Walk the upper village and work out that the walls and the roofs are the same material — black slate, all of it

Find out about the Reyes de Patones, hereditary kings of a village of seven households, who lasted until about 1750

Work out when and how the kingdom ended: a request for a mayor, and independence from Uceda in 1769

Walk the hillside above, low rocky scrub of rockrose and broom with pine on the north face

Go down to Patones de Abajo, where everyone actually lives now, and see the difference

Patones de Arriba had seven households in 1555 and, by long tradition, its own hereditary kings — the Reyes de Patones — who lasted until about 1750, when the villagers asked to have a mayor instead. In 1769 Patones was granted the status of a village independent of Uceda, which is probably when the last king disappeared. It is the best example of arquitectura negra in the Madrid region: everything, walls and roofs both, built of black slate, on a rocky scrub hillside an hour from the capital.

Patones de Arriba: The Village With Its Own Kings

Find out who Cisneros was and how far one man from this town reached

Go into the church of Santa María Magdalena and work out why a town this size has one that size

Walk the surviving stretches of the town wall

Find the Canal de Isabel II works and work out what Madrid drank before it was built

Go up into the Sierra Norte behind, which is the emptiest corner of the Madrid region

Cardinal Cisneros — regent of Castile, founder of the university at Alcalá and the man behind the Complutensian Polyglot Bible — was born in Torrelaguna in 1436, and the town he came from is a compact walled place at the foot of the Sierra Norte with a church out of all proportion to it. It is also where the Canal de Isabel II starts: the 19th-century engineering that finally gave Madrid a reliable water supply, taking the Lozoya and carrying it seventy kilometres to the city, which is the single change that made the modern capital possible.

Torrelaguna: Cisneros and the Canal

Walk the wall — 11th-century Moorish work restored in the 15th, and one of very few surviving circuits in the Madrid region

Work out how the defence works: the wall on one side, the Lozoya wrapped round the other three

Go into the Picasso museum and work out how a small town on the Lozoya came to hold a collection like it

Find out who Eugenio Arias was — a barber from Buitrago, and a friend of Picasso

Go into the 15th-century Gothic-Mudéjar castle and the church of Santa María del Castillo of 1321 beside it

Buitrago is one of very few towns in the Madrid region to have kept its walls — Moorish, 11th century, restored in the 15th — on a peninsula with the Lozoya running round it. Inside is a Gothic-Mudéjar castle of the 15th century, the church of Santa María del Castillo of 1321 in the same style, and, unexpectedly, a Picasso museum. Eugenio Arias was a barber from this town and a friend of the artist, and the collection he built is now held here.

Buitrago del Lozoya: Walls and a Picasso Collection

Book and walk the free guided visit through the beech wood

Find the map in the reception hut that shows where the other beech woods lie, and why this one is not the most southerly

Look for where the wood meets the Jarama on the border with Guadalajara

The Hayedo de Montejo is 250 hectares of beech forest on the slopes of the Sierra de Ayllón, on the Jarama at the northern tip of the Madrid region. Beech belongs in central Europe, and it survives here from the end of the last Ice Age because the hillside stays in shade and catches the moisture from the air. It was declared a Natural Site of National Interest in 1974 and in 2017 joined UNESCO's Primeval Beech Forests of the Carpathians and Other Regions of Europe. Visits are free, guided and limited. It is often called the most southerly beech wood in Europe; it is not even the most southerly in Spain, and the guides explain why.

The Hayedo de Montejo: A Beech Wood Left Over From the Ice Age

Go into the church and find the grille made by the Carthusian monk Francisco de Salamanca

Find the walnut choir stalls carved in the 16th century, removed to Madrid in 1883 and returned here in 2003

Work out the founding story: a king who burned a charterhouse in France and built this one in 1390 to make up for it

Walk the Lozoya valley around it, which is inside the Sierra de Guadarrama National Park

Cross to the Puente del Perdón and find out what it was called that for

Henry II of Castile ordered a charterhouse built in the Lozoya valley in 1390 — the first Carthusian foundation in Castile — and tradition says he chose that order because his own army had burned one of their houses during the war in France. He picked the spot himself, beside a hermitage already there, and told his son, the future John I, exactly where. It was Carthusian until 1835. The grille separating the congregation from the monks was made by a Carthusian monk, Francisco de Salamanca, and the 16th-century walnut choir stalls were taken to Madrid in 1883 and only brought back in 2003.

El Paular: The First Carthusian House in Castile

Walk the upper gallery and work out that this is a palace wearing a castle's clothes

Find the Romanesque-Mudéjar church swallowed inside the structure when the castle was built over it

Work out Juan Guas's Isabelline Gothic and its Hispano-Muslim borrowings, and what he did next at Guadalajara

Find the remains of the old castle nearby, which this one replaced

Walk up into La Pedriza behind it, the granite massif the family's land ran into

The Mendoza were one of the most powerful families in late medieval Castile, and in the last third of the 15th century they replaced their old fortress at Manzanares with something grander — a palace-fortress built over a Romanesque-Mudéjar church, which was simply absorbed into the structure. Juan Guas drew it in Isabelline Gothic with Hispano-Muslim influence, and it is regarded as the precursor of his Palacio del Infantado at Guadalajara, where he settled the template for a Catholic Monarchs palace. It stands under the Guadarrama beside the Santillana reservoir, with La Pedriza behind it.

Manzanares el Real: A Castle Built to Be Lived In

Hike from the Puerto de Cotos to the summit of Peñalara at 2,428m, the highest peak of the Sierra de Guadarrama

Look down on the glacial lake and cirques beneath Peñalara, protected as the Peñalara Natural Park in 1990 and now part of the Sierra de Guadarrama National Park, declared on 26 June 2013

Visit the Peñalara visitor centre at the Puerto de Cotos, the start of the summit path

Peñalara is the highest point of the Sierra de Guadarrama at 2,428 metres, an hour from Madrid, and the corries beneath it hold the southernmost glacial lakes in Europe of their kind — cut by ice that has been gone for ten thousand years. It was protected as a natural park in 1990 and folded into the Sierra de Guadarrama National Park when that was declared on 26 June 2013. The path from Cotos is short and popular, which is why numbers to the summit are now limited at busy times.

Peñalara and the Glacial Cirques

Stand at a surface hot spot and work out that the rock under your boots is 100 to 120°C after nearly three hundred years

Watch the borehole demonstration — water poured into the ground comes back as a geyser because it is about 600°C at thirteen metres

Work out the scale of 1730: six years of eruption, nine villages buried, a quarter of the island under lava

Take the Ruta de los Volcanes coach through the Montañas del Fuego, which is the only way into the core of the park

Find out what happened in 1824 and why Lanzarote is not a closed book volcanically

The eruption that started at Timanfaya in 1730 went on for six years. Nine villages were buried — Tingafa, Mancha Blanca, Las Maretas, Santa Catalina, Jaretas, San Juan, Peña de Plomos, Testeina and Rodeos — the lava covered a quarter of Lanzarote and filled the surrounding farmland with ash. It erupted again in 1824. The ground has still not finished cooling: surface hot spots reach 100 to 120°C and it is around 600°C thirteen metres down, which the park demonstrates by pouring water into a borehole and setting off a geyser.