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A “numerical recycling machine” takes any positive integer as input and then applies the following procedure:
Example. Suppose we start with the number 15. In Step 1 we compute 15 × 15 = 225 and send this result to Step 2. In Step 2 we compute 2 + 2 + 5 = 9 and send this result back to Step 1. Back in Step 1, we compute 9 × 9 = 81 and send this to Step 2, and so on. 15 → 225 → 9 → 81 → . . . Explore how this recycling machine works for different input values. What patterns do you notice? What conjectures do you have about how the machine works in general? |
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Follow-up ChallengesNew Machine #1. Suppose we change Step 1 of the procedure so that we cube the number (instead of squaring). How would this change the overall behavior of the machine?
New Machine #2. Suppose we change Step 2 of the procedure as follows: If the digit-sum has more than one digit, find its digit sum. Continue until the result has just one digit and pass this single digit back to Step 1. For instance, if we get the number 256 in Step 2, we find the digit sum 2+5+6 = 13. The sum has more than one digit and so we do the digit-sum step again: 1 + 3 = 4. We send the result 4 back to Step 1. How would this change the overall behavior of the machine? |
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