Problem 014


Project Euler

Longest Collatz sequence

Problem 14

The following iterative sequence is defined for the set of positive integers:

nn/2 (n is even)
n → 3n + 1 (n is odd)

Using the rule above and starting with 13, we generate the following sequence:

13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1

It can be seen that this sequence (starting at 13 and finishing at 1) contains 10 terms. Although it has not been proved yet (Collatz Problem), it is thought that all starting numbers finish at 1.

Which starting number, under one million, produces the longest chain?

NOTE: Once the chain starts the terms are allowed to go above one million.


Using recursion, if the value of the number is 1, then return 1 term. Otherwise, return 1 term + the however many number of terms for the new value (depending if the number is odd or even). With dynamics programming, the algorithm can speed up by using an array to remember the longest number of terms for each previously calculated number and minimize redundancy at the expense of memory.

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