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""" |
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Project Euler Problem 44: Pentagon Numbers |
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========================================== |
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.. module:: solutions.problem44 |
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:synopsis: My solution to problem #44. |
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The source code for this problem can be |
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`found here <https://bitbucket.org/nekedome/project-euler/src/master/solutions/problem44.py>`_. |
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Problem Statement |
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################# |
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Pentagonal numbers are generated by the formula, :math:`P_n = \\frac{n(3n-1)}{2}`. The first ten pentagonal numbers are: |
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.. math:: |
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1, 5, 12, 22, 35, 51, 70, 92, 117, 145, \\dots |
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It can be seen that :math:`P_4 + P_7 = 22 + 70 = 92 = P_8`. However, their difference, :math:`70 - 22 = 48`, is not |
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pentagonal. |
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Find the pair of pentagonal numbers, :math:`P_j` and :math:`P_k`, for which their sum and difference are pentagonal and |
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:math:`D = |P_k - Pj|` is minimised; what is the value of :math:`D`? |
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Solution Discussion |
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################### |
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This solution is really a brute-force search with some sensible ordering constraints to remove redundant work that would |
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otherwise occur. |
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Iterate through a two-dimensional collection of pentagonal numbers, :math:`\\lbrace (p_j, p_k) \\rbrace`, and simply |
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test whether both the numbers :math:`p_j + p_k` and :math:`|p_j - p_k|` are pentagonal. |
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By iterating over :math:`p_k` first, and in ascending order, and then :math:`p_j` second in descending order from |
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:math:`p_{k-1}` to :math:`p_1` we will guarantee that :math:`p_k \\gt p_j`. Therefore, we will only consider positive |
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values for :math:`p_k + p_j` and :math:`p_k - p_j`. |
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Further, by initialising :math:`p_j` to a value as close to :math:`p_k` as possible and then increasing the distance |
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between :math:`p_j,p_k` at each iteration, the first candidate :math:`(p_j, p_k)` satisfying the search conditions will |
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have minimal distance :math:`|p_j - p_k|` for the current value of :math:`p_k`. |
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Solution Implementation |
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####################### |
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.. literalinclude:: ../../solutions/problem44.py |
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:language: python |
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:lines: 51- |
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""" |
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from lib.sequence import Pentagonals |
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def solve(): |
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""" Compute the answer to Project Euler's problem #44 """ |
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pentagonals = Pentagonals() |
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visited_pentagonals = [] # will reverse-iterate over this list {p_j} |
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pentagonals_set = set() # for testing whether sums or differences of pentagonal numbers are pentagonal numbers |
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for p_k in pentagonals: |
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for p_j in reversed(visited_pentagonals): |
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if (p_k - p_j) in pentagonals_set and (p_k + p_j) in pentagonals: |
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return p_k - p_j # p_k > p_j => |p_k - p_j| = p_k = p_j |
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visited_pentagonals.append(p_k) |
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pentagonals_set.add(p_k) |
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expected_answer = 5482660 |
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This check looks for lines that are too long. You can specify the maximum line length.