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MATH 1473 Bonus Worksheet: Hailstone Sequences Want to see something cool? Take any positive whole number (called a natural number). If the number is odd, multiply it by 3 and add 1. If the number is even, divide it by two. For example, the number 5 is odd, so multiply it by 3 and add 1: 5 × 3 + 1 = 16 Let’s try it again. Since 16 is even, divide it by 2: 16 ÷ 2 = 8 Again, since 8 is even: 8 ÷ 2 = 4. **4 is even, so: 4 ÷ 2 = 2. And 2 is even: 2 ÷ 2 = 1. What happens if you try this process on the number 1? That’s right! 1 is an odd number, so 1 × 3 + 1 = 4 . Notice that the process starts to repeat itself now (go back to the ** above). This process now starts to go 4 to 2 to 1 to 4 to 2 to 1 to 4 to . . . The list of numbers that are created by this process is called a Hailstone Sequence, because it goes up and down repeatedly just like a hailstone in a cloud before falling to the earth. Using the work that we did above, the Hailstone sequence for the number 5 is 5, 16, 8, 4, 2, 1. Always start with the original number, and stop after reaching 1 the first time to avoid repetition. So, starting with the number 5 and following the directions given above, this process leads us back to the number 1 (after 5 steps). Try this same process starting with the number 7. Show that this process for the number seven also reaches the number 1 (show your work below). Hailstone sequence for the number 7: Find the Hailstone sequences for each of the numbers printed below. 256 23 29 100 Did all of these Hailstone sequences end in 1? Hailstone sequences are involved in one of the greatest unsolved problems in mathematics today, usually called the Collatz Problem or the 3k+1 problem. Using the internet and other sources, write and explain (in your own words) four interesting and relevant points regarding this problem. Be sure to label your sources if necessary. You may have to seek out several different sources! 1) Source:______________________________________ 2) Source:______________________________________ 3) Source:______________________________________ 4) Source:______________________________________