hamstrung by the pandemic and years of ​international sanctions.

Outside the exhibition hall, North Korean soldiers displayed their martial-art skills while an air force squadron flew overhead, leaving behind streaks of red, blue and yellow smoke​, photos released through state news media showed​. Paratroopers descended from the sky with a Worker’s Party flag.

“We are a nuclear power with self-reliance,” a large banner said. Another banner read, “We are a great missile power.”

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North Korea Missile Tests Are Part of a Familiar Strategy

SEOUL — The signals are confusing. One day, North Korea is ​​raising hopes for dialogue with South Korea, and the next, it is firing missiles or showing off the latest weaponry in its nuclear arsenal.

In the past week alone, North suggested the possibility of inter-Korean summit talks and said it would reopen communication hotlines with its neighbor. It also fired long-range cruise missiles, trotted out what it called ​its first hypersonic missile and, on Thursday, tested a new antiaircraft missile. Earlier in September, it launched ballistic missile​s​ from ​a ​train​ rolled out of a mountain tunnel, on the same day that it called the South’s president, Moon Jae-in, “stupid.”

Once again, North Korea is turning to a well-honed, two-pronged strategy, designed to let it flex its military muscles without risking retaliation or nixing the chances for dialogue.

In the absence of talks with Washington, the missile tests reminded the world that North Korea is developing increasingly sophisticated weaponry capable of delivering nuclear warheads. But individually, these short-range or still-under-development missiles don’t amount to a direct threat to the United States.

met with then-President Donald J. Trump three times between 2018 and 2019, becoming the first North Korean leader to hold a summit with a​ sitting American ​president. But ​his diplomatic efforts failed to lift crippling sanctions the United Nations imposed on his impoverished country after its nuclear and I.C.B.M. tests​. Soon the pandemic hit, further hamstringing the North’s economy.

​American and South Korean officials had hoped that the North’s deepening economic troubles, caused by the double whammy of sanctions and the pandemic, would make North Korea more amenable to dialogue.

So far, Mr. Kim has proved them wrong.

Since his talks with Mr. Trump collapsed in early 2019, he has vowed to slog through the economic difficulties while expanding his nuclear arsenal​, his country’s single best diplomatic leverage and deterrent against what it considers American threats to topple its government. By demonstrating his country’s growing military capabilities, Mr. Kim has also sought to legitimize his rule at a time when he has been able to deliver little on the economic front to his long-suffering people.​

The antiaircraft missile test on Thursday indicated that ​the North is building a weapon similar to Russia’s S-400, one of the most potent air-defense systems in the world, according to Kim Dong-yub, an expert on North Korean weapons at the University of North Korean Studies.

The Biden administration has repeatedly urged North Korea to​ return to talks without preconditions. But Mr. Kim said he would not restart negotiations until he was convinced that ​Washington was ready to ease sanctions and its “hostile policy,” including the joint annual military exercises it conducts with South Korea.

an arms race in the region.

Mr. Kim can’t really attempt shocking provocations like the ones he conducted in 2017 — three I.C.B.M. tests and a nuclear test — that brought the Trump administration to the table. Such tests would sharply raise tensions, invite more U.N. sanctions and potentially invoke the ire of China by ruining the mood for the Beijing Winter Olympics in February.​

desperate to put his Korean Peninsula peace process, his signature foreign policy, back on track before his single, five-year term ​ends ​in May.

“It’s our government’s destiny” to pursue dialogue with the North, Mr. Moon told reporters last week, referring to his efforts to build peace through his three meetings with Mr. Kim in 2018 and his efforts to help arrange the summit meetings between Mr. Kim and Mr. Trump.

This week, Mr. Kim also offered conciliatory words toward South Korea.

“We have neither aim nor reason to provoke South Korea and no idea to harm it,” he said.

North Korea was wooing South Korea while shunning talks with Washington, said Cheong Seong-chang, director of the Center for North Korean Studies at the Sejong Institute in South Korea. Other analysts said North Korea was leaning on South Korea to help bring Washington to dialogue.

On Thursday, Sung Kim, the U.S. special representative for North Korea, met with his counterparts from Japan and South Korea ​and indicated that ​Washington would support humanitarian aid to North Korea as an incentive for dialogue.

Analysis doubted that it would be enough.

“I am not sure that the old way of providing humanitarian shipments​ as an incentive​ will work this time, given the North’s reluctance to accept outside help ​during the pandemic,” said Professor Yang of the University of North Korean Studies. “North Korea wants the United States to address more fundamental issues ​concerning its well-being​. It wants clearer commitment​s ​from the United States to easing sanctions and guaranteeing its security.”

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Japan Faces Big Problems. Its Next Leader Offers Few Bold Solutions.

TOKYO — With the world’s oldest population, rapidly declining births, gargantuan public debt and increasingly damaging natural disasters fueled by climate change, Japan faces deep-rooted challenges that the longstanding governing party has failed to tackle.

Yet in choosing a new prime minister on Wednesday, the Liberal Democratic Party elected the candidate least likely to offer bold solutions.

The party’s elite power brokers chose Fumio Kishida, 64, a stalwart moderate, in a runoff election for the leadership, seeming to disregard the public’s preference for a maverick challenger. In doing so, they anointed a politician with little to distinguish him from the unpopular departing leader, Yoshihide Suga, or his predecessor, Shinzo Abe, Japan’s longest-serving prime minister.

