Answer to Riddle #84: Twiddled Bolts

84. Two identical bolts are placed together so that their helical grooves intermesh. If you move the bolts around each other as you would twiddle your thumbs, holding each bolt firmly by the head so that it does not rotate and twiddling them in the direction shown, ill the heads a) move inward, b) move outward, or c) stay the same distance a from each other?
Interlocking Bolts orbiting each other

An unusual puzzle this because your ability to answer depends on not your pure intelligence but your ability to visualize things in the abstract. I read this puzzle the other day, and I've been thinking of a way to explain it, I think I've come up with a good plan, let's see if you agree:

Before reading the answer can I interest you in a clue?

Consider the arrangement below: a special bolt with just a flange on the end interlocked with a regular bolt

It shows rather than two interlocking bolts, one bolt and a sort of specially designed thing with a singular thread in a circle around it.

It's reasonably easy to tell by intuition that given the orbital direction that the stud at the top (currently,) will move to the right.

Now consider the arrangement below... a special bolt with just a flange on the end interlocked with a regular bolt, the other way round from before

The stud will also move to the right.

So if both are moving to the right, at the same speed, then there is no relative motion, (save the rotation,) at all.

Finally as the convincer I've included a video...

I know, I know, I twiddled the bolts in the opposite direction. Hopefully you can see that if the don't move relative to ach other when twiddled one way they wont when twiddled the other.

Puzzle comes from a book, My Best Mathematical Puzzles by Martin Gardner

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