Khan Academy Transformation Puzzle

Question

This problem on Khan Academy has me stumped. It is part of the transformation puzzles exercise part 3. As the questions are randomized, here it is:

While solving the previous transformation puzzles, I found some interesting things properties about the solutions, you'd have a main transformation "sandwiched" between some other transformations that were inverses of each other. That is, you could do something like:

1. Rotate 90 degrees

2. Dilate by factor of 2

3. [Main transformation here]

4. Dilate by factor of 1/2

5. Rotate -90 degrees

Additionally, all solutions always dilated by a factor of 2 or 4, and rotated by a multiple of 90 degrees. I think this might have to do with the fact that at any other value for angle, either the sine or the cosine would produce an irrational.

Also, looking at this problem, the shape does not appear to be symmetric. And since the end product is simply a translation, it would imply that there would be an even number of reflections (or none at all) in the solution. (Is this correct reasoning?)

However, I'm not quite sure how to solve it. I got fairly close with two ways:

Method 1:

• Rotation by -90

• Reflection

• Dilation by 2

• Rotation by 180

• Dilation by 1/2

• Rotation by 90

Method 2:

• Rotation by 90
• Dilation by 1/2
• Rotation by 180
• Reflection
• Rotation by 180
• Dilation by 2
• Rotation by 90

But none of them are correct. Could someone shine some light on the solution, and preferably a technique to use to solve these puzzles?

Answer 1

R - Rotation, positive values -counter clockwise, negative - clockwise

D - Dilation

M - Reflection

1. R +90
2. D -3/2
3. R +180
4. D -2/3
5. R +90

Or, using only positive dilations with factors $\lambda=2^k, \, k\in \mathbb{Z}$:

1. R +90
2. D 1/2
3. R +180
4. M
5. D 2
6. M
7. R +180
8. M
9. R +180
10. D 1/2
11. R +180
12. M
13. R+180
14. D 2
15. R +90