Elders in the party, which has had a near monopoly on power in the decades since World War II, made their choice confident that, with a weak political opposition and low voter turnout, they would face little chance of losing a general election later this year. So, largely insulated from voter pressure, they opted for a predictable former foreign minister who has learned to control any impulse to stray from the mainstream party platform.

slowly emerges from six months of pandemic restrictions that have battered the economy.

Taro Kono, an outspoken nonconformist whose common touch has made him popular with the public and with rank-and-file party members. Mr. Kishida prevailed in the second round of voting, in which ballots cast by members of Parliament held greater weight than ballots cast by other party members.

He will become prime minister when Parliament holds a special session next week, and will then lead the party into the general election, which must be held by November.

In his victory speech on Wednesday, Mr. Kishida acknowledged the challenges he faces. “We have mountains of important issues that lie ahead in Japan’s future,” he said.

They loom both at home and abroad. Mr. Kishida faces mounting tensions in the region as China has grown increasingly aggressive and North Korea has started testing ballistic missiles again. Taiwan is seeking membership in a multilateral trade pact that Japan helped negotiate, and Mr. Kishida may have to help finesse a decision on how to accept the self-governed island into the group without angering China.

As a former foreign minister, Mr. Kishida may have an easier time managing his international portfolio. Most analysts expect that he will maintain a strong relationship with the United States and continue to build on alliances with Australia and India to create a bulwark against China.

But on the domestic front, he is mostly offering a continuation of Mr. Abe’s economic policies, which have failed to cure the country’s stagnation. Income inequality is rising as fewer workers benefit from Japan’s vaunted system of lifetime employment — a reality reflected in Mr. Kishida’s campaign promise of a “new capitalism” that encourages companies to share more profits with middle-class workers.

close to 60 percent of the public is now inoculated. But Mr. Kishida has offered few concrete policies to address other issues like aging, population decline or climate change.

In a magazine questionnaire, he said that he needed “scientific verification” that human activities were causing global warming, saying, “I think that’s the case to some extent.”

Given the enduring power of the right flank of the Liberal Democratic Party, despite its minority standing in the party, Mr. Kishida closed what daylight he had with these power brokers during the campaign.

He had previously gained a reputation as being more dovish than the influential right wing led by Mr. Abe, but during the leadership race, he expressed a hawkish stance toward China. As a parliamentary representative from Hiroshima, Mr. Kishida has opposed nuclear weapons, but he has made clear his support for restarting Japan’s nuclear power plants, which have been idled since the triple meltdown in Fukushima 10 years ago.

And he toned down his support for overhauling a law requiring married couples to share a surname for legal purposes and declared that he would not endorse same-sex marriage, going against public sentiment but hewing to the views of the party’s conservative elite.

“I think Kishida knows how he won, and it was not by appealing to the general public, it was not by running as a liberal, but courting support to his right,” said Tobias Harris, a senior fellow at the Center for American Progress in Washington. “So what that’s going to mean for the composition of his cabinet and his priorities, and what his party’s platform ends up looking like, means he could end up being pulled in a few different directions.”

resigned last fall because of ill health. He had led the party for eight consecutive years, a remarkable stint given Japan’s history of revolving-door prime ministers. When he stepped down, the party chose Mr. Suga, who had served as Mr. Abe’s chief cabinet secretary, to extend his boss’s legacy.

Sanae Takaichi — a hard-line conservative who was seeking to become Japan’s first female prime minister — to revitalize his base in the party’s far right, analysts and other lawmakers said he helped steer support to Mr. Kishida in the runoff.

As a result, Mr. Kishida may end up beholden to his predecessor.

“Kishida cannot go against what Abe wants,” said Shigeru Ishiba, a former defense minister who challenged Mr. Abe for the party leadership twice and withdrew from running in the leadership election this month to support Mr. Kono.

“I am not sure I would use the word ‘puppet,’ but maybe he is a puppet?” Mr. Ishiba added. “What is clear is he depends on Abe’s influence.”

During the campaign for the party leadership, Mr. Kishida appeared to acknowledge some dissatisfaction with the Abe era with his talk of a “new capitalism.” In doing so, he followed a familiar template within the Liberal Democratic Party, which has been adept at adopting policies first introduced by the opposition in order to keep voters assuaged.

“That’s one of the reasons why they have maintained such longevity as a party,” said Saori N. Katada, a professor of international relations at the University of Southern California. “Kishida is definitely taking that card and running with it.”

Makiko Inoue, Hikari Hida and Hisako Ueno contributed reporting.

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German Elections Live Updates: Social Democrats Have Narrowly Beaten Merkel’s Party

preliminary official results reported early Monday.

The federal German election agency posted the results at 4:30 a.m. local time.

The close outcome means the Social Democrats, with only 25.7 percent of the vote, must team up with other parties to form a government. And in the complex equation that can be required in Germany to form a government, it is possible that if the winning party fails to get others on board, the party that placed second could wind up leading the country.

It could take weeks if not months of haggling to form a coalition, leaving Europe’s biggest democracy suspended in a kind of limbo at a critical moment when the continent is still struggling to recover from the pandemic and France — Germany’s partner at the core of Europe — faces divisive elections of its own next spring.

Sunday’s election signaled the end of an era for Germany and for Europe. For over a decade, Ms. Merkel was not just chancellor of Germany but effectively the leader of Europe. She steered her country and the continent through successive crises and in the process helped Germany become Europe’s leading power for the first time since World War II.

