In these later examples (starting with the easy puzzle of the OP, and your 3d examples), I find that I do the process in two stages.
Unfocus my sight until the third image shows up in the middle at the correct size (as a blurry mess). Then try to focus the center image.
which is one reason why stereo movies have struggled. (That plus some people get sick... Having both a flat and 3-d movie in two different theaters comes across as money grubbing to the consumer but it is really a money sink to the theater.)
You can also tell if your head's level, just by crossing your eyes. If the two images are diagonal to each other, then your eyes/head aren't level. I have no idea what the possible use for that would be.
It makes me wonder if the wall-eyed version could be useful for eye health.
I've often heard when doing computer work, you should focus on something 20 meters away for 20 seconds, every 20 minutes.
Doing a wall-eyed magic eye seems like the same thing physically, your focal point is much further away.
Would be cool to have some software that lets you overlap two coding windows, so you have a 3d terminal.
https://triaxes.com/docs/3DTheory-en/522ParallelCrosseyedvie...
which some people struggle with, somebody posted a
https://en.wikipedia.org/wiki/Autostereogram
to HN yesterday which some people get and others don't. (That's different from the "cross-eyed stereogram" because one of them involves having two images and the other one has one image with two images hidden in it)