Image classification is a classic example of such a task. How exactly would you go about writing an algorithm to tell the difference between a picture of a cat and a picture of a dog?
Well, this might be cheating, but I would apply a bunch of different filters for things like edge-detection, etc. Then I would come up with a statistical model that, for each feature, gave the likelihood that there image under consideration was a dog. Then I would aggregate all those results into a final likelihood.
Not trying to be sarcastic, I just can't think of any way other than the ML way.
To further the point: what filters would you choose? What features could you choose heuristically to distinguish between the two? They both have fur, they both have four legs, they both have two eyes, they both come in a wide variety of colors and patterns... Most dogs have an elongated snout but not all of them (pugs, bulldog, etc.).
I would be extremely impressed if someone developed an algorithm that could accomplish this task without using any type of statistical/machine learning.
Not trying to be sarcastic, I just can't think of any way other than the ML way.