Cheers erupted at the Social Democratic Party’s headquarters when the exit polls were announced early Sunday evening. A short while later, supporters clapped and chanted “Olaf! Olaf!” as Olaf Scholz, their candidate, took the stage to address the crowd.

“People checked the box for the S.P.D. because they want there to be a change of government in this country and because they want the next chancellor to be called Olaf Scholz,” he said.

The campaign proved to be the most volatile in decades. Armin Laschet, the candidate of Ms. Merkel’s Christian Democrats, was long seen as the front-runner until a series of blunders compounded by his own unpopularity eroded his party’s lead. Olaf Scholz, the Social Democratic candidate, was counted out altogether before his steady persona led his party to a spectacular 10-point comeback. And the Greens, who briefly led the polls early on, fell short of expectations but recorded their best result ever.

The Christian Democrats’ share of the vote collapsed with only 24.1 percent of the vote, heading toward the worst showing in their history. For the first time, three parties will be needed to form a coalition — and both main parties are planning to hold competing talks to do so.

Nevertheless, Mr. Laschet appeared at his party headquarters an hour after the polls closed, declaring the outcome “unclear” and vowing to try to form a government even if his party came in second.

Credit…Pool photo by Clemens Bilan

The progressive, environmentalist Greens appeared to make significant gains since the 2017 election but seemed to fall short of having a viable shot at the chancellery. That positions the Greens, as well as the business-friendly Free Democrats, to join the next government. They will play a key role in deciding what the next German government could look like, depending on which of the larger parties they would like to govern with.

On the outer edge of the political spectrum, support for the far-right Alternative for Germany, or AfD, appeared roughly unchanged, while the Left party appeared to be hovering on the 5 percent threshold needed to win seats in Parliament.

In mid-October the election agency will present the official final results.

Credit…Michele Tantussi/Reuters

BERLIN — What do a traffic light, the Jamaican flag and a kiwi have in common?

Those watching German politics closely will know all three are nicknames for potential governing coalitions.

In the weeks following the election, the parties will try to form a coalition government that has a majority in the German Parliament. The winning party in the election will have the first chance to try to form that coalition, but if it doesn’t succeed the chance goes to the runner up.

For the first time since the founding of the federal republic 72 years ago, it looks as though it will take at least three parties to form a stable government.

Here’s how things might play out:

Traffic Light Coalition 🚦: This could be the most likely combination. Its name derives from the parties that would be included, the Social Democrats (red), the free market liberal Free Democrats (yellow) and the Greens (uh, green).

Jamaica Coalition 🇯🇲: If Chancellor Angela Merkel’s conservative Christian Democratic Union (black) should take the lead, Germany might be looking at a Jamaica coalition — named after the black, green and yellow of the Jamaican flag. That bloc would consist of the conservatives, the Greens and the Free Democrats.

And the kiwi 🥝? That would be a duo of the conservatives and the Greens, who have worked together in several state governments, but on current polling are unlikely to command a national majority.

Given the relatively low polling of the once-mighty Christian Democrats and Social Democrats, the topic of possible coalitions has dominated news coverage for weeks in Germany. For the past five years, the two big parties have governed Germany together in a “Grand Coalition,” but they don’t want to repeat that and it might not have a majority in any case.

The Social Democrats and the Greens have governed Germany together before — a prosaically named “Red-Green coalition” was in power from 1997 until 2005 — and have signaled their willingness to work together again. But this time they are not expected to win the seats necessary to get a majority on their own.

Seeing their popularity slip, Merkel’s conservatives and much of the conservative media have warned that an ascendant Social Democrats would turn to the far-left party, Die Linke, to round out their numbers.

Credit…Pool photo by Tobias Schwarz

They call it the “Elephant Round”: After the polls close and as the votes are being counted on Sunday, all of the heavy-hitting party leaders sit down together, live on public television, to discuss the outcome that is shaping up.

Those who are winning will exclaim, those who are losing will explain and smaller parties will jockey for position in a new government, cozying up to potential partners or coolly shunning others.

For Germans watching at home, the event, which is scheduled to start at 8:15, is a chance to read the tea leaves about their future government.

For the politicians sitting in the brightly lit studio, the round offers them a chance to try to set the tone for the weeks of negotiations that are expected to follow, given that none of the parties running are expected to win enough votes to allow them to govern alone. Leaders of the smaller parties use the opportunity to make their first demands and draw their lines in the sand.

It is a chance for grandstanding and, occasionally, for grinning. That happened famously in 2005, when Chancellor Gerhard Schröder’s Social Democrats lost by a small margin to Angela Merkel’s Christian Democratic Union. He nevertheless tried to claim victory, on grounds that his party had done much better than predicted in the polls. “We’ve won,” Ms. Merkel replied with a controlled smile. “And after a couple of days of reflection, the Social Democrats will realize that, too.”

This year, fate may be in the favor of the Social Democrats. Ms. Merkel is stepping aside after 16 years in power and Olaf Scholz, her vice chancellor and finance minister, led the polls in the final weeks of the race. His campaign portrayed him as coolheaded and in control. Come Sunday night, Germans will be watching to see whether he can keep that up when faced with the “elephants.”

In Germany, political parties name their candidates for chancellor before campaigning begins, and most of the focus falls on the selections who have a realistic chance of winning.

