Ten days in Mexico: three cities, two flights, three thousand years

Book nothing today. Whatever the flight has done to you, the historic centre in the evening is the right cure: it is flat, it is floodlit and none of it needs a ticket.

Soldiers lower the enormous flag in the middle of the main square at six every evening and march it into the palace, which takes a good twenty minutes and is the best free spectacle in the country. The cathedral on the north side has been sinking unevenly into the old lakebed for four centuries, and the engineers who stabilised it in the 1990s left the tilt visible.

Finish above it. The tower two streets west opened in 1956 and has an observation deck near the top; go up as the lights come on.

Start at the palace on the east side of the square, early, with your passport. Entry is free and the point of it is the staircase, where Diego Rivera painted the whole history of Mexico as one continuous crowd, conquest at the bottom and revolution at the top.

Two minutes north is the excavation where the Aztec capital came back up. Electrical workers hit a carved disc here in 1978, and the dig that followed found not one temple but seven nested inside each other, each built over the last. You walk down through the layers on a raised path.

The afternoon belongs to the museum in the park, which is the single most important building on this trip. Three thousand years of Mesoamerican civilisation around one courtyard, arranged so that the Aztec sun stone waits at the end of the main hall. Two hours is the minimum; three is honest.

Fifty kilometres northeast, and worth the early start. Buses leave the northern terminal about every twenty minutes and take an hour; the site opens at nine and the coaches arrive from eleven, along with the hawkers selling jaguar whistles.

Nobody knows who built it or what they called themselves. By the time the Aztecs arrived it had been abandoned for centuries, and the names on the map today, the Pyramid of the Sun, the Avenue of the Dead, are the ones they guessed at. At its height it was among the largest cities on earth.

The walk up the avenue between the two great pyramids is the experience, and there is no shade on any of it. Whether either pyramid can be climbed changes from year to year, so ask at the gate rather than planning on it.

An hour in the air and you are in a different country. Oaxaca sits far lower, is considerably warmer, and is small enough that the taxi from the airport takes twenty five minutes and puts you two streets from everything.

Spend the afternoon on the square. It has a bandstand, enormous laurel trees, arcades on all four sides and a permanent slow churn of buskers, vendors and people sitting for hours. Take a table and let the jet lag go.

Then eat in the market two blocks south, where the corridor down the eastern side is a hall of charcoal grills. You buy meat by weight from a butcher's counter and hand it over the coals, and it comes back three minutes later with tortillas. Do not wear anything you mind smelling of woodsmoke.

Go early. The site sits on a ridge above the city with no shade, it opens at eight, and both the shuttles and the heat build through the morning. A taxi is twenty minutes and there is an hourly bus from town.

The Zapotecs levelled the top of a mountain by hand, from about 500 BC, and built a plaza on it the length of several football pitches. At its height around seventeen thousand people lived up here, which made it one of the largest cities in the Americas of its day. It was already long abandoned when the Spanish arrived.

Two to three hours, walking the perimeter rather than cutting across. The platforms at each end are the viewpoints, and the whole plan only resolves from the top of the southern one.

One block in the middle of Oaxaca holds the best three hours of it. The Dominican church comes first and it is a shock after the plain exterior: every surface gilded and modelled, with the family tree of the order sprawling across the ceiling inside the door.

The convent beside it is now the state museum, and the thing to reach is the room holding the contents of a tomb dug out of Monte Alban in the 1930s: hundreds of objects in gold, jade, turquoise and carved bone, left by Mixtec nobles who reused a Zapotec tomb centuries after it was built. It is the richest hoard of its kind ever found.

Behind the whole complex is the garden, on ground the army held until the 1990s. Entry is by guided tour at fixed times only, so it is worth checking the board the day before.

The longest travelling day of the trip, and still only a half of one. Oaxaca to Mérida usually routes through Mexico City, so it is a morning rather than an hour, and you land somewhere flat, hot and green in the afternoon.

Everything you need on the first evening is on one square. Laurel trees, the stone double seats made for courting couples, and buildings on all four sides worth walking into. The cathedral is the austere one, which is rare in Mexico and is partly Franciscan taste and partly what the revolution took away.

Across the square is the house the conqueror of Yucatán built himself in 1549, with two Spanish soldiers carved over the door, each standing on the head of a Maya. Nothing else in the country states the case that plainly.

An hour south on a good road, and half a day at most. Uxmal is quieter than the famous site east of the city and considerably more beautiful: long horizontal facades covered in geometric mosaic, assembled from thousands of cut stones without mortar.

The first thing you meet is a pyramid with rounded sides that exists nowhere else in the Maya world, thirty five metres of it. The story says a dwarf magician raised it overnight; archaeologists count five rebuildings, which took rather longer.

Behind it the site opens into courtyards and terraces, and the long palace on the southern one is the masterpiece: an upper facade of cut stone with the hooked nose of the rain god stacked at the corners, because there are no rivers in this part of Yucatán and everything depended on rain.

Leave at seven. The drive east takes a little under two hours, the gates open at eight, and the first hour is the only quiet one: from ten the coaches arrive from the coast and the avenues between the buildings fill with stalls.

The pyramid in the middle is a calendar. Ninety one steps on each of four staircases plus the top platform make three hundred and sixty five, and at the equinoxes the stepped corner throws a shadow down the northern balustrade to meet the carved serpent head at the bottom.

The rest of the site is larger than people expect: the biggest court for the ball game the Maya built, a round observatory whose windows line up with Venus, and a sinkhole at the end of a causeway that took offerings and bodies for centuries.

Then swim. There is a famous cenote three kilometres away, a circular shaft with vines hanging into it, and after four hours in that heat it is the best thing that will happen to you all trip.

A short last morning walking north. Start at the state government building on the square, whose upper floor carries twenty seven large canvases on the history of the peninsula, painted over more than a decade. It is free and takes half an hour.

Then out along the Paseo de Montejo, the avenue the rope barons laid out in imitation of a Paris boulevard when henequen made Yucatán rich. The grandest of the mansions is now the regional archaeology museum, and it is worth going in as much for the house as for the Maya rooms.

Lunch on a side street, then the airport, which is twenty minutes south. Ten days, three cities, four ancient ones, and the country will still owe you a second trip.