Problem 42834. Integer complexity (Large numbers)

Created by Peng Liu

Solve Later

{"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","targetMode":"","relationshipId":"rId1","target":"/matlab/document.xml"},{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/output","targetMode":"","relationshipId":"rId2","target":"/matlab/output.xml"}],"parts":[{"partUri":"/matlab/document.xml","relationship":[],"contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?>\n<w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Inspired by</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"http://www.mathworks.com/matlabcentral/cody/problems/42831-integer-complexity\"><w:r><w:t>Problem 42831</w:t></w:r></w:hyperlink><w:r><w:t>, this problem aims to calculate the</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://en.wikipedia.org/wiki/Integer_complexity\"><w:r><w:t>integer complexity</w:t></w:r></w:hyperlink><w:r><w:t> for large numbers. The</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t>integer complexity</w:t></w:r><w:r><w:t> of a natural number n is defined as the least number of 1’s required to express n using only the two operations + and × and parentheses.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>Example</w:t></w:r><w:r><w:t>: the number 11 may be represented using 8 ones:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[    11 = (1 + 1 + 1) × (1 + 1 + 1) + 1 + 1.]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>However, it has no representation using 7 or fewer ones. Therefore, its complexity is 8.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Your solution will be scored based on its running time. No cheating please.</w:t></w:r></w:p></w:body></w:document>"},{"partUri":"/matlab/output.xml","contentType":"text/xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\" ?><embeddedOutputs><metaData><evaluationState>manual</evaluationState><layoutState>code</layoutState><outputStatus>ready</outputStatus></metaData><outputArray type=\"array\"/><regionArray type=\"array\"/></embeddedOutputs>"}]}

Solve

Solution Stats

32 Solutions
10 Solvers

Last Solution submitted on May 18, 2021

Last 200 Solutions

Problem Comments

Solution Comments

Solution 906063

2 Comments

2 Comments

Rafael S.T. Vieira on 19 Sep 2020

I was obtaining 34 for 7e4 too, but it's not a problem of the test suite. It just means that your code is not testing all possible combinations (and there are probably other numbers that will fail too). There is no known algorithm to generate the list of integer complexity (https://oeis.org/A005245), which means brute force is the only way to be sure. And brute force finds 32 for 7e4.

Rafael S.T. Vieira on 19 Sep 2020

It someone doesn't believe 7e4 = (1+1)*(1+1)*(1+1)*(1+1)*(1+ (1+1)*(1+1+1)*(1+1+1)*(1+1+1)*(1+1+1)*(1+1+1)*(1+1+1)*(1+1+1)) (32 ones). Or 2^4*(1 + 2*3^7).

Problem Recent Solvers10

Problem Tags

complexity integer number theory

Community Treasure Hunt

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

Start Hunting!

Problem 42834. Integer complexity (Large numbers)

Solution Stats

Last 200 Solutions

Problem Comments

Solution Comments

Problem Recent Solvers10

Suggested Problems

More from this Author30

Problem Tags

Community Treasure Hunt

Problem 42834. Integer complexity (Large numbers)

Solution Stats

Last 200 Solutions

Problem Comments

Solution Comments

Problem Recent Solvers10

Suggested Problems

More from this Author30

Problem Tags

Community Treasure Hunt

Players