Traditionally, those have been the candidates of the center-right Christian Democrats (Chancellor Angela Merkel’s party) and those of the center-left Social Democrats. For the first time this year, the candidate for the environmentalist Greens is viewed as having a real shot at the chancellery.

Here are the leading hopefuls:

Credit…Laetitia Vancon for The New York Times

Age: 40

Current position: Co-leader of the Green Party

About her: Ms. Baerbock aims to shake up the status quo. She is challenging Germans to deal with the crises that Ms. Merkel has left largely unattended: decarbonizing the powerful automobile sector; weaning the country off coal; and rethinking trade relationships with strategic competitors like China and Russia.

“This election is not just about what happens in the next four years, it’s about our future,” Ms. Baerbock told a crowd in Bochum, a western German town, this summer.

Ms. Baerbock, who has not a position in government, has started off on a promising note, but her campaign has struggled as she has been a frequent target of disinformation efforts. She has also been accused by rivals of plagiarism and of padding her résumé, and her Green Party has been faulted for not being able to capitalize on environmental issues in the wake of flooding this summer.

Even so, there is almost no combination of parties imaginable in the next coalition government that does not include the Greens. That makes Ms. Baerbock, her ideas and her party of central importance to Germany’s future.

“We need change to preserve what we love and cherish,” she told the crowd in Bochum. “Change requires courage, and change is on the ballot on Sept. 26.”

Credit…John Macdougall/Agence France-Presse — Getty Images

Age: 60

Current position: Leader of the Christian Democratic Union; governor of the state of North Rhine-Westphalia

About him: Mr. Laschet has run North Rhine-Westphalia, Germany’s most populous state, since 2017 — a credential he has long said qualifies him to run the country. As the leader of the Christian Democratic Union, Ms. Merkel’s party, he should have been the natural heir to the chancellor. But his gaffe-prone campaign has struggled to find traction among Germans. Extraordinary flooding this summer in the region he runs exposed flaws in his environmental policies and disaster management. He was caught on camera laughing during a solemn ceremony for flood victims.

But Mr. Laschet is known for comebacks, and for surviving blunders.

Among his influences is his faith. At a time when more and more Germans are quitting the Roman Catholic Church, Mr. Laschet is a proud member. Another influence is Aachen, Germany’s westernmost city, where he was born and raised. Growing up in a place with deep ties to Belgium and the Netherlands, Mr. Laschet has been integrated into the larger European ideal all of his life.

Credit…Gordon Welters for The New York Times

Age: 63

Current position: Vice chancellor of Germany and federal finance minister

About him: When Olaf Scholz asked his fellow Social Democrats to nominate him as their candidate for chancellor, some inside his own camp publicly wondered if the party should bother fielding a candidate at all. What a difference a few months make. Today, Mr. Scholz and his once moribund party have unexpectedly become the favorites to lead the next government.

During the campaign, Mr. Scholz has managed to turn what has long been the main liability for his party — co-governing as junior partners of Ms. Merkel’s conservatives — into his main asset: In an election with no incumbent, he has styled himself as the incumbent — or as the closest thing there is to Ms. Merkel.

“Germans aren’t a very change-friendly people, and the departure of Angela Merkel is basically enough change for them,” said Christiane Hoffmann, a prominent political observer and journalist. “They’re most likely to trust the candidate who promises that the transition is as easy as possible.”

He has been photographed making the chancellor’s hallmark diamond-shaped hand gesture — the “Merkel rhombus” — and used the female form of the German word for chancellor on a campaign poster to convince Germans that he could continue Ms. Merkel’s work even though he is a man.

The symbolism isn’t subtle, but it is working — so well in fact that the chancellor herself has felt compelled to push back on it — most recently in what might be her last speech in the Bundestag.

Credit…Laetitia Vancon for The New York Times

It has been said that Germans are sometimes so organized that chaos reigns. Germany’s election system is no exception. It is so complex that even many Germans don’t understand it.

Here’s a brief primer.

Not exactly. Unlike in the United States, voters don’t directly elect their head of government. Rather, they vote for representatives in Parliament, who will choose the next chancellor, but only after forming a government. More on that later.

The major parties declare who they would choose for chancellor, so Germans going to the polls today know who they are in effect voting for. This year the candidates most likely to become chancellor are Olaf Scholz of the Social Democrats or Armin Laschet of the Christian Democrats. Annalena Baerbock, a Green, has an outside chance.

Any German citizen 18 or over. They don’t need to register beforehand.

Everyone going to the polls today has two votes. The first vote is for a candidate to be the district’s local representative. The second vote is for a party. Voters can split their votes among parties and often do. For example, a person could cast one vote for a Social Democrat as the local member of Parliament, and a second vote for the Christian Democrats as a party.

Parliament has 598 members, but could wind up with many more because of a quirk in the system. The top vote-getter in every district automatically gets a seat in Parliament. These candidates account for half of the members of Parliament. The remaining seats are allocated according to how many second votes each party receives.

But parties may be allocated additional seats according to a formula designed to ensure that every faction in Parliament has a delegation that accurately reflects its national support. So Parliament could easily wind up with 700 members.

Also: A party that polls less than 5 percent doesn’t get any seats at all.

It is very unlikely that any party will wind up with a majority in Parliament. The party that gets the most votes must then try to form a government by agreeing to a coalition with other parties. That has become mathematically more difficult because of the rise of the far-right Alternative for Germany party and the far-left Linke party.

