I want to buy Greenland, said President Trump. No way, said the Danes and Greenlanders, who share control over the giant frozen island and its rich mineral treasures. Then I’m not going to visit your queen, shot back the self-proclaimed master of the real estate deal, who can’t stand being rebuffed. “Is this some sort of joke?” tweeted Helle Thorning-Schmidt, a former Danish prime minister, speaking for everyone.

That’s the gist of it, one of the more astounding plays by a president who finds new ways to amaze, alienate and infuriate almost daily.

To be fair to the president, acquiring Greenland would be nice for the United States. The island sits atop a trove of rare-earth metals, a category whose mining and export is increasingly dominated by China. It also has national security importance to the United States, which maintains its northernmost missile-warning, space surveillance and deepwater seaport at the Thule Air Base on Greenland’s northwestern coast. China has tried repeatedly to get a foothold on the island, but has been blocked so far by Denmark.

Mr. Trump is not the first American president to seek to buy Greenland; Harry Truman tried and failed in 1946. And earlier presidents did acquire quite a bit of other territories through purchases: Thomas Jefferson concluded the Louisiana Purchase with France in 1803; Andrew Johnson bought Alaska in 1867; and Woodrow Wilson picked up the Danish West Indies, now the United States Virgin Islands, from Denmark in 1917.