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A ball-and-stick model may make spatial relationships clear at a glance that would only be apparent to experienced people well able to visualize in 3D without one.

There's an example from my days in graduate school that I like to use. One of the research groups in the department had been trying to synthesize a certain highly substituted ring system, but had been unable to do so (they had gotten close, but never managed the last couple of steps). Let me remind you, these were people who held at the least a Bachelor of Science in Chemistry, under the direction of a Ph.D., so it's not like they were completely clueless.

And then someone had the notion to construct a model of the system. Whereupon they immediately gave it up as hopeless, because then they could see why it wasn't working: there simply wasn't room to stick those last few substituents on; they physically wouldn't fit.

Thinking in 3D is a very special skill, and it's not always trivial to keep all the permutations straight in your head; it's relatively easy to convince yourself "Yeah, but maybe if this were twisted that way" until a model is staring you in the face with two atoms that need to occupy the same space no matter how you contort the rest of the structure to avoid it.

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Q: What advantage of a ball-and-stick model over a structural formula?
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