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## 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?