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62 lines
1.8 KiB
Python
62 lines
1.8 KiB
Python
"""
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Problem 14: https://projecteuler.net/problem=14
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Collatz conjecture: start with any positive integer n. Next term obtained from
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the previous term as follows:
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If the previous term is even, the next term is one half the previous term.
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If the previous term is odd, the next term is 3 times the previous term plus 1.
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The conjecture states the sequence will always reach 1 regardless of starting
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n.
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Problem Statement:
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The following iterative sequence is defined for the set of positive integers:
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n → n/2 (n is even)
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n → 3n + 1 (n is odd)
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Using the rule above and starting with 13, we generate the following sequence:
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13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1
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It can be seen that this sequence (starting at 13 and finishing at 1) contains
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10 terms. Although it has not been proved yet (Collatz Problem), it is thought
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that all starting numbers finish at 1.
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Which starting number, under one million, produces the longest chain?
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"""
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from __future__ import annotations
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COLLATZ_SEQUENCE_LENGTHS = {1: 1}
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def collatz_sequence_length(n: int) -> int:
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"""Returns the Collatz sequence length for n."""
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if n in COLLATZ_SEQUENCE_LENGTHS:
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return COLLATZ_SEQUENCE_LENGTHS[n]
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next_n = n // 2 if n % 2 == 0 else 3 * n + 1
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sequence_length = collatz_sequence_length(next_n) + 1
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COLLATZ_SEQUENCE_LENGTHS[n] = sequence_length
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return sequence_length
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def solution(n: int = 1000000) -> int:
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"""Returns the number under n that generates the longest Collatz sequence.
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>>> solution(1000000)
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837799
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>>> solution(200)
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171
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>>> solution(5000)
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3711
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>>> solution(15000)
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13255
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"""
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result = max((collatz_sequence_length(i), i) for i in range(1, n))
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return result[1]
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if __name__ == "__main__":
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print(solution(int(input().strip())))
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