Here is a fascinating (unsolved) problem known as the hailstone problem (or
the Ulam Conjecture or the
Collatz Conjecture). It involves sequences in two different ways. First,
choose a positive integer \(N\) and call it \(a_{0} .\) This is the seed of a
sequence. The rest of the sequence is generated as follows: For \(n=0,1,2,
\ldots\)
$$a_{n+1}=\left\\{\begin{array}{ll}
a_{n} / 2 & \text { if } a_{n} \text { is even } \\
3 a_{n}+1 & \text { if } a_{n} \text { is odd. }
\end{array}\right.$$
However, if \(a_{n}=1\) for any \(n,\) then the sequence terminates.
a. Compute the sequence that results from the seeds \(N=2,3\), \(4, \ldots, 10 .\)
You should verify that in all these cases, the sequence eventually terminates.
The hailstone conjecture (still unproved) states that for all positive
integers \(N,\) the sequence terminates after a finite number of terms.
b. Now define the hailstone sequence \(\left\\{H_{k}\right\\},\) which is the
number of terms needed for the sequence \(\left\\{a_{n}\right\\}\) to terminate
starting with a seed of \(k\). Verify that \(H_{2}=1, H_{3}=7,\) and \(H_{4}=2\).
c. Plot as many terms of the hailstone sequence as is feasible. How did the
sequence get its name? Does the conjecture appear to be true?
