Problem 53990. Classify product/digit-sum sequences

Created by ChrisR

Solve Later

{"parts":[{"partUri":"/matlab/document.xml","contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?><w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:hyperlink w:docLocation=\"https://www.mathworks.com/matlabcentral/cody/problems/53120\"><w:r><w:rPr><w:u/></w:rPr><w:t>Cody Problem 53120</w:t></w:r></w:hyperlink><w:r><w:t> involved a sequence in which a term is computed by multiplying the previous two terms and adding the digits of the product. In that problem the first two terms of the sequence were 1 and 2. The next four terms were 2, 4, 8, and 5.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>What happens if the first two terms are changed? It turns out that these product/digit-sum sequences can be sorted into five groups. For reasons that will likely be apparent to those who solved </w:t></w:r><w:hyperlink w:docLocation=\"https://www.mathworks.com/matlabcentral/cody/problems/53120\"><w:r><w:rPr><w:u/></w:rPr><w:t>Cody Problem 53120</w:t></w:r></w:hyperlink><w:r><w:t>, the sequence there (and others like it) is assigned the number 163. The other four types of sequence are assigned the numbers 1, 9, 26, and 62. </w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Write a function to classify the product/digit-sum sequences given the first two terms </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"a\"/></w:customXmlPr><w:r><w:t>a</w:t></w:r></w:customXml><w:r><w:t> and </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"b\"/></w:customXmlPr><w:r><w:t>b</w:t></w:r></w:customXml><w:r><w:t>. To encourage solvers to find the pattern, loops are banned, and to allow for large inputs, </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"a\"/></w:customXmlPr><w:r><w:t>a</w:t></w:r></w:customXml><w:r><w:t> and </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"b\"/></w:customXmlPr><w:r><w:t>b</w:t></w:r></w:customXml><w:r><w:t> are specified as strings. </w:t></w:r></w:p></w:body></w:document>","relationship":null}],"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","target":"/matlab/document.xml","relationshipId":"rId1"}]}

<div style = "text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; "><div style="block-size: 186px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407.5px 93px; transform-origin: 407.5px 93px; vertical-align: baseline; "><div style="block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 31.5px; text-align: left; transform-origin: 384.5px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><a target='_blank' href = "https://www.mathworks.com/matlabcentral/cody/problems/53120"><span style=""><span style="text-decoration-line: underline; ">Cody Problem 53120</span></span></a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 309.642px 7.66667px; transform-origin: 309.642px 7.66667px; unicode-bidi: normal; "><span style=""> involved a sequence in which a term is computed by multiplying the previous two terms and adding the digits of the product. In that problem the first two terms of the sequence were 1 and 2. The next four terms were 2, 4, 8, and 5.</span></span></div><div style="block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 31.5px; text-align: left; transform-origin: 384.5px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 383.883px 7.66667px; transform-origin: 383.883px 7.66667px; unicode-bidi: normal; "><span style="">What happens if the first two terms are changed? It turns out that these product/digit-sum sequences can be sorted into five groups. For reasons that will likely be apparent to those who solved </span></span><a target='_blank' href = "https://www.mathworks.com/matlabcentral/cody/problems/53120"><span style=""><span style="text-decoration-line: underline; ">Cody Problem 53120</span></span></a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 103.075px 7.66667px; transform-origin: 103.075px 7.66667px; unicode-bidi: normal; "><span style="">, the sequence there (and others like it) is assigned the number 163. The other four types of sequence are assigned the numbers 1, 9, 26, and 62. </span></span></div><div style="block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 21px; text-align: left; transform-origin: 384.5px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 262.017px 7.66667px; transform-origin: 262.017px 7.66667px; unicode-bidi: normal; "><span style="">Write a function to classify the product/digit-sum sequences given the first two terms </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">a</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.66667px; transform-origin: 15.5583px 7.66667px; unicode-bidi: normal; "><span style=""> and </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">b</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 93.35px 7.66667px; transform-origin: 93.35px 7.66667px; unicode-bidi: normal; "><span style="">. To encourage solvers to find the pattern, loops are banned, and to allow for large inputs, </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">a</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.66667px; transform-origin: 15.5583px 7.66667px; unicode-bidi: normal; "><span style=""> and </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">b</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 77.4px 7.66667px; transform-origin: 77.4px 7.66667px; unicode-bidi: normal; "><span style=""> are specified as strings. </span></span></div></div></div>

Solve

Solution Stats

14 Solutions
9 Solvers

Last Solution submitted on May 07, 2023

Last 200 Solutions

Problem Comments

2 Comments

2 Comments

Rafael S.T. Vieira on 26 Nov 2022

I didn't find a key, but a predominance between the five lists: 9 is the most powerful. 4 or 5 tests are enough, but some have made matrices with more than 25 possible combinations.

Rafael S.T. Vieira on 26 Nov 2022

Tip FYI: If the sum of the digits of a number is divisible by 3, then the number is also divisible by 3, and the same rule exists for 9. All numbers have similar laws, it seems. A number divisible by 5 ends with 0 or 5, and so on...

Solution Comments

Show comments

Problem Recent Solvers9

Problem Tags

Community Treasure Hunt

Find the treasures in MATLAB Central and discover how the community can help you!

Start Hunting!

Problem 53990. Classify product/digit-sum sequences

Solution Stats

Last 200 Solutions

Problem Comments

Solution Comments

Problem Recent Solvers9

Suggested Problems

More from this Author206

Problem Tags

Community Treasure Hunt

Problem 53990. Classify product/digit-sum sequences

Solution Stats

Last 200 Solutions

Problem Comments

Solution Comments

Problem Recent Solvers9

Suggested Problems

More from this Author206

Problem Tags

Community Treasure Hunt

Players