2007 Begets 2016 Begets 1008 Begets 504 Begets 252....
Step 1: Pick a whole number.
Step 2: If the number is even, divide by 2...but if it is odd, add 9.
Step 3: Repeat Step 2 using this new number.
Question 1: Investigate what happens for various starting numbers...how can you be systematic?
Question 2: What predictions can you make, given a particular starting number?
Hint: Try it......
Solution Commentary: What conjectures/predictions did you make?
Compare them with these conjectures posed by past students:
- For multiples of 3, you will always get back to the repeating pattern of 12, 6, 3,...
- For multiples of 9, you will always get back to the repeating pattern of 9, 18, 9,...
- 10, 5, 14, 7, 16, 8, 4, 2, 1 cycles through all the numbers and you eventually get back to the starting number
- 11 and its multiples cycles through all the numbers but you do not get back to the starting number (I think this is true for primes greater than 11 as well)
- Numbers that are doubles of each other have the same repeating sequence
- 0 is not interesting
Can any of these conjectures be verified...or proven?
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