Added solution for Project Euler problem 101

[fpringle]

Describe your change:

Project Euler problem 101: Optimum polynomial

If we are presented with the first k terms of a sequence it is impossible to say with certainty the value of the next term, as there are infinitely many polynomial functions that can model the sequence.

As an example, let us consider the sequence of cube numbers. This is defined by the generating function,
u_n = n³: 1, 8, 27, 64, 125, 216, ...

Suppose we were only given the first two terms of this sequence. Working on the principle that "simple is best" we should assume a linear relationship and predict the next term to be 15 (common difference 7). Even if we were presented with the first three terms, by the same principle of simplicity, a quadratic relationship should be assumed.

We shall define OP(k, n) to be the nth term of the optimum polynomial generating function for the first k terms of a sequence. It should be clear that OP(k, n) will accurately generate the terms of the sequence for n ≤ k, and potentially the first incorrect term (FIT) will be OP(k, k+1); in which case we shall call it a bad OP (BOP).

As a basis, if we were only given the first term of sequence, it would be most sensible to assume constancy; that is, for n ≥ 2, OP(1, n) = u₁.

Hence we obtain the following OPs for the cubic sequence:

OP(1, n) = 1: 1, 1, 1, 1, ...
OP(2, n) = 7n−6: 1, 8, 15, ...
OP(3, n) = 6n²−11n+6: 1, 8, 27, 58, ...
OP(4, n) = n³: 1, 8, 27, 64, 125, ...
Clearly no BOPs exist for k ≥ 4.

By considering the sum of FITs generated by the BOPs (indicated in red above), we obtain 1 + 15 + 58 = 74.

Consider the following tenth degree polynomial generating function:

u_n = 1 − n + n² − n³ + n⁴ − n⁵ + n⁶ − n⁷ + n⁸ − n⁹ + n¹⁰

Find the sum of FITs for the BOPs.

• Add an algorithm?
• Fix a bug or typo in an existing algorithm?
• Documentation change?

Checklist:

• I have read CONTRIBUTING.md.
• This pull request is all my own work -- I have not plagiarized.
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• This PR only changes one algorithm file. To ease review, please open separate PRs for separate algorithms.
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• If this pull request resolves one or more open issues then the commit message contains Fixes: #{$ISSUE_NO}.

[ghost]

Click here to look at the relevant links ⬇️

🔗 Relevant Links

Repository:

• Contributing guidelines
• Project Euler solution guidelines

Python:

• Type hints
• doctest
• unittest
• pytest

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[dhruvmanila]

Please update all the one letter variable to be a bit more descriptive. That would help the reader understand what the variable means.

[ghost]

Click here to look at the relevant links ⬇️

🔗 Relevant Links

Repository:

• Contributing guidelines
• Project Euler solution guidelines

Python:

• Type hints
• doctest
• unittest
• pytest

Automated review generated by algorithms-keeper. If there's any problem regarding this review, please open an issue about it.

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[ghost]

Click here to look at the relevant links ⬇️

🔗 Relevant Links

Repository:

• Contributing guidelines
• Project Euler solution guidelines

Python:

• Type hints
• doctest
• unittest
• pytest

Automated review generated by algorithms-keeper. If there's any problem regarding this review, please open an issue about it.

---

algorithms-keeper commands and options

algorithms-keeper actions can be triggered by commenting on this PR:

• @algorithms-keeper review to trigger the checks for only added pull request files
• @algorithms-keeper review-all to trigger the checks for all the pull request files, including the modified files. As we cannot post review comments on lines not part of the diff, this command will only modify the labels accordingly.

NOTE: Commands are in beta and so this feature is restricted only to a member or owner of the organization.

[dhruvmanila]

I think you accidentally added the latest commit from master in this pull request. You should rebase instead of merge. If you want to keep your pull request in sync with master: git rebase master <your PR branch name>

[dhruvmanila]

LGTM! 🎉