Inger Stojberg, the integration minister, confirmed that 68.8 percent of the 2,400 people who took the test in June had failed it. But she was unrepentant about the test’s difficulty, telling Politiken that being Danish is “very special” and that “citizenship is something you have to earn.”

“Too many did not prepare properly,” she said.

The new test comes as the Danish government has clamped down on immigration, including by introducing a law requiring recently arrived refugees to hand over valuables, such as gold or jewelry, to help pay for the costs of lodging them.

The test replaced a version introduced by the previous center-left government in 2014, and critics like Mr. Nielsen say it is undeniably harder. Applicants must now get 80 percent of the answers correct to pass, up from 73 percent previously.

Even without the citizenship exam, the requirements to qualify for Danish citizenship are arduous and include passing an oral and written exam in Danish, a difficult language with an abundance of vowel sounds that linguists say makes it hard to learn. Applicants must also be able to prove that they have been able to support themselves financially for four and a half of the past five years.

In Denmark, as in several European countries, a far-right populist party has been appealing to voters by warning against the perils of immigration. The Danish People’s Party has been a central proponent of the tougher citizenship test. Mr. Rasmussen’s governing center-right party does not have a majority in Parliament and often needs the support of lawmakers from the People’s Party to pass legislation.

With its generous welfare state, strong tradition of egalitarianism and Scandinavian quality of life, Denmark is an attractive country for many foreigners. The country was recently ranked as the world’s happiest for the third time since 2013.

As to the questions that started this article, Geranium in Copenhagen was the restaurant that got three stars; there are 98 municipalities; and the constellation was Cassiopeia.