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Here’s a number game:
Note 5
Problem D1
Draw a picture of this algorithm and use your picture to answer the following questions:
Problem D2
Build an algorithm that lets you find the number someone picked in Problem D1 if you know his or her final result. Try it out.
Problem D3
Here’s another number game:
Use the algorithm above to answer these questions:
Problem D4
Try to draw a picture of the algorithm in Problem D3. How is it different from the algorithm in Problem D1?
Problem D5
Here’s another number game:
Use the algorithm above to answer these questions:
Problem D6
What strategies did you use on Problems D3(f) and D3(g)? Why are these different from the strategies used in Problems D1(f) and D1(g)?
Problem D7
Describe another number game that has the same kind of outputs as the game of Problem D5.
Note 5
Groups: Work on Problems D1-D7 and discuss what everyone finds.
The problem with the algorithm in Problem D3 is the step “add your original number.” It’s not clear how to model that with machines. Even if we are able to adapt the machine metaphor to work, we can’t “undo” that step of the process.
Consider going through a symbolic representation of the algorithm to see that it does not share this weakness with the machine, because you can combine terms, essentially creating an equivalent algorithm that you can undo:
[(n + 3) x 2 + n – 4] x 3 = (3n + 6 – 4) x 3
(This is just one of many equivalent algorithms.)
Problem D1
Problem D2
This algorithm would undo the steps of the original by using inverse operations taken in reverse order. That would be:
You should find that this algorithm undoes the one in Problem D1. Inverse algorithms like this are frequently incorporated in mathematical magic tricks.
Problem D3
a. | The final number will be 33. |
b. | The final number will be 51 |
c. | The final number will be 114 |
d. | The final number will be 348. |
e. | The final number will be 9n + 6 [n >> n + 3 >> 2n + 6 >> 3n + 6 >> 3n + 2 >> 9n + 6]. |
f. | This is difficult without completing D3(e), but solving 9n + 6 = 48 suggests that Mary’s original number was 4 2/3. |
g. | Solving 9n + 6 = 30 shows that Joe’s original number was 2 2/3. |
Problem D4
It’s pretty tough to draw a picture of the machine for “add the original number”! Specifically, the original number is a variable, so the results of that machine vary depending on what you put in to start. A machine labeled “+ n” might work, but notice how different that is from one that reads “+ 3”.
Problem D5
Problem D6
You almost need to know a formula for the entire algorithm before solving D3(f) and D3(g). In Problem D1, a “reverse” strategy will work, taking the algorithm step-by-step in reverse. But in Problem D3, the “add the original number” step is impossible unless you know the original number! So you could guess until you get it, or come up with a formula for the entire algorithm.
Problem D7
Here’s one: Pick a number, multiply it by 9, add 18, divide by 3, add 6, divide by 3 again, then subtract the original number. The answer will always be 4.