I always just wrote this little tale off as "good at reading people", and honestly that's still my assumption. After reading Blindsight though, I think it's a good allegory for possible intelligence without consciousness. What if the horse just has the ability to perform those kind of calculations when incentivized, but has no concept of what it's actually doing beyond responding to stimulus.

Then again I knew a horse that would recreationally lick electric fences, so probably not that. Interesting thought though.

Turns out it was really good at reading people. It would notice the subtle changes in facial expressions as it tapped the correct number of times. When the horse and participant were blocked by a partition, it couldn't math anymore.