This is an old one. One of the first programming problems I ever solved. Not very difficult to get it right, not very difficult to make it fast.
The 3n + 1 Problem is problem 100 in the UVa Online Judge. Even though I include the problem description in this post, I encourage you to visit the UVa Online Judge because there you will be able to submit your solution to get it judged.
Consider the following algorithm.
Given the input 22, the following sequence of numbers will be printed
22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1.
It is conjectured that the algorithm above will terminate for any
integral input value. This conjecture is the Collatz
Despite the simplicity of the algorithm, it is unknown whether this
conjecture is true. It has been verified, however, for all integers
n such that
0 < n < 1,000,000 (and, in fact, for many more numbers
Given an input
n, it is possible to determine the number of numbers
printed (including the 1). For a given
n this is called the
n. In the example above, the cycle length of 22 is
For any two numbers
j you are to determine the maximum cycle
length over all numbers between
The input will consist of a series of pairs of integers
one pair of integers per line. All integers will be less than
1,000,000 and greater than
You should process all pairs of integers and for each pair determine
the maximum cycle length over all integers between and including
You can assume that no operation overflows a 32-bit integer.
For each pair of input integers
j you should output
j, and the maximum cycle length for integers between and including
j. These three numbers should be separated by at least one
space with all three numbers on one line and with one line of output
for each line of input. The integers
j must appear in the output
in the same order in which they appeared in the input and should be
followed by the maximum cycle length (on the same line).
Consider the following solution. Function
collatz_len computes the
cycle length of a given integer. Each recursive call corresponds to an
assignment in the algorithm of the problem statement. The numbers in
the left margin indicate execution count for a random
We reduce the number of steps taken to solve the random input by
memoizing the cycle lengths. The following solution memoizes calls
n less than 1,000,000. Memoization reduces the number of calls
one order of magnitude because the calls to
from ~11,400,000 steps to ~900,000 and we added ~3,000,000 steps for
initialization in line 26.
Our solution ranks 183 in the UVa Online Judge. There are submissions with reported runtimes of 0.000 seconds. I have yet to figure out a faster solution.
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I love to explain and answer questions on programming problems. I publish a new programming problem and its solution every month. Did I mention that I love to answer questions?