The mainstream parties have ruled out coalitions with either of those parties because of their extreme positions. But it will be a struggle for the remaining parties to find enough common ground to cobble together a majority. The process could take months.

Credit…Lena Mucha for The New York Times

Voter turnout in Germany — as a measure of the people visiting polling stations — was down on Sunday when compared to the last election in 2017, officials said. But the number is misleading. Participation could be extraordinarily high once mail-in ballots are counted.

By 2 p.m., 37 percent of eligible voters had cast ballots in person, election officials said, down from 41 percent during the same period in 2017. But at least 40 percent of Germans were expected to vote by mail because of the coronavirus, potentially pushing turnout above the 76 percent recorded in 2017.

Despite the decrease in in-person voting nationwide, there were long lines at polling stations in Berlin, where voters were also choosing candidates for the local government. Some polling places reportedly ran out of ballots and had trouble getting more because many streets were closed because of the Berlin Marathon, which was expected to attract almost 30,000 participants.

With Chancellor Angela Merkel poised to step down after 16 years in office, the stakes are high. Polls showed a close race between the Social Democrats and the Christian Democratic Union, Ms. Merkel’s party, which could encourage turnout. Voting sites remain open until 6 p.m. local time.

The high number of mail-in ballots is not expected to delay the results in the same way that occurred in the United States presidential elections last year, when close races in some states were not decided for days. German officials will only count mail-in ballots that had arrived by Sunday, and should have a good idea by midnight at the latest of which party prevailed.

Credit…Pool photo by Martin Divisek

The Alternative for Germany, or AfD, which shocked the nation four years ago by becoming the first far-right party to win seats in Parliament since World War II, suffered a slippage in support Sunday but also solidified its status as a permanent force to be reckoned with.

“We are here to stay, and we showed that today,” Tino Chrupalla, co-leader of the party, told party members gathered on the outskirts of Berlin.

Early results showed the party with 11 percent of the votes, down from almost 13 percent in 2017. The AfD is likely to no longer be the largest opposition party in Parliament.

If those results hold in final tallies, that will still give the AfD a sizable delegation in Parliament, and the vote showed that the party has a core constituency even when immigration, its main issue, was not a major topic in the campaign.

At the AfD’s post-election gathering Sunday, activists took comfort in the poor showing by the Christian Democrats, the party of Chancellor Angela Merkel, who compete with the AfD for conservative voters. “The C.D.U. got what they deserved,” said Alexander Gauland, the leader of the AfD delegation in Parliament.

Alternative for Germany held its election party at an event space 45 minutes by subway from central Berlin, perhaps in an effort to discourage counter-demonstrators. Several dozen protesters gathered across the street from the AfD event, holding signs accusing the party of being fascist. But they were probably outnumbered by the police.

As AfD activists ate potato salad and wurst from a buffet, the prevailing view seemed to be that the party’s candidates would have done better if the media and the other parties hadn’t ganged up on them.

“We had to campaign against everyone,” said Daniela Öeynhausen, who appears to have won a seat in the state Parliament of Brandenburg. “It was still an impressive two-digit result considering the unfair attacks.”

Julian Potthast, who said he believed he had won election to a district council in a neighborhood of Berlin, portrayed the party — whose rhetoric has been linked to attacks on immigrants or people perceived as non-Germans — as itself the victim of violence. He said that his vehicle was vandalized and that graffiti was sprayed on his home.

The party was unfairly portrayed as fascist, he complained. But he also conceded the party might have made mistakes, for example in its stance against restrictions to limit the spread of the coronavirus. “It’s not as good as we hoped,” Mr. Potthast said. “We have to look very carefully at why we lost votes.”

Credit…Thomas Kienzle/Agence France-Presse — Getty Images

Chancellor Angela Merkel will not disappear Sunday night after the votes are counted.

Until a new government is formed, a process that can take several weeks to several months, she will remain in office as head of the acting, or caretaker, government.

Ms. Merkel announced in the fall of 2018 that she would not run again and she gave up leadership of her party, the Christian Democratic Union. After that, her position as chancellor was weakened as members of the C.D.U. jockeyed to replace her. She had hoped to stay out of the election campaign, but as the conservative candidate, Armin Laschet, started to flounder, she made several appearances aimed at bolstering support for him.

Ms. Merkel is expected to try to take a similarly hands-off approach to steering the caretaker government — if world events allow. The last two years of her fourth and final term in office has seen the deadly coronavirus pandemic, what she herself has called “apocalyptic” flooding in western Germany and the chaotic withdrawal from Afghanistan.

Once the new chancellor is sworn in, Ms. Merkel will vacate her office in the imposing concrete building that dominates Berlin’s government district for good.

But, after the last election, in 2017, it took 171 days — or nearly six months — to form a new government, which means she is likely to be around for a while.

What she will do next remains to be seen. In response to that question in repeated interviews, she has said that first and foremost she will take some time off to reflect and reorient herself before making her next move.

“I will take a break and I will think about what really interests me, because in the past 16 years, I haven’t had the time to do that,” she said in July, after receiving an honorary doctorate from Johns Hopkins University.

“Then I will maybe read a bit, and then my eyes might close because I am tired and I will sleep a bit,” she said, with a smile: “And then we’ll see where I emerge.”

Credit…Sebastian Kahnert/picture alliance, via Getty Images

BERLIN — German election officials are expecting mail-in ballots to break records in Sunday’s federal election. At least 40 percent and possibly a majority of ballots will arrive by mail, according to Georg Thiel, head of the agency in charge of counting the votes.

