Question: Who was the reigning king when the Builth Castle was built and who was also known as "Malleus Scotorum"?

Context articles: Milntown Castle
Builth Castle
Edward I of England
Cambridge Castle
Saint Margaret of Scotland
Disputation of Paris
Achanduin Castle
Tysilio
Chibana Castle
Queenborough Castle

Milntown Castle was an early 16th-century castle which was situated near Milton, in Easter Ross, in the Scottish Highlands.
 It was built by the Munro of Milntown family, a cadet branch of the Clan Munro.
 In 1656, the castle and estate was sold to George Mackenzie, 1st Earl of Cromartie also known as George MacKenzie of Tarbet.
 He renamed the estate New Tarbat after Tarbat Castle (now more commonly known as Ballone Castle), the family's original seat near Portmahomack.
 Mackenzie had the Milntown Castle pulled down and only part of the basement survives.
 He then replaced the castle with a new mansion built nearby.
 When the new mansion was built, the old Milntown Castle was remodelled as part of the garden.
 That mansion was itself demolished and in 1787 was replaced with a Georgian house (now known as Tarbat House) by his descendant John Mackenzie, Lord MacLeod.
Builth Castle (Welsh: "Castell Llanfair-ym-Muallt" ) was a castle built under King Edward I, just outside Builth Wells, Powys, Wales.
 At one time it was an impressive stone-built castle but all the masonry has been removed over the years and all that remains are the mound on which it stood, the ditches and embankments.
Edward I (17/18 June 1239 – 7 July 1307), also known as Edward Longshanks and the Hammer of the Scots (Latin: "Malleus Scotorum" ), was King of England from 1272 to 1307.
 He spent much of his reign reforming royal administration and common law.
 Through an extensive legal inquiry, Edward investigated the tenure of various feudal liberties, while the law was reformed through a series of statutes regulating criminal and property law.
 Increasingly, however, Edward's attention was drawn towards military affairs.
Cambridge Castle, locally also known as Castle Mound, is located in Cambridge, Cambridgeshire, England.
 Originally built after the Norman conquest to control the strategically important route to the north of England, it played a role in the conflicts of the Anarchy, the First and Second Barons' Wars. Hugely expanded by Edward I, the castle then fell rapidly into disuse in the late medieval era, its stonework recycled for building purposes in the surrounding colleges.
 Cambridge Castle was refortified during the English Civil War but once again fell into disuse, used primarily as the county gaol.
 The castle gaol was finally demolished in 1842, with a new prison built in the castle bailey.
 This prison was demolished in 1932, replaced with the modern Shire Hall, and only the castle motte and limited earthworks still stand.
 The site is open to the public daily and offers views over the historic buildings of the city.
Saint Margaret of Scotland (c. 1045 – 16 November 1093), also known as Margaret of Wessex, was an English princess of the House of Wessex.
 Margaret was sometimes called "The Pearl of Scotland".
 Born in exile in the Kingdom of Hungary, she was the sister of Edgar Ætheling, the shortly reigned and uncrowned Anglo-Saxon King of England.
 Margaret and her family returned to the Kingdom of England in 1057, but fled to the Kingdom of Scotland following the Norman conquest of England in 1066.
 In 1070 Margaret married King Malcolm III of Scotland, becoming Queen of Scots.
 She was a very pious Roman Catholic, and among many charitable works she established a ferry across the Firth of Forth in Scotland for pilgrims travelling to St Andrews in Fife, which gave the towns of South Queensferry and North Queensferry their names.
 Margaret was the mother of three kings of Scotland, or four, if Edmund of Scotland, who ruled with his uncle, Donald III, is counted, and of a queen consort of England.
 According to the "Vita S. Margaritae (Scotorum) Reginae" ("Life of St. Margaret, Queen (of the Scots)"), attributed to Turgot of Durham, she died at Edinburgh Castle in Edinburgh, Scotland in 1093, merely days after receiving the news of her husband's death in battle.
 In 1250 Pope Innocent IV canonized her, and her remains were reinterred in a shrine in Dunfermline Abbey in Fife, Scotland.
 Her relics were dispersed after the Scottish Reformation and subsequently lost.
 Mary, Queen of Scots at one time owned her head, which was subsequently preserved by Jesuits in the Scottish College, Douai, France, from where it was subsequently lost during the French Revolution.
The Disputation of Paris, also known as the Trial of the Talmud, took place in 1240 at the court of the reigning king of France, Louis IX (St. Louis).
 It followed the work of Nicholas Donin, a Jewish convert to Christianity, who translated the Talmud and pressed 35 charges against it to Pope Gregory IX by quoting a series of alleged blasphemous passages about Jesus, Mary or Christianity.
 Four rabbis defended the Talmud against Donin's accusations.
Achanduin Castle, (also known as Achadun Castle and Acha-Dun), is a castle, now in ruins, located about 5 kilometres west of Achnacroish on the north-western coastline of the Isle of Lismore, in Argyll and Bute, Scotland.
 The castle overlooks Loch Linnhe and Bernera Island.
 The ruins are thought to date back to the thirteenth century.
 Achanduin Castle had long been thought to have been built by the Bishop of Argyll, though recent research has proved this to be unlikely.
 The castle was likely built by the MacDougalls around 1290 who held it throughout the fourteenth century.
 The castle was also thought to have been held by the Bishops of Argyll until the mid sixteenth century.
 It is a scheduled ancient monument.
Saint Tysilio (also known as/confused with Saint Suliac; Latin: "Tysilius, Suliacus" ; died 640 AD) was a Welsh bishop, prince and scholar, son of the reigning King of Powys, Brochwel Ysgithrog, maternal nephew of the great Abbot Dunod of Bangor Iscoed and an ecclesiastic who took a prominent part in the affairs of Wales during the distressful period at the opening of the 7th century.
Chibana Castle (知花城 , Chibana jō , Okinawan: "Tibana gushiku") , also known as Goeku Castle (越来城 , Goeku jō , Okinawan: "Gweeku gushiku") was a Ryukyuan "gusuku" on Okinawa Island.
 Located in the Chibana district of Okinawa City, the castle site sits on a hill and is surrounded by jungle.
 The castle was in use during the reign of the First Shō Dynasty and served as the residence of the Aji of Goeku Magiri.
 The castle, and Goeku Magiri, was given to the Ryukyuan general Uni-Ufugusuku by King Shō Taikyū in return for defeating the Aji of Katsuren Castle, Amawari.
 After the fall of the First Shō Dynasty, Uni-Ufugusuku was expelled from the castle and later committed suicide.
 His tomb, and that of his wife, was built at the bottom of the hill that Chibana Castle is atop.
 In the early 1960s, a structure was built on top of the hill to commemorate the castle and serve as a tourist attraction, from which all of the former Goeku Magiri can be seen.
 It has been neglected in recent years and has been allowed to become overgrown and hazardous (i.e. crumbling stairs).
Queenborough Castle, also known as Sheppey Castle, is a 14th century castle, the remnants of which are in the town of Queenborough on the Isle of Sheppey, Kent in England.
 The castle and the associated planned town were built on the orders of King Edward III from 1361 and named in honour his wife, Queen Philippa.
 It was the first concentric castle to be built in England, and the only royal castle to be new-built in England during the Late Middle Ages.
 Overlooking the Swale, then an important waterway approaching the River Medway, Queenborough Castle formed part of the country's coastal defences until 1650 when it was declared to be unfit for use and was almost completely demolished shortly afterwards.
 The site is now a public park and the only visible remains are some low earthworks.