A Scientific American bolt puzzle | Hacker News Reader

A Scientific American bolt puzzle

leancrew.com

62 pointszdw31 comments

I grew up on Martin Gardner in the 70's and 80's, so I'm always happy to see him referenced. My favorite physics puzzle of his is: If you heat a solid iron torus, does the radius of the hole in the middle grow or shrink?

Love that one, and the closely related hole in a large metal plate. Something about that reframing flips my intuition on it's head.

Grow, of course. I mean, come on.

Some intuition says one must consider the outer iron expanding into the hole. Some thought allows the realization that the only direction for expansion is outward.

Or consider that, assuming the expansion is uniform, it's applying a scale transform to the object and otherwise leaving the shape exactly the same. All aspects of the shape will therefore grow larger.

This is a novel way to think about it that hadn’t occurred to me. Thanks!

[Edit: removed spoiler]

You can gain a bit of intuition about growing and shrinking circles by imagining them to be squares with _very_ rounded corners, or equivalently, imagining what happens if you lengthen them by inserting straight lengths in four places, turning them into squares with very rounded corners.

For example, the old question of: if you have a rope lying on the surface of the earth all the way around a great circle, and make it 10m longer, how high could it be above the surface?

Even though the square-with-corners thing helps me visualize it quite perfectly, I _still_ find it weird that that the answer is 1.5m!

Or you could imagine an infinitely thin torus.

Anyone who's opened a tight jar lid by putting it under hot water should know the answer to this instinctively.

This thought experiment has at least one very useful practical application, and the timing of your comment is quite the coincidence. I'm getting ready to heat up a differential pinion bearing retainer (Ford 9" style) in a toaster oven right now...

Interesting to see this presented as a puzzle- I've never thought about it academically, but knew the answer immediately from hands on experience. Slipping metal bearings over metal shafts when the inner diameter of the bearing is less than the outer diameter of the shaft is a piece of cake if you throw the bearing into a toaster oven for 15-20 minutes first. It'll drop right on right down to the base no problems. When they're unheated you wouldn't be able to get them on and down even with a significant amount of brute force hammering.

Switching the direction that you're twiddling the bolts would have to change the direction of any movement. But by symmetry, clockwise and counterclockwise twiddling are identical (looking down on the head of each bolt, one is always moving clockwise and one is always moving counterclockwise). So there must be no in/out movement at all.

This is similar to how I guessed the answer - it's a symmetric system, so there's no reason why it would be one direction over the other and so logically there would be no movement.

No there is not that symmetry, because the helix is handed, and the result could depend on the rotation direction with respect to that handedness.

Relative to the handedness, one bolt is always moving with it, one bolt is always moving against it. Switching direction doesn't change that. So switching direction can't change whether it moves inward or outward.

Couldn't you transform twiddling one way into twiddling the other way by reflecting through a mirror and turning the bolts 180 degrees end-over-end? If you start with the bolts moving North, the mirroring makes them go South, and turning them around makes them go North again, but now you're twiddling the other way. So the bolts would have to go North no matter which way you twiddle them, so either twiddling is not reversible (which is a repugnant proposition) or they must have a speed of 0.

Moreover, the problem as posed doesn't specify the direction of twiddling, which kind of gives away the answer.

The arrows in the picture specify the direction of twiddling. And the text says "see illustration".

It stays stationary because you're effectively tightening one bolt and loosening the other. Imagine the point of contact between the bolts is a line drawn on a stationary nut instead. Is that line moving clockwise or counterclockwise around the bolt? One way tightens, the other way loosens. And it's opposite for the two bolts.

When you twiddle your thumbs, your thumbnails always point the same direction relative to each other. So the bolts relative rotation will maintain that. Imagine a stripe on the face towards us of the bolts, and twiddling action will not rotate the stripe in an absolute or relative sense.

For the two bolts, this is equivalent to rotating both bolts so the stripe stays in the same relative position, as if one were rotated around the other or they were twisted in the same absolute direction. If you are concerned about symmetry violation because of the direction of the threads, you can reverse the threads on both bolts, with the same result. You can try this with a couple of bolts and a couple of rubber bands to keep them in the same relative position. The illustration is a hint that the motion is equivalent to rotating both in the same absolute direction (near side moving up.) Then view that system from the head and note the direction each would be moving, away or towards you.

If you have to undo a bolt or nut from behind, the bolt head moves in a reverse direction from your viewpoint, just as if you were to view a glass clock from behind.

I met Martin Garder and spent a morning with him, one on one. I have a photo of the two of us together, and on the back his notes about our meeting, written in his own hand, given to me by his son.

Amazing experience for someone who grew up reading his books.

Lovely man, sharp as a tack right to the end, and sadly missed.

spoiler alert, directly quoted from article:

>The heads of the twiddled bolts move neither inward nor outward. The situation is comparable to that of a person walking up an escalator at the same rate that it is moving down. - Martin Gardner

>I don’t find it an especially helpful analogy. Why is the twiddling of bolts like a person walking up an escalator?

the bolts and the up-the-down escalator are a good comparison because if you analyze what happens from one bolt's perspective, and then the other bolt's perspective, one is walking up (in) and the other down (out).

(as a nit, I don't like when people breezily "reword" what somebody says while analyzing it: are "comparable", "analogous", and "like" exactly synonymous with one another?)

On your nit, I think the author's use of language is good.

If two things are "comparable", they must be "like" each other in some way. Otherwise they could be contrasted, but never compared. The author's question is equivalent to "in what way are these two bolts equivalent to the person and the escalator?"

I also found it a useful analogy. Or rather, that was the way I thought of it.

First I had to think about what would happen if the top bold, instead of rotating around a bolt, rotated around a band that wrapped straight around the other shaft. It stood to reason that the bolt would move backward away from the band, as the band moved along the spiral. (Whether it moved outward or inward ended up being irrelevant.)

Then I went back to it being two bolts, and visualized the movement from the head of each. In one, the other bolt moved clockwise, and in the other, the bolt move counter-clockwise. So it stood to reason that one bolt would want to move away from the other, and the other move towards. So they would stay stationary.

I know the answer because I've actually done this with real bolts (out of boredom — before I knew Martin Gardner had a puzzle about it).

Likewise I also know the answer to which direction the spool rolls if you set it on a table and slowly pull on the thread.

Here is my guess. If you twiddle your thumbs in the screw-in direction, the heads move together at twice the rate that a head would move when screwed into a fixed thread. And in the screw-out direction, the heads would move apart at twice the rate of dissassembling a bolt from a fixed threaded hole.

Ooh I’m pleased I got this answer right imagining it. But I may have imagined it incorrectly — in my mind I was worried that there would be pressure from rotation at the point of contact. But as I write it, I realize that the question “pressure in which direction?” Shows that’s fairly unlikely.

Having done this IRL before, my mental model is that the other screw is just like a nut, except only touching at one point, so the behavior is the same as screwing in a bolt normally.

>so the behavior is the same as screwing in a bolt normally

...if you are simultaneously screwing out the other bolt normally, because the bolts screw neither in nor out relative to one another.

I found it to be much easier in the rotating frame of reference where both bolts are spinnimg in place in the same direction

But I've played around with bolts a lot as a kid with Meccano etc so the answer is second nature.

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