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# The "aha!" problems 8: Reflect on this

### #1

Posted 22 May 2014 - 12:22 PM

A ray of light encounters a pair of angled mirrors from the left, as shown, a distance * d* from one mirror and at an angle

*from the other mirror. If angle*

**a***is exactly 22.5*

**a**^{o}, how close will the ray get to their intersection at point

*before it eventually exits again to the left? The drawing may not be to scale.*

**O***The greatest challenge to any thinker is stating the problem in a way that will allow a solution.*

- Bertrand Russell

### #2

Posted 22 May 2014 - 04:54 PM Best Answer

### #3

Posted 22 May 2014 - 06:01 PM

Spoiler for Hmm...

Wow ... talk about a "bait-and-switch." And this is such a nice analysis ...

Our Puzzle Quality Control Department has informed us that we cannot

claim an "aha!" status for a puzzle that admits to a straightforward constructive

solution, as this one does, brilliant and insightful though it may be.

They inform us that it was incorrect on our part to specify the angle * a* at all.

Ouch. We do apologize.

Can the puzzle still be answered?

*The greatest challenge to any thinker is stating the problem in a way that will allow a solution.*

- Bertrand Russell

### #4

Posted 23 May 2014 - 02:03 AM

Still, for a coveted bonanova Gold Star, can RG's answer be shown to be completely independent of

**?**

*a*It's a pretty proof.

*The greatest challenge to any thinker is stating the problem in a way that will allow a solution.*

- Bertrand Russell

### #5

Posted 23 May 2014 - 02:19 AM

### #6

Posted 23 May 2014 - 02:44 AM

Spoiler for straight light coming thru

You have it for submultiples of pi radians.

Can we generalize to arbitrary

**?**

*a**The greatest challenge to any thinker is stating the problem in a way that will allow a solution.*

- Bertrand Russell

### #7

Posted 23 May 2014 - 02:49 AM

hi bonanova,

i have given it for any arbitrary a (both when its sub multiple and when its not).

only case to be added is, if a is integer multiple of pi, then the ray doesnt return at all

### #8

Posted 23 May 2014 - 05:11 AM

Maybe I misunderstood. You may have the answer without actually saying it.

You're certainly on the right track. The question what is the distance of closest approach to * O*,

and there is no condition that it must coincide with the incident ray when it leaves.

I read your answer to depend on * a*. Did you mean that?

*The greatest challenge to any thinker is stating the problem in a way that will allow a solution.*

- Bertrand Russell

### #9

Posted 23 May 2014 - 08:17 AM

### #10

Posted 23 May 2014 - 03:03 PM

Yes, bonanova. I think the answer depends on a (e.g. if a is obtuse the ray doesnt exit on left and never hits the top mirror). It coinicides with the icident ray if a is a factor of pi/2 radian

Everything in your post is correct. But remember what the question is.

Whatever value * a* has, how close does the ray come to the point

*?*

**O**

The phrase "before it exits to the left" is appropriate to * a* being acute, as in the figure.

If you want to include * a* being obtuse, so that the ray would not exit to the left,

that just gives you "forever" as the time frame to consider the ray's point of closest approach.

Although of course you wouldn't need "forever" to determine it.

In other words, the wording is not meant to be tricky: assume * a* to be acute if you like.

*The greatest challenge to any thinker is stating the problem in a way that will allow a solution.*

- Bertrand Russell

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