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 . This is the seed of a sequence. The rest of the sequence is generated as follows: For n=0, 1, 2, ...
. However, if =1 for any n, then the sequence terminates. a. Compute the sequence that results from the seeds N=2, 3, 4, ...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 , which is the number of terms needed for the sequence to terminate starting with a seed of k. Verify that =1, , and . 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?
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Work in Excel is below. The cell is open, so that the command is visible. The table shows the sequences for seeds , until termination. We see that in this example, every sequence has terminated before 21st term (the letter stays for "Not any more :)" and indicates the termination).
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There are many unsolved problems in mathematics. One famous unsolved problem is known as the Collatz problem, or the 3n+1 problem. This problem was created by L. Collatz in 1937. Although the procedures in the Collatz problem are easy to understand, the problem remains unsolved. Search the Internet or a library to find information on the Collatz problem. Write a short report that explains the Collatz problem. In your report, explain the meaning of a “hailstone” sequence. b. Show that for each of the natural numbers 2, 3, 4, ... , 10, the Collatz pocedure generates a sequence that “returns” to 1.
Prof. Martinez stimulates interest in homework by requiring two students each to draw a problem at random and solve the problem drawn. On a certain day, ten problems were assigned. Anita is to draw first and Al second. They had solved nine problems but had not solved one. Find the probability that (a) Anita draws the unsolved problem. (b) neither Anita nor Al draws the unsolved problem. (c) Al does not draw the unsolved problem.