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Fidgeting with a Purpose (Weekly Brain Potion)

 
 
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Fidgeting with a Purpose (Weekly Brain Potion)
LensmireJohn 发表于 2017年12月8日 Friday 16:00
 

Phillip just got a new fidget spinner, with ends colored red, blue, and green, as pictured below:

FidgetOriginal

He quickly got bored with just spinning it, but noticed something interesting. He saw that if he spun the spinner $120^\circ$, the fidget spinner, ignoring the colors, would look the same, as we can see below.

RotationExample

He also realized that the same was true if he flipped the spinner horizontally, as shown below:

FlipExample

Phillip explored what arrangements he could get, using only the two operations:

  • Rotate $120^\circ$
  • Flip horizontally

and came up with the $6$ arrangements below (for example (A) is the original and (B) is the result of rotating by $120^\circ$ once)

6Arrangements

Phillip was a little careless, and forgot exactly how he achieved each of the arrangements. Can you help him by determining the steps/operations (starting with the original and only using the two operations described above) to reach each of the arrangements? Is there only one way to reach each arrangement? What is the minimum number of operations needed?

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