Download MATH 1473 Bonus Worksheet: Hailstone Sequences

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:______________________________________