1. ## Cognac 2

This will probably one of my last puzzles for a bit a I would like to see some new original offerings from other Loungers if possible.

Jezza Bear has worked out what was going wrong in his original problem in Cognac and has corrected his plans (below). In the diagram you can see a cog assembly on a specially arranged track. There are two cogs, one smaller than the other and are spot welded together. Side on they look like the rectangles on the right. The puzzle does not require you to know the diameters, just that one is smaller than the other. The black fuzzy lines are specially designed tracks that the cogs run on.

Jezza Bear is confused though and here are his ramblings

"Now If I turn the cog assembly a quarter turn A to B, both cogs run along the trackthe same distance, I can see that with the crosshairs I drew on the side...that is fair enough. But say I had two seperate cogs one with twice the diameter as the other one and I rolled them along a table by 1 quarter turn each, the larger one would travel twice the distance, wouldn't it?

Why should they travel the same distance when they are spot welded together.

That lot in Woodys Lounge always seem to know better, lets ask them"

So here is Jezza Bears Question:

"Why when the two cogs are welded together they travel the same distance but when seperated the larger one travels further as expected?"

2. ## Re: Cognac 2

<img src=/S/rofl.gif border=0 alt=rofl width=15 height=15> Tricky !

I have seen this one before so I am not going to answer here.

But to clarify: <span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">The two cogs are tack welded together such that their axis points are EXACTLY the same - Correct?</span hide>
<img src=/S/cheers.gif border=0 alt=cheers width=30 height=16>

3. ## Re: Cognac 2

<span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold"> PM me with a precis of your response, I would like to hear it</span hide> <img src=/S/evilgrin.gif border=0 alt=evilgrin width=15 height=15>

Done !

5. ## Re: Cognac 2

Suspense......Was Scott right?....Would he have won a [choccy bar] if he had answered in the Forum <img src=/S/question.gif border=0 alt=question width=15 height=15>

<img src=/S/grin.gif border=0 alt=grin width=15 height=15>

6. ## Re: Cognac 2

<img src=/S/whisper.gif border=0 alt=whisper width=29 height=17>I haven't received it yet.....I think he is using P.I.P.

<span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">P.I.P. - Pigeon Internet Protocol----- a lot of west highland terriers use it I have been told</span hide>

7. ## Re: Cognac 2

Actually ....

WORSE !!!!

I accidently send the PM to BigAl and he responded to let me know I sent it off to the wrong person.

I have corrected the problem and re-sent to you. I could have <img src=/w3timages/censored.gif alt=censored border=0> I put your name in correctly - but could not verify it after the last "Preview".

BTB-Per my family, I AM a PIP !! <span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold"><big>P</big>ain <big>I</big>n <big>P</big>osterior !!!</span hide>

8. ## Re: Cognac 2

Oh Yes - Please be advised that HisHighness <span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">AKA the "Avatar" under my User Name</span hide> was not sure how to take the <span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold"> " ...a lot of west highland terriers ..." </span hide> comment.

He is a little sensitive since he is <span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold"> a Fourteen Year Old Norwich Terrier with family abroad !!!</span hide> <img src=/S/grin.gif border=0 alt=grin width=15 height=15>

He was already upset with me because I refused to let him have any of the [choccy bar] I won last time - not good for .....
<img src=/S/hmmn.gif border=0 alt=hmmn width=15 height=15> <img src=/S/whisper.gif border=0 alt=whisper width=29 height=17>HisHighness really doesn't like to be called a D - O - G ! As my mother used to say, "D - o- g - s are people to you know" and HisHighness took this to heart.

9. ## Re: Cognac 2

I would say that <span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">because they are welded together, they have no choice but to travel the same distance.</span hide> ? <img src=/S/shrug.gif border=0 alt=shrug width=39 height=15>

10. ## Re: Cognac 2

Wow...were you waiting for me to do my a.m. log in <img src=/S/hiding.gif border=0 alt=hiding width=70 height=24>

A colleague has come up with the same idea, and I told her that although that is the logical answer and you cannot refute it, there is a reason, when I told her she had an eureka moment....

11. ## Re: Cognac 2

Then the only other logical answer is <span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">that the tracks moved relative to each other, 'cos you never said nuffin' about them being fixed</span hide>?

12. ## Re: Cognac 2

Who said the answer was logical <img src=/S/evilgrin.gif border=0 alt=evilgrin width=15 height=15>

The tracks don't move.....BTW

Time for a clue:

<span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">Lets us move away from the concept of circles and redraw the diagram replacing the circular wheels with hexagons, see what happens to the small cog when you turn the hexagons 60 degrees....</span hide>

13. ## Re: Cognac 2

Once more unto the breach, dear friends, once more;

There has been no more movement on this one for quite a bit and shall provide you with an answer of sorts and a proof, please feel free to argue amongst yourselves but I think the ghost of Galilieo may want to have a chat <img src=/S/invisible.gif border=0 alt=invisible width=15 height=15>

The general context of this is infinitesimals originally argued in the using the method of disproof to prove the argument <img src=/S/dizzy.gif border=0 alt=dizzy width=15 height=15>

In an earlier clue to Leif I said Lets us move away from the concept of circles and redraw the diagram replacing the circular wheels with hexagons, see what happens to the small cog when you turn the hexagons 60 degrees....(see below)

I have therefore constructed another diagram, replacing the circles with hexagons. Look at the cutaway of an enlarged vertex of the hexagon and you will see that I have transposed an equilateral triangle (in red) and the height is [um] highlighted in blue.

Assume that the vertivcal distance between each hexagon is 1 unit therefore the length of the sides of the triangle is 1.154

Therefore each time the hexagon turns the smaller hexagon lifts off the upper track to a maimum height of 0.154 units.

Think of a circle, it has an infinite number of straight sides and therefore every angular turn it takes the inner cog will lift a small amount from the upper track, infact an infinite number of times which are infinitesimal in size. We "see" the inner cog roll along the upper track, it does not slip but it skips but we can't see it skip because each movement is infinitesimal in size.

Anyway....enjoy....you can argue amongst yourselves now <img src=/S/evilgrin.gif border=0 alt=evilgrin width=15 height=15>

14. ## Re: Cognac 2

Oh why not... <img src=/S/evilgrin.gif border=0 alt=evilgrin width=15 height=15> I scratched my head on this for a moment and thought that maybe it was just semantics in the difference between slip and skip (or is it just one letter?) But then I noticed that you were using an argument for a circle that can only apply to polygons. You said

"Think of a circle, it has an infinite number of straight sides and therefore every angular turn it takes the inner cog will lift a small amount from the upper track, infact an infinite number of times which are infinitesimal in size"

But this is not true. A circle is defined a the locus of points in a plane equidistant from the center. There are NO points, even infinitessimally, further from the center that would cause the smaller wheel to lift up.

You cannot have the situation of the two wheels as drawn without slippage. If both tracks were indeed cogs, you could not move the circles at all.

15. ## Re: Cognac 2

As I said, this is my last puzzle for a time.....I am not going to argue semantics of language with you....I was hoping to leave it at that. It is very difficult to write succinct answers to these puzzles. It would mean having to write a whole set of paragraphs and diagrams to explain this and I suppose this is not possible in this environment, that is why I am not doing these types of puzzles any longer.

If you have any comments and you want to discuss further you have my public email in my control panel, you can email me and I can look and review the answer in this thread.

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