Although actual tallies will only be known after polls close, the authorities have seen requests for mail-in ballots grow this year as the pandemic fuels anxiety about crowded polling stations.

Mail-in balloting has been permitted in Germany for more than 60 years. When it was first allowed, in the 1957 election, only 5 percent of voters used the option; during the last federal election in 2017, 29 percent chose to mail in their choice. Vote counters are set up to handle a doubling of that number — nearly 60 percent — this year, Mr. Thiel said.

The postal service in Germany is one of the quickest and most reliable in the world, with letters usually delivered within a day to anywhere in the country. Still, an official warned voters last week that if they wanted their ballot to be counted, it should be in the mail by Thursday; only ballots received by 6 p.m. on Sunday — when polls close — will be tallied.

The populist Alternative for Germany party, segments of which have parroted former President Donald J. Trump’s claims of manipulated mail-in ballots in the U.S., has used slogans like “the mailbox is not a ballot box” to try to dissuade voters from using the option. But those concerns do not appear to have resonated with the electorate.

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The new Parliament will ultimately determine who succeeds Angela Merkel, who has served as chancellor for 16 years. Early exit polling has suggested a tight race between the Christian Democratic Union, Ms. Merkel’s center-right party, and the center-left Social Democrats.CreditCredit…Lena Mucha for The New York Times

Sixty million people are eligible to vote in the German national election on Sunday. There won’t be a new government that night, or the next day — it could take the rival parties weeks or even months to settle on a coalition with a parliamentary majority. But the ballots are tallied quickly, and the new shape of Germany’s political landscape is likely to be visible within hours.

Here’s what Election Day will look like, and what to watch for.

8 a.m. local time: Polls opened. Candidates are not allowed to campaign on this day, but some may be seen casting ballots.

6 p.m. (noon Eastern): Polling stations close. Not long after, the first exit polls should be available. These polls can be within percentage points of the final result. But this year, because the race is tight, it could be a few more hours before a clear picture emerges. Mail-in ballots, which have been part of Germany’s voting system since 1957, are expected to play an outsized role given the pandemic, as they did in the U.S. presidential election. Only mail-in ballots received by 6 p.m. Sunday will be counted.

Around 6:15 p.m.: The first projections based on actual counted ballots will be released. These get updated throughout the evening until a fairly clear picture emerges of which party is winning.

8:15 p.m.: The heads of all the major parties meet to discuss successes and failures of their campaigns, and they will signal who they would be willing to work with in a coalition government. This discussion is called the “Elephant Round,” and it lasts an hour.

8 p.m. to midnight: Nearly all votes should be counted.

Early, early morning: The election authorities release something they call the official temporary results. These usually come between 2 a.m. and 3 a.m. — though during the last national election, they didn’t arrive until 5:30 a.m.

Credit…Bernd von Jutrczenka/Picture Alliance, via Getty Images

During her 16 years as Germany’s chancellor, Angela Merkel has become an international avatar of calm, reason and democratic values for the way she handled crises that included a near financial meltdown of the eurozone, the arrival of more than a million migrants and a pandemic.

Today Germany is an economic colossus, the engine of Europe, enjoying prosperity and near full employment despite the pandemic. But can it last?

That is the question looming as Ms. Merkel prepares to leave the political stage after national elections on Sunday. There are signs that Germany is economically vulnerable, losing competitiveness and unprepared for a future shaped by technology and the rivalry between the United States and China.

During her tenure, economists say, Germany neglected to build world-class digital infrastructure, bungled a hasty exit from nuclear power, and became alarmingly dependent on China as a market for its autos and other exports.

The China question is especially complex. Germany’s strong growth during Ms. Merkel’s tenure was largely a result of trade with China, which she helped promote. But, increasingly, China is becoming a competitor in areas like industrial machinery and electric vehicles.

Economists say that Germany has not invested enough in education and in emerging technologies like artificial intelligence and electric vehicles. Germans pay some of the highest energy prices in the world because Ms. Merkel pushed to close nuclear power plants, without expanding the country’s network of renewable energy sources enough to cover the deficit.

“That is going to come back to haunt Germany in the next 10 years,” said Guntram Wolff, director of Bruegel, a research institute in Brussels.

Credit…Lena Mucha for The New York Times

WÜLFRATH, Germany — Hibaja Maai gave birth three days after arriving in Germany.

She had fled the bombs that destroyed her home in Syria and crossed the black waters of the Mediterranean on a rickety boat with her three young children. In Greece, a doctor urged her to stay put, but she pressed on, through Macedonia, Serbia, Hungary and Austria. Only after she had crossed the border into Bavaria did she relax and almost immediately go into labor.

“It’s a girl,” the doctor said when he handed her the newborn bundle.

There was no question in Ms. Maai’s mind what her daughter’s name would be.

“We are calling her Angela,” she told her husband, who had fled six months earlier and was reunited with his family two days before little Angela’s birth on Feb. 1, 2016.

“Angela Merkel saved our lives,” Ms. Maai said in a recent interview in her new hometown, Wülfrath, in northwestern Germany. “She gave us a roof over our heads, and she gave a future to our children. We love her like a mother.”

Chancellor Angela Merkel is stepping down after her replacement is chosen following Germany’s Sept. 26 election. Her decision to welcome more than a million refugees from Syria, Iraq, Afghanistan and elsewhere in 2015 and 2016 stands as perhaps the most consequential moment of her 16 years in power.

