Problem 620

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albert
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Problem 620

Post by albert » Sun Feb 11, 2018 1:14 pm

"The circles can overlap."

One could leave this out, as in the sequel the circles are represented by gears, which obviously can't overlap.
Also the drawing suggest that ghost gears that are somehow occupying the same space are not allowed.
So it started out as circles out that roll off each other without sliding, but the remainder to makes it into a Diophantine problem is murky.

Or have I misunderstood the problem totally?

Groetjes Albert

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RobertStanforth
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Re: Problem 620

Post by RobertStanforth » Sun Feb 11, 2018 1:34 pm

Hi Albert,

The 'planet' gears can overlap if they occupy different locations in the $z$ axis (i.e. the direction coming out of the page).

Steppenwolf99
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Re: Problem 620

Post by Steppenwolf99 » Sun Feb 11, 2018 2:58 pm

What arrangements are counted as distinct? It doesn't look like two arrangements of gears that differ only in orientation are counted as distinct.

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RobertStanforth
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Re: Problem 620

Post by RobertStanforth » Sun Feb 11, 2018 3:24 pm

Rotations and reflections do not give rise to distinct arrangements.

Radewoosh
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Re: Problem 620

Post by Radewoosh » Sun Feb 11, 2018 6:42 pm

Can two planets (with this same circumference) have exactly the same position?

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RobertStanforth
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Re: Problem 620

Post by RobertStanforth » Sun Feb 11, 2018 9:03 pm

Radewoosh wrote:
Sun Feb 11, 2018 6:42 pm
Can two planets (with this same circumference) have exactly the same position?
No, the four planets must all be distinct: no two may have the same size and position.

abcwuhang
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Re: Problem 620

Post by abcwuhang » Mon Feb 12, 2018 1:18 pm

As an epicyclic gear train allows the planets to rotate, what is the meaning of perfectly meshing of gears? Could you show how to drive the whole system? Thanks.
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RobertStanforth
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Re: Problem 620

Post by RobertStanforth » Mon Feb 12, 2018 7:21 pm

abcwuhang wrote:
Mon Feb 12, 2018 1:18 pm
As an epicyclic gear train allows the planets to rotate, what is the meaning of perfectly meshing of gears? Could you show how to drive the whole system? Thanks.
Each of the six gears rotates about its own axis. The axes themselves don't move.

Steppenwolf99
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Re: Problem 620

Post by Steppenwolf99 » Wed Feb 14, 2018 7:37 pm

I left this alone for 2 days and finally solved it on my second attempt today. A comment: some of the clarifications on this section should be included in the problem description. The trouble I had was more about parsing semantics and I made a couple of wrong assumptions which could have been avoided if I had come back to this page earlier. Thanks for an elegant problem.

hexadoodle
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Re: Problem 620

Post by hexadoodle » Thu Feb 22, 2018 10:36 pm

A correct configuration of gears will ALWAYS have their underlying circles as tangent to each other as described in the problem, correct?

How important is the shape of the teeth? If they were more square or more rounded, would it affect the answer? What if the teeth stuck out further or lesser? (Is this what is meant by "pitch" in the problem?)

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Animus
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Re: Problem 620

Post by Animus » Thu Feb 22, 2018 10:51 pm

hexadoodle wrote:
Thu Feb 22, 2018 10:36 pm
A correct configuration of gears will ALWAYS have their underlying circles as tangent to each other as described in the problem, correct?
Yes.
How important is the shape of the teeth? If they were more square or more rounded, would it affect the answer? What if the teeth stuck out further or lesser? (Is this what is meant by "pitch" in the problem?)
With perfect meshing teeth two connecting gears, represented by their touching circles, will move perfectly in sync, with their angular velocties having the invers ratio of the number of their teeth.
In the real world, perfectly meshing teeth are difficult to construct, and yes, the hight and the form ot the teeth are both important, but that does not affect the answer in this idealized problem.

MuthuVeerappanR
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Re: Problem 620

Post by MuthuVeerappanR » Tue Feb 27, 2018 10:49 am

The question introduces $c, s, p$ and $q$ as circumference of the circles but in the second para says they are the numbers of teeth. Am confused about the interpretation of these values in the final summation. I'm already confused about whether the 'module' is something implicitly given or not?

Many people have solved it so far. Am assuming I'm missing something. But still, can anyone clarify??

Thanks.
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RobertStanforth
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Re: Problem 620

Post by RobertStanforth » Tue Feb 27, 2018 6:56 pm

The gears have a pitch (i.e. tooth-to-tooth distance) of 1cm. Hence a circle's circumference in centimetres is equal to its number of teeth when it's considered as a gear.

amidar1
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Re: Problem 620

Post by amidar1 » Sun Mar 04, 2018 5:55 pm

I have been beating my head against a wall with this problem since it was published three weeks ago. In the last couple of days, I feel like I have started to get some traction, but sadly I keep seeing the red X with the despondent little man in it.

I would post some digits or some additional data points that my program is giving me beyond G(20) = 205, but with still fewer than 100 solvers, I am worried that some would feel that it is too soon to post additional information.
Is anyone willing to let me PM them some numbers so you can tell me if I am at least in the ball park?

Thanks,
bunderscore

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RobertStanforth
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Re: Problem 620

Post by RobertStanforth » Mon Mar 05, 2018 7:29 am

amidar1 wrote:
Sun Mar 04, 2018 5:55 pm
I would post some digits or some additional data points that my program is giving me beyond G(20) = 205, but with still fewer than 100 solvers, I am worried that some would feel that it is too soon to post additional information.
Is anyone willing to let me PM them some numbers so you can tell me if I am at least in the ball park?
Thank you for not posting partial/intermediate values.
Please be patient: people are still competing for the remaining leaderboard places without external assistance.
Good luck with the problem!
Robert

amidar1
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Re: Problem 620

Post by amidar1 » Tue Mar 13, 2018 4:48 pm

RobertStanforth wrote:
Mon Mar 05, 2018 7:29 am
Thank you for not posting partial/intermediate values.
Please be patient: people are still competing for the remaining leaderboard places without external assistance.
Good luck with the problem!
Robert
Understood. Thank you, Robert, for considering.
This problem was quite a journey for me!

Cheers,
bunderscore

sevenkplus
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Re: Problem 620

Post by sevenkplus » Thu Mar 22, 2018 4:50 pm

Oh I wasted too much time figuring out what the problem want us to do...

I want to post some clarifications so that future people will have less struggle.
1. Although the shape of teeth does not change answer, the number of teeth matters. Suppose we have a teeth every 0.5cm instead of 1cm, the answer will be different.
2. Circumference is not radius, which is obvious but may be forgotten sometimes.

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