\/posts\/1098\/revisions"}],"predecessor-version":[{"id":1124,"href":"https:\/\/cetinkayakoc.net\/blog\/wp-json\/wp\/v2\/posts\/1098\/revisions\/1124"}],"wp:attachment":[{"href":"https:\/\/cetinkayakoc.net\/blog\/wp-json\/wp\/v2\/media?parent=1098"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/cetinkayakoc.net\/blog\/wp-json\/wp\/v2\/categories?post=1098"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/cetinkayakoc.net\/blog\/wp-json\/wp\/v2\/tags?post=1098"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}