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# Random math VI

+31
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+1879

Transformation puzzle

Feb 15, 2017
edited by EighthMersennePrime  Feb 15, 2017

#11
+2169
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The third paragraph in post #6 gives an outline of my strategy. Reiterating: the only thing I could instantly discern was it would need an odd number of reflections. Other than that, it is just an application of intuitive skill developed over time with practice.

I agree; the strategy is one of trial and error, but note that your error count will drop quickly with practice.

------

I almost forgot I’m the protégée of a masterful troll.   I should have said that your error count will drop quickly with practice, unless you are a dolt

P.S. Don’t forget to give me points. Melody said my solution was “Brilliant” and worth a hundred points. So far, I only have five measly points on my solution post.

You know, it really is fun being a troll!

Feb 25, 2017
edited by GingerAle  Feb 25, 2017

#1
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Hi 8MP

I have played with this question a lot.

I have learned to use GeoGebra better which I am quite pleased about but I am no closer to a solution.

I have aske more senior mathematicians to help. So far we have nothing.

I did get a response from someone that i have great respect for.

He said he toyed with it in Matematica and come up with zilch.

Is it possible that there is an error in the question?  Maybe it is not doable...

Feb 19, 2017
#2
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It's on Khan Academy, and I've tried LOTS of times, and I don't think it's impossible since KA is a well funded and popular site, so if it was then someone would of already reported it, oh, and also I can't reveal answer because the reveal answer button is gone including the get a hint which just says "there is no hint" when I try clicking it.

EighthMersennePrime  Feb 19, 2017
#3
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Solution for Transformation puzzle.

After playing with this for a while, I found it is solvable in 4 steps.

1) Reflect over y=x from (-1.5,-1.5) to (1.5, 1.5)

2) Dilate to scale factor of 2 about (-5, 0)

3) Rotate by 90° about (5, 0)

4) Dilate to scale factor of 1/2 about (-5, 0)

Feb 22, 2017
#4
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Here is Ginger's answer.

It is brilliant.

Thanks Ginger, I'd give you 100 points if I could!

1) Reflect over y=x from      Why did you add this bit?  (-1.5,-1.5) to (1.5, 1.5)

2) Dilate to scale factor of 2 about (-5, 0)

3) Rotate by 90° about (5, 0)           anticlockwise

4) Dilate to scale factor of 1/2 about (-5, 0)

How did you do it.   I mean can you give me any insite to your thinking process?

Melody  Feb 22, 2017
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+113567
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Hi 8MP and Ginger,

Since Ginger has not expalined how she worked out this answer I have decided to give it a go. :)

Maybe this is what she did....

1) The reflection.  Maybe this was the best place to start,  it seem right because it would orient the star so that when it was rotated anticlockwise it would have D on the top.

2) Now  just looked at the B point. I'd  want to move it so that (afterwards) when it was rotated 90 anticlockwise the B point would end up on the negative x axis. But there obviously had to be a dilation to get B into the correct position first.

Now when B was rotated about (5,0) it would need to be on the x axis.

So I draw the line x=5.  B would need to be moved to a point on that line.

3) I'd then my attention to the dilation that would be needed.

I drew a line from the centre of the dilation (-5,0) through B’ and subtended it till it intersected with x=5

That point of intersection would need to be the new B value.  i.e.   B’’

4) B’ was the point (0,9.5)

I could see from the diagram that B” was the point (5,19)

So it was clear that B’’ had to be twice as far as B’ from the centre of the dilation.

So I needed a dilation factor of 2

Now I did the dilation and the 3rd star was born.

3)  Now I just had to rotate it 90 degrees anticlockwise to get B onto the x axis.

Lastly Ginger had to counter the enlargement so she implemented a dilation of 0.5 centred at (-5,0)

Here is the pic

Is this how you thought about it too Ginger?

Feb 23, 2017
#6
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Thank you, Melody.

… can you give me any insite to your thinking process?

This may be more difficult than solving the puzzle.

While some of the math i used fits your description, I can’t say I solved it by any advanced mathematical derivation, though, someday I truly hope to do that.  I’ve solved several of these types of puzzles and I enjoy doing them just like a crossword puzzle.  I did this one as a diversion from studying for midterms.