It changed Europe, changed Germany, and above all changed the lives of those seeking refuge, a debt acknowledged by families who named their newborn children after her in gratitude.

The chancellor has no children of her own. But in different corners of Germany, there are now 5- and 6-year-old girls (and some boys) who carry variations of her name — Angela, Angie, Merkel and even Angela Merkel. How many is impossible to say. The New York Times has identified nine, but social workers suggest there could be far more, each of them now calling Germany home.

Credit…Clemens Bilan/EPA, via Shutterstock

Never before has the issue of climate change played such a role in a German election.

Though it still remained unclear who will lead Germany, nearly every party pledged to put climate change near the top of the agenda for the next government.

Despite entering office in 2005 with ambitions to reduce carbon emissions, four successive governments under Chancellor Angela Merkel failed to significantly reduce Germany’s carbon footprint. It remains in the top 10 of the world’s most polluting countries, according to the World Bank.

It has been young climate activists who have succeeded in bringing the climate debate to the forefront of Germany’s political discussion. This year, they successfully took the government to court, forcing a 2019 law aimed at bringing the country’s carbon emissions down to nearly zero by 2050 to be reworked with more ambitious and detailed goals to reduce emissions through 2030.

On Friday, people of all ages marched through the center of Berlin, then rallied on the lawn before the Reichstag, where Germany’s Parliament meets. Thousands turned out for similar protests in other cities across the country.

They were joined by Greta Thunberg, the 18-year-old climate activist who started the Fridays for Future protests in Stockholm in 2018 by skipping school as a way of shaming the world into addressing climate change, made a guest appearance at a protest in Berlin. Future Fridays were a staple in Germany until the pandemic hit.

“Yes, we must vote and you must vote, but remember that voting will not be enough,” she told the crowd, urging them to stay motivated and keep up the pressure on politicians.

“We can still turn this around. People are ready for change,” she said. “We demand the change and we are the change.”

Credit…Fabian Bimmer/Reuters

BERLIN — In the prelude to Sunday’s federal election, one of the strangest questions faced by Armin Laschet, governor of Germany’s most populous state and one of the front-runners, was what his dragon name would be.

Mr. Laschet, apparently nonplused, exhaled loudly. “No idea,” he answered. “What kind of names do dragons have?”

As the vote neared and the competition to replace Chancellor Angela Merkel increasingly turned on the candidates’ characters, the contenders submitted themselves to an exhaustive schedule of interviews, debates and town hall-style discussions — including some inquiries from children. In fact, many of the most memorable moments were prompted by the younger questioners.

On one program, “Can You Do the Chancellery,” each of the main candidates was given 30 minutes to teach a classroom of 8- to 13-year-olds. During their separate sessions leading the class, candidates answered questions and had to explain complex themes (like global taxation or global warming) on a whiteboard.

Pauline and Romeo, the children who asked Mr. Laschet about dragons, were part of a segment on a late-night talk show. The two, both 11, threw Mr. Laschet no softballs. Among other things, they asked if he was planning on quitting smoking (a question he dodged, though he did offer that he did not inhale) and about a far-right candidate in his party.

When the 10-minute segment aired this month, Mr. Laschet was widely panned for his performance. (Two other candidates, Annalena Baerbock of the Greens and Olaf Scholz of the Social Democrats, survived Pauline and Romeo without making any headlines.)

But Mr. Laschet was not the only one to struggle. Tino Chrupalla, co-chairman of the populist Alternative for Germany party, also had a tough time with a younger interrogator.

In a publicly broadcast interview, Mr. Chrupalla told a teenage reporter called Alexander that his party wanted to see more German poems and songs being taught in classrooms. But when Alexander asked him what his favorite German poem was, Mr. Chrupalla struggled to name one.

Credit…Wolfgang Rattay/Reuters

Unusually long lines at polling stations on Sunday caused several Berlin voting locations to remain open for hours after the 6 p.m. closing deadline. That extension may add hours to the time it will take Germany to tally the votes.

The culprit seems to have been a combination of higher-than-expected in-person voting, missing or wrong ballots, and a road-blocking marathon that delayed restocking supplies.

Paco Mallia, 18, who looked forward to voting for the first time, turned back when he saw the long line at his polling station in the central neighborhood of Moabit on Sunday morning.

When he returned just before closing time, the line remained long, but an election worker assured Mr. Mallia that he would get to vote.

At other polling stations in the city, handwritten notes informed voters that as long as they stood in line by 6 p.m. they could cast a ballot.

Mr. Mallia decided to stay. “This election is kind of a big deal for me,” he said.

Although delays were reported in other jurisdictions, Berlin — where residents also voted in state and local elections — seems to have been hardest hit.

Dirk Behrendt, a Green Party city official, demanded an investigation into the delays.

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More Power Lines or Rooftop Solar Panels: The Fight Over Energy’s Future

The nation is facing once in a generation choices about how energy ought to be delivered to homes, businesses and electric cars — decisions that could shape the course of climate change and determine how the United States copes with wildfires, heat waves and other extreme weather linked to global warming.

On one side, large electric utilities and President Biden want to build thousands of miles of power lines to move electricity created by distant wind turbines and solar farms to cities and suburbs. On the other, some environmental organizations and community groups are pushing for greater investment in rooftop solar panels, batteries and local wind turbines.

