Consider the following operation: if a number is even, divide by 2. If the number is odd, multiply by 3 and add 1. By repeating this operation, you get the Collatz sequence for your starting number. Does the sequence always eventually hit 1? The Collatz conjecture speculates the answer is yes, but no one knows for sure. Drag the slider for a to whatever starting value you like. Then slide n over to the 1 value, and slowly increase n. This will give you the values of the Collatz sequence, as well as a graph of those values.
1. Play around with the applet a bit. Based off the graph, how can you tell when the sequence reaches 1? Based off the graph, how can you tell what the largest number in the sequence is? 2. Try every possible value of a from 1 to 10. Which produces the longest sequence before reaching 1? 3. Try values of a from a = 11 to a = 100. Are there any that produce an especially long sequence? Try ``zooming out" if the sequence goes off the page. Try to find at least three starting numbers with really long sesequences. 4. Do any of the graphs for the long sequences look the same? Why might this be?