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Yeah I think on say, a proper road the horse would win at any distance.

the road adds some infrastructure - i think at that point the person should get a bike?

1990: For a few joyous years there was the Man v Horse v Bike competition in Mid-Wales - https://youtu.be/vFlglZUIKO8

Well that’s just not fair. Even the fastest human on earth can’t be expected to keep up with a horse riding a bike.

There's probably some upper limit to it, i am not sure a horse could live through the ultra marathon moab race in the western us.