PyPE solution to PE problem 45
PyPE solution to PE problem 45
Blueprint information
- Status:
- Complete
- Approver:
- None
- Priority:
- Undefined
- Drafter:
- None
- Direction:
- Needs approval
- Assignee:
- None
- Definition:
- Approved
- Series goal:
- None
- Implementation:
-
Implemented
- Milestone target:
- None
- Started by
- Scott Armitage
- Completed by
- Scott Armitage
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ProjectEule
===
Triangle, pentagonal, and hexagonal numbers are generated by the following
formulae:
Triangle T_(n)=n(n+1)/2 1, 3, 6, 10, 15, ...
Pentagonal P_(n)=n(3n-1)/2 1, 5, 12, 22, 35, ...
Hexagonal H_(n)=n(2n-1) 1, 6, 15, 28, 45, ...
It can be verified that T_(285) = P_(165) = H_(143) = 40755.
Find the next triangle number that is also pentagonal and hexagonal.
Solution
--------
We have a starting point (H_143), and we have a formula for generating
triangular, pentagonal, and hexagonal nubmers. Since the formula for
hexagonal numbers generates the shortest list, we start counting from
here and calculate each subsequent H_n.
Note that all hexagonal numbers are also triangular numbers, and so we
only need to check if each resulting H_n is also pentagonal. If it is,
we return H_n as our answer; otherwise, we keep going.
The test in ispentagonal is derived from the P(n) function above, solving
for the roots n for a given x. If the roots n are integer, then x is P_n.
Answer
------
1533776805
