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Since the invention of hard road surfaces, a centaur would need to wear metal shoes like any other horse-like creature.
Can you see any barrier to a suitably qualified centaur
(a) being a blacksmith/farrier and
(b) shoeing him or herself when the need arises?
Notes
If possible I'd like answers from people who have knowledge of horses and their anatomy - and especially farriers of course!
I included the biology tag because anatomy and veterinary science were not available.
Short answer: A centaur can't reach his own hooves, but can reach the hooves of others by lying down or kneeling. One centaur can maintain the hooves of another.
Better answer: There are a few reasons why horses wear shoes: to keep their hooves from cracking on paved roads; to gain better grip on cobblestones, in slippery conditions, or during races; and fashion (some horse shoes have a leather portion against the hoof to make them look nice for horse shows.) Centaurs would wear shoes for the exact same reasons.
If centaurs built a city, I doubt they would pave the roads, per se. They would probably install something akin to a polo pitch, which could be tended by horses, goats, etc. who would eat the grass and keep the pitch clear for centaurs to gallop through.
Hale and hearty centaurs would likely not need shoes or a farrier because they would take plenty of exercise in the fields where they don't need shoes and will wear down the hooves enough that they don't need trimming. Sick centaurs who can't run around for a time would grow hooves that would need trimming. The need for a farrier would probably be associated with poor health and would thus be stigmatized.
For humans, long fingernails are associated with a life of leisure because one cannot work manual labor and maintain long nails. You might think this would be the same for centaurs and their hooves but it only applies to their fingernails. The hooves are a centaur's toenails and nobody has anything good to say about long toenails. In fact, excessively grown hooves can cripple a horse or centaur.
Because shoes can only be installed by another centaur, they would be essential indicators of civility and sociability - you can't have shoes if you can't get someone willing to put them on you. Shoes would be a status symbol. The fancier and more difficult they are to put on, the better, because it means that someone else took a lot of time to service your feet.
Barefoot centaurs would include: barbarians/wild centaurs; low class, poor, or rural centaurs; those who prefer to live a simpler life (hippy centaurs); and higher class centaurs who are counter-cultural for some reason (maybe in protest).
Rich, high status centaurs would flaunt their wealth and status with elaborate, difficult shoes. Military centaurs would have studded shoes for difficult terrain. Athletes would have specialize shoes for their sport. (Race horses wear lightweight shoes for racing. These wear out after just one use. Their only purpose is to reinforce the foot for the duration of the race.) Injured centaurs would wear remedial shoes to repair their gait and help them heal.
In both cases, size is a problem.
In terms of being a blacksmith/farrier, the issue is height. If a centaur is a "normal" horse's body with a "normal" human torso in place of neck/head, since the center of a horse's torso is about 4 feet off the ground, that puts normal shoulder height for the trunk about 6 feet up. It would be an enormous strain on the farrier to have to bend over far enough to get access to the hooves.
And the second part, can a centaur shoe himself, is worse. Unless the standard centaur is based on a pony body, the body is just too damn big. E.g. this Testing Two Methods of Estimating Horse Weight page gives a rough estimate for length for a 1300 lb horse as somewhere in excess of 5 feet. From hips to shoulders on a standard human is in the vicinity of 2 feet. Even assuming a massively flexible spine on the human part, allowing a 180 degree twist, a standing centaur can't reach its own rear hooves. And trying to do it lying down, with the torso bending down and around the stomach while the rear legs are pulled forwards, would seem to require an improbable hip/spine flexibility.
This is a little lateral but i think it applies
I don't think centaurs would use horseshoes. Horseshoes are put on horses because they are stupid. You need to add something that fits snug, and the horse will not remove.
I think centaurs would wear shoes more like humans. They should put them on in the morning and then take them of at night. With their hands they could handle lacing the shoe.
Reaching the back feet may be hard, but it is defiantly easier than nailing in a horseshoe even if you do need to do it more than once. Perhaps a tool could exist to help lace the shoes if aid from others is not available
I think it is very unlikely that a sentient creature would like to have something nailed on to it, even if it is not painfull
Is a centaur who insists on shoeing his own feet like a human who insists on cutting his own hair? Or more like one who insists on doing their own dental work?
At any rate, I've never shoed a horse, or indeed even done my own cobbling work, but if I was the last centaur on Earth--er, Narnia?-- I'd probably try to use molds to get an idea of the shape of my hooves. If I had access to plaster-of-paris, that'd be ideal, but maybe even mud would work. I'd step into the goo while it was soft, wait for it to set up, then use the resulting mold as a model for my farrier work. If I really wanted to get fancy, I would use that "negative" mold to cast a positive copy of my foot by pouring yet more goop in it and waiting for that to set.
I could use that to shape the shoe, but I don't know how I'd affix the shoe to my hooves. Maybe if I had access to a power drill I could mount it sticking up out of the floor, and very carefully drill out a hole, but I still don't know how I'd get the nail in, unless I just stomped it in. Even so, I've never heard of a centaur with a power drill.
I think in a society where centaurs exist it seems likely that shoes would look different. Probably something akin to ski boots that they could step into and have click into place and then be removed with a bit of cross legged stepping on switches at the back to release the clasps and step out again. More ornate shoes with decorative bits that go up the leg like ribbons and such would probably require a stool to lift a leg on to so the human arms could more reasonably reach these and would be limited to the forequarters as opposed to the hind.
They could have a machine that had all 8 drill bits sticking up vertically in the correct horseshoe nail locations, and were only half an inch or so long. It would have a simple controller attached to a wire. This way they could just turn it on, step on it with each foot, then just step on the horse shoes that probably have all 8 nails built in. The nails would be slightly thicker than the holes.
Other idea that I just thought of, you get a magnetic plate hammered into your hoove one time, and that way you can just put a shoe on. When it wears out, slide a new one on.
There are a lot more to horseshoes than just nail one iron shoes.
Hipposandals are tieon or buckled horse shoes, and are easily removable, a big bonus ofr an intelligent species.
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most shoes these days are glue one shoes which reduce the risk of splitting the hoof, since centaurs are intelligent I doubt shoes need to be nailed on. Perhaps for soldiers or slaves.
Then you have woven shoes, wooden wide bogshoes for soft ground, anti-fungal copper shoes, there is a large variety of horse shoes and I expect an intelligent hooves species to come up with many more just as humans have come up with thousands of types of shoes for different things.
Well, having seen the very useful answers of others, an idea has just occurred to me.
Surely the centaur farrier could use long-handled tools in conjunction with a mirror. The only problem about this, is making a hammer that can work around corners.
I think that the drilling and stomping idea that has been put forward is possible. In these days, all sorts of electronic and electrical devices could be designed, as well as using a personalised jig for each hoof.
Even hand-driven tools that were floor-mounted would have been possible in pre-industrial times with the necessary ingenuity.
