1. Spur-gear differential; and,

2. Shishi-odoshi.

Both of these are saturating mechanical devices that can be used to build NAND gates; the latter, I think, would be very pleasing, if exceedingly slow.

For the spur-gear differential, you'd need to up-scale the output by a factor of 2 (since the output is half-speed), and use a locking wedge to build a one-way gear out of one of the spur-gear differentials. However, it has the nice property that the logic is made entirely out of a single element: the spur-gear differential.

Similarly, for the shishi-odoshi: you're going to have to do a bit of analysis (drilling a hole in the bottom part of the bamboo ladle), to figure out the in-flow and out-flow to build the basic AND gate, and then balancing out the NOT gate, to build your basic NAND. This is, obviously, very finicky; but, I supposed, that'd be quite a bit of the charm of a Zen computer garden?

Also, this 1950s Naval Training film explaining the fundamentals of how mechanical fire control computers work to solve complex problems is excellent. https://www.youtube.com/watch?v=s1i-dnAH9Y4

Will robots (which will hopefully soon be available) be able to do it?

Exhalation really gets me thinking about what it means to be sentient & self-aware. If the neurons in our brains could, even in theory, be simulated by logic gates then, equally in theory, a Turing machine could be sentient. I can even imagine a bunch of rocks being sentient: https://xkcd.com/505/