Question

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?

Short answer

Question: Compute the sequence that results from the seeds N=7 and explain whether the sequence terminates or not. Answer: For seed N=7, the resulting sequence is: 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1. Yes, the sequence terminates as it reaches the number 1.

Step-by-step solution

Seed N = 2

\(a_0 = 2\) Since \(a_0\) is even, \(a_1 = a_0 / 2 = 1\). The sequence terminates. The resulting sequence is: \(2, 1\).

Seed N = 3

\(a_0 = 3\) Since \(a_0\) is odd, \(a_1 = 3a_0 + 1 = 10\). The remaining sequence, given this start, is as follows (using the recursive formula): \(3, 10, 5, 16, 8, 4, 2, 1\).

Seed N = 4

\(a_0 = 4\) Since \(a_0\) is even, \(a_1 = a_0 / 2 = 2\). The remaining sequence is: \(4, 2, 1\). Continue with the same procedure for other seed values:

Seed N = 5

The sequence is: \(5, 16, 8, 4, 2, 1\).

Seed N = 6

The sequence is: \(6, 3, 10, 5, 16, 8, 4, 2, 1\).

Seed N = 7

The sequence is: \(7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1\).

Seed N = 8

The sequence is: \(8, 4, 2, 1\).

Seed N = 9

The sequence is: \(9, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1\).

Seed N = 10

The sequence is: \(10, 5, 16, 8, 4, 2, 1\). In each case, the sequence terminates. #b. Hailstone sequence H_k# To calculate \(H_k\), we will count the number of terms needed for the sequence to terminate for seed values \(k = 2, 3, 4\).

Hailstone sequence H_k for k = 2

From the previous task, the sequence for \(k=2\) was \(2, 1\). It terminates in 1 step. Therefore, \(H_2 = 1\).

Hailstone sequence H_k for k = 3

The sequence for \(k=3\) was \(3, 10, 5, 16, 8, 4, 2, 1\). It terminates in 7 steps. Therefore, \(H_3 = 7\).

Hailstone sequence H_k for k = 4

The sequence for \(k=4\) was \(4, 2, 1\). It terminates in 2 steps. Therefore, \(H_4 = 2\).

c. Plotting the hailstone sequence

To plot the hailstone sequence, plot the values of \(H_k\) for various values of \(k\). Although we can't draw a plot in this response, you could plot the values on graph paper or use graphing software. The sequence got its name due to its seemingly random behavior, which is reminiscent of hailstones falling from the sky. The hailstone conjecture remains unproved, but so far, no counterexample has been found, and it appears to hold true for all positive integers tested.

Key concepts

Collatz Conjecture

The Collatz Conjecture hails from the realm of mathematical puzzles, presenting an uncomplicated rule that belies its complex behavior. Imagine starting with any positive integer, let's call it and following two simple steps: First, if the number is even, divide it by two. Second, if the number is odd, triple it and add one. Continue this process repeatedly and the belief is that no matter the initial number, you will eventually reach the number 1.

This conjecture tantalizes mathematicians and hobbyists alike because despite extensive testing, no one has yet managed to prove it rigorously nor find a counterexample that would topple it. As of my knowledge cutoff in March 2023, the Collatz Conjecture remains one of math's enduring enigmas, encouraging a global community to explore it both for fun and for pushing the boundaries of mathematical theory.

Recursive Sequence

When dealing with the Collatz Conjecture, we encounter a recursive sequence, which is a sequence of numbers where each term after the first is defined as a function of the preceding terms. In simple terms, to find any number in the sequence, you use the number before it in a predefined way.

In the hailstone problem, for example, the current number in the sequence dictates whether it's halved or altered through the rule of 'triple plus one,' based on whether it's even or odd. This self-referential method creates a chain of numbers that can be very predictable in some instances and surprisingly unpredictable in others, adding to the allure of such sequences in mathematical study.

Mathematical Conjecture

The notion of conjecture in mathematics sits between hypothesis and theorem. It's a proposition that is suspected to be true based on heuristic arguments and limited evidence but has not yet been proven or disproven. Regarding the Collatz Conjecture, it remains one of the most famous conjectures in mathematics because of its deceptively simple appearance and the inability of mathematicians to prove it despite it holding true for all tested integers.

Mathematical conjectures like these invite rigorous investigation and often require sophisticated tools from various branches of mathematics, pushing the boundaries of understanding and challenging the skills of those who attempt to resolve them.

Sequence Termination

The term 'sequence termination' refers to the point at which a sequence arrives at an end, often according to a predefined condition. For the hailstone problem or the Collatz Conjecture, termination happens when the sequence arrives at the number 1. At this point, since 1 is odd, applying the 'triple plus one' rule would yield 4, and from there, a quick succession of divisions by two leads you back to 1, creating an infinite loop.

The conjecture suggests that all positive integers will lead to sequences that terminate at 1, although we don't have a proof for it yet. Such behaviors provoke deep inquiry into the nature of mathematical sequences and the principles governing their behaviors, which are pivotal for comprehending much more complex systems in mathematics and applied sciences.