There is an intense policy struggle taking place in Washington and state capitals about the choices that lawmakers, energy businesses and individuals make in the next few years, which could lock in an energy system that lasts for decades. The divide between those who want more power lines and those calling for a more decentralized energy system has split the renewable energy industry and the environmental movement. And it has created partnerships of convenience between fossil fuel companies and local groups fighting power lines.

At issue is how quickly the country can move to cleaner energy and how much electricity rates will increase.

senators from both parties agreed to in June. That deal includes the creation of a Grid Development Authority to speed up approvals for transmission lines.

Most energy experts agree that the United States must improve its aging electric grids, especially after millions of Texans spent days freezing this winter when the state’s electricity system faltered.

“The choices we make today will set us on a path that, if history is a barometer, could last for 50 to 100 years,” said Amy Myers Jaffe, managing director of the Climate Policy Lab at Tufts University. “At stake is literally the health and economic well-being of every American.”

The option supported by Mr. Biden and some large energy companies would replace coal and natural gas power plants with large wind and solar farms hundreds of miles from cities, requiring lots of new power lines. Such integration would strengthen the control that the utility industry and Wall Street have over the grid.

batteries installed at homes, businesses and municipal buildings.

Those batteries kicked in up to 6 percent of the state grid’s power supply during the crisis, helping to make up for idled natural gas and nuclear power plants. Rooftop solar panels generated an additional 4 percent of the state’s electricity.

become more common in recent years.

Some environmentalists argue that greater use of rooftop solar and batteries is becoming more essential because of climate change.

After its gear ignited several large wildfires, Pacific Gas & Electric began shutting off power on hot and windy days to prevent fires. The company emerged from bankruptcy last year after amassing $30 billion in liabilities for wildfires caused by its equipment, including transmission lines.

Elizabeth Ellenburg, an 87-year-old cancer survivor in Napa, Calif., bought solar panels and a battery from Sunrun in 2019 to keep her refrigerator, oxygen equipment and appliances running during PG&E’s power shut-offs, a plan that she said has worked well.

“Usually, when PG&E goes out it’s not 24 hours — it’s days,” said Ms. Ellenburg, a retired nurse. “I need to have the ability to use medical equipment. To live in my own home, I needed power other than the power company.”

working to improve its equipment. “Our focus is to make both our distribution and transmission system more resilient and fireproof,” said Sumeet Singh, PG&E’s chief risk officer.

But spending on fire prevention by California utilities has raised electricity rates, and consumer groups say building more power lines will drive them even higher.

Average residential electricity rates nationally have increased by about 14 percent over the last decade even though average household energy use rose just over 1 percent.

2019 report by the National Renewable Energy Laboratory, a research arm of the Energy Department, found that greater use of rooftop solar can reduce the need for new transmission lines, displace expensive power plants and save the energy that is lost when electricity is moved long distances. The study also found that rooftop systems can put pressure on utilities to improve or expand neighborhood wires and equipment.

Texas was paralyzed for more than four days by a deep freeze that shut down power plants and disabled natural gas pipelines. People used cars and grills and even burned furniture to keep warm; at least 150 died.

One reason for the failure was that the state has kept the grid managed by the Electric Reliability Council of Texas largely disconnected from the rest of the country to avoid federal oversight. That prevented the state from importing power and makes Texas a case for the interconnected power system that Mr. Biden wants.

Consider Marfa, an artsy town in the Chihuahuan Desert. Residents struggled to stay warm as the ground was blanketed with snow and freezing rain. Yet 75 miles to the west, the lights were on in Van Horn, Texas. That town is served by El Paso Electric, a utility attached to the Western Electricity Coordinating Council, a grid that ties together 14 states, two Canadian provinces and a Mexican state.

$1.4 million, compared with about $1 million to Donald J. Trump, according to the Center for Responsive Politics.

In Washington, developers of large solar and wind projects are pushing for a more connected grid while utilities want more federal funding for new transmission lines. Advocates for rooftop solar panels and batteries are lobbying Congress for more federal incentives.

Separately, there are pitched battles going on in state capitals over how much utilities must pay homeowners for the electricity generated by rooftop solar panels. Utilities in California, Florida and elsewhere want lawmakers to reduce those rates. Homeowners with solar panels and renewable energy groups are fighting those efforts.

Despite Mr. Biden’s support, the utility industry could struggle to add power lines.

Many Americans resist transmission lines for aesthetic and environmental reasons. Powerful economic interests are also at play. In Maine, for instance, a campaign is underway to stop a 145-mile line that will bring hydroelectric power from Quebec to Massachusetts.

New England has phased out coal but still uses natural gas. Lawmakers are hoping to change that with the help of the $1 billion line, called the New England Clean Energy Connect.

This spring, workmen cleared trees and installed steel poles in the forests of western Maine. First proposed a decade ago, the project was supposed to cut through New Hampshire until the state rejected it. Federal and state regulators have signed off on the Maine route, which is sponsored by Central Maine Power and HydroQuebec.

But the project is mired in lawsuits, and Maine residents could block it through a November ballot measure.

set a record in May, and some scientists believe recent heat waves were made worse by climate change.

“Transmission projects take upward of 10 years from conception to completion,” said Douglas D. Giuffre, a power expert at IHS Markit. “So if we’re looking at decarbonization of the power sector by 2035, then this all needs to happen very rapidly.”

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