A SciAm bolt puzzle
March 3, 2025 at 6:26 AM by Dr. Drang
A week or so ago, Scientific American republished this Martin Gardner puzzle from 1958:
Two identical bolts are placed together so that their helical grooves intermesh (see illustration). 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, will the heads move inward, move outward or remain the same distance from each other? The problem should, of course, be solved without resorting to an actual test.
Illustration by Amanda Montañez for Scientific American.
The “twiddle your thumbs” description is very good. because a key feature of the bolts’ movement is that even though they orbit one another, they don’t rotate. The vertical faces on their heads, for example, remain vertical throughout the motion.
Also important is that the bolts are identical, otherwise their threads wouldn’t mesh. Here’s a drawing from ANSI Standard ASME B1.1, Unified Inch Screw Threads, which shows the standard thread profile:
If the pitch, , the major diameter, , and minor diameter, , of the two bolts match—which they’ll only do if the bolts are identical—the bolts will fit together and can slide around each other without binding. If the dimensions don’t match, the bolts won’t fit together.
(By the way, although the B1.1 standard is for threads measured in inches, this thread profile applies to metric threads as well. The difference is that the pitch and diameters for metric threads will be given in millimeters.)
Gardner’s answer is
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.
This is certainly true, but I don’t find it an especially helpful analogy. Why is the twiddling of bolts like a person walking up an escalator? A person and an escalator are two very different things, but the two bolts are identical.
Let’s look carefully at the geometry. Here’s a stylized drawing of the two bolts. The centers of the thread peaks are drawn as black helices; the centers of the thread valleys are drawn as blue helices; and the shank is drawn as a translucent cylinder so you can see the threads on the opposite sides of the bolts. I’ve left off the bolt heads because they don’t play any role in the engagement.
For what it’s worth, I’ve used the dimensions of a particular bolt to make the drawing: a 1″ coarse thread (UNC) bolt. The major diameter is 1″, the minor diameter is 0.8647″, and the thread pitch is 0.125″. Any bolt would do, but I wanted to use a real bolt. I’ll explain how I made the drawings in a later post.
We’re looking at the bolts directly from the side in an orthographic projection, so there are no perspective effects. This will help us see how the bolts move (or don’t move) axially as they orbit each other. One thing this view makes clear is that peaks and valleys are opposite one another; directly across from every peak is a valley.
In the current position, the bottom of the top bolt is engaged with the top of the bottom bolt. The peaks fit into the valleys. We’ll mark one peak of the top bolt and the valley of the bottom bolt it fits into.
Now imagine each bolt going through half an orbit, the top bolt swinging around behind and the bottom bolt coming up the front. Because neither bolt rotates, what’s now the top of the top bolt will end up in contact with what’s now the bottom of the bottom bolt. If you follow the black (peak) line of the top bolt from the purple dot along the near side up to the top, that’s what will remain in contact during the half-orbit. Similarly, if you follow the blue (valley) line of the bottom bolt from the purple dot along the far side (by looking through the translucent shank) down to the bottom, that’s what will remain in contact during the half-orbit.
In other words, the two points marked in the figure below—still drawn in the original configuration—are the points that will be in contact after the half-orbit.
The dashed vertical line shows that these two points are aligned before the half-orbit. And they must be aligned after the half-orbit because they’re in contact. Since they’re aligned at both the beginning and the end of the motion, the bolts don’t advance or retreat—they stay in the same left/right position. You could make the same argument using a quarter-orbit, a full orbit, or any multiple of an orbit. Because the geometry of the threads on the two bolts is identical, and because peaks and valleys are opposite one another, the bolts don’t go left or right during the twiddling motion.
Gardner wants you to work out the problem without grabbing a pair of bolts and trying it out. That would be cheating. But after solving the puzzles, it’s nice to see it work on real bolts.