Over time, with practice, this became intuitive to me by drawing on my innate and developed artistic skills.  When I first started doing these it seemed to take forever before I could visualize a solution with the limits imposed.  I’m much faster now, but sometimes I never figure them out, though I never really give up.  I will say that while artistic skills are a big help, I think just by playing with these types of puzzles most will naturally learn the spatial relationships and start solving them relatively quickly.

For this puzzle, the first thing I noticed was that it would need to be reflected an odd number of times because the images are mirrored.  After that the process became common to all transformation puzzles: mentally project how each move will affect its orientation and how that orientation relates to the final desired position; how the dilation will influence the sweeping arc around the fixed point.

My mentor –our beloved troll, Lancelot Link, recognized, early on, my skills in imagining spatial perspectives. (Such skills are common to us genetically enhanced chimps. http://web2.0calc.com/questions/problem-solve-this-given-to-primary-school-children-in-china#r6)  Knowing this, he set out to teach me visually, which included geometry and general physics, whenever possible.  In one of his first comments he included a photo of a garden hose shooting a horizontal stream of water arcing to the ground.   He said every time he sees a hose or a fountain shooting streams of water he sees projectile and gravity formulas.  For my first physics assignment: study the stream of continuous projectiles (water drops) emitted from a garden hose:  take measurements with varying valve settings, plug them into the appropriate equations and solve.  I think by doing this I learned the equations much faster and still remember them long after the lessons are over, compared to just learning them by rote.

From this I began to understand how parabolas relate the information of falling objects in a gravity field. These were baby steps, for sure. But now when I’m watering my garden, I can estimate the velocity of the water by how far it travels after it exits the hose.  I don’t see the formulas every time I use a hose or see a fountain, but they are becoming (monkey) footnotes to many observations.

Lancelot also suggested I learn to parachute to help me understand air resistance and terminal velocity. I actually did that --it was quite fun. Though, I drew the line at bungee jumping to help me understand Hooke’s law. This chimp was not ready to swing from a long vine. I’m a chimp, not blŏŏdy Tarzan.

All of this opened a new world for me. Now, I’m a genetically enhanced chimp with an education.  I can see the next step in my evolution – even if I’m not there yet.  Oh! And a most important thing: I’m still alive.

These coordinates   ((-1.5,-1.5) to (1.5, 1.5)) are just arbitrary points to clarify the reflection. They conform to Khan Academy solutions for example problems when y=x reflections are used.

Feb 23, 2017
#7
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Thanks Ginger, that is all really interesting.

I have not seen transformation puzzles like these before, I can see how they become intuitive.

Is there are site address or some where I can get more of them?

I really like doing that one and I enjoy drawing with Geogebra too. :)

I guess there is a site somewhere.  If you do not know of it then I guess Mr Google probably will. ://

I like the practical physics lessons too.

Next time I have to use the hose I am sure I will be thinking about the projection path of the water :)

Melody  Feb 23, 2017
#8
+2169
+10

Here is the Khan site where this puzzle came from.  (It’s puzzle #3, but there are several #3s, The number has to do with the relative complexity.) Most have an interactive graphing feature for solving.

The puzzles cycle as you solve them or attempt to.  Refresh the page for the next puzzle.  New ones are added periodically.

Feb 23, 2017
#9
+113567
0

Thanks very much Ginger.

I shall take a good look :)

Melody  Feb 23, 2017
#10
+1879
+10

Is there a strategy for solving these puzzles? Or is it just trial and error.

Feb 25, 2017
#11
+2169
+11

The third paragraph in post #6 gives an outline of my strategy. Reiterating: the only thing I could instantly discern was it would need an odd number of reflections. Other than that, it is just an application of intuitive skill developed over time with practice.

I agree; the strategy is one of trial and error, but note that your error count will drop quickly with practice.

------

I almost forgot I’m the protégée of a masterful troll.   I should have said that your error count will drop quickly with practice, unless you are a dolt

P.S. Don’t forget to give me points. Melody said my solution was “Brilliant” and worth a hundred points. So far, I only have five measly points on my solution post.

You know, it really is fun being a troll!

GingerAle Feb 25, 2017
edited by GingerAle  Feb 25, 2017
#12
+1879
+13

Thanks.

There you go :) I gave 5 points to each answer in this post (20 points total unless you reply to this which will then be 25)

EighthMersennePrime  Feb 26, 2017