Problem 620

[albert]

"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

[RobertStanforth]

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]

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

[RobertStanforth]

Rotations and reflections do not give rise to distinct arrangements.

[Radewoosh]

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

[RobertStanforth]

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]

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.

[RobertStanforth]

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]

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]

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?)

[Animus]

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]

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.

[RobertStanforth]

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]

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

[RobertStanforth]

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]

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]

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.

[dani.spivak]

I've been reading this problem for a while now and I don't understand what it's asking. Can someone explain what I'm looking at with the given arrangement? I cannot figure out what the constraints on the shapes of the gears are, or what it means that they mesh perfectly.
In particular, if I assume that all circles must be tangent, and that the points of tangency have to be a half-integer number of centimetres apart, then I get g(16,5,5,6)=0. Why is this assumption wrong?

[Animus]

Hi Dani

In particular, if I assume that all circles must be tangent, and that the points of tangency have to be a half-integer number of centimetres apart, then I get g(16,5,5,6)=0. Why is this assumption wrong?

Take a close look at the illustration given, showing a valid configuration. You can see that the teeth do not interlock with their counterpart teeth in the same relative position at the various touching points of the gears.
Thus, the second part of your assumption is false, as it implies an additional (false) restriction about the relative distance of the inner gears. You need to come up with another restriction resulting from the fact that all gears are connected.

For example, imagine a sticky chewing gum that got into the moving gears and happens to switch gears at some of the tangent touching points (leading to different paths). What can be said about the total length traveled, when the gum eventually happens to return to its starting point?

[dani.spivak]

I see my mistake, the gears do not have to be centred on each other at the same time. I think I have an idea of what you're getting at with the piece of gum analogy, but I'd like to clarify if the following conditions capture the problem:
1) The circles must all be tangent to each other
2) there exists an initial positioning of gear ends every cm on their circumferences, such that when they turn at a rate of 1cm/s (with the inner one in the opposite direction), at each point of tangency exactly 0.5 seconds pass between the gear end of one circle passing and the gear end of the other circle passing.