Begin with a positive integer known as the seed To grow more

Begin with a positive integer, known as the \"seed.\" To \"grow\" more numbers, use the following rules: If a number is even, divide it by 2. If a number is odd, multiply it by 3 and add 1. Example: 3, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, etc. Why are they called \"hailstone numbers\"? (Use a web search engine and write the answer in your own weeds rather than turning in a printout.) Draw a Row diagram foe the seeds 1, 2, 3 through 26, (Write small and plan ahead for space.) Does 27 eventually flow into the 3-cycle 4, 2, 1 ? If so, how much larger is the flow diagram that includes 27? Prove that infinitely many positive integers flow into 4, 2, 1. Prove or disprove that every positive integer flows into 4, 2, 1. (Worth your favorite beverage plus 10^100 points!) Has anyone (on earth) given a proof as in # 5 ? (Perform a web search for \"Collaiz conjecture\" or \"3x + 1 problem.\" Report in your own words.) Consider negative integer seeds. Draw a flow diagram for seeds -1, -2, -3 through -32. How does it differ from the flow diagram for positive seeds? Can a negative seed flow to a positive number? Can a negative seed flow to -infinity? Change the rules and make experimental observations, For example. What happens if the second rule is changed to \"if odd, multiply by 5 and add 1\"?

Solution

1.

Let\'s try a few numbers to see what happens on applying the rule given in the question
n=3; 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, ...
n=4; 2, 1, 4, 2, 1, ...
n=5; 16, 8, 4, 2, 1, 4, 2, 1, ...
n=6; 3, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, ...
n=7; 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, ...

We can observe that the numbers in such sequence bounce up and down which is why they are called hailstorm number.