<a href="https://www.nytimes.com/2019/12/16/world/europe/uk-election-brexit.html" target="_blank" rel="noopener noreferrer">The U.K. Election Explained, in One Number</a>  <font color="#6f6f6f">The New York Times</font>

THE INTERPRETEr

Votes for the pro-Brexit Conservatives had 10 times the effective power of votes for the anti-Brexit Liberal Democrats. Thank the electoral system known as “first past the post.”

Jo Swinson, the former leader of the Liberal Democrats, early Friday in Bearsden, Scotland, before learning that she had lost her Parliament seat to the Scottish National Party.Credit…David Cheskin/Getty Images

LONDON — The answer to Brexit, the Conservative Party’s election victory and everything in British politics is (with apologies to Douglas Adams) 336,038.

That number is what you get when you divide the 3,696,423 total votes cast nationally for the Liberal Democrats party in last week’s election by the 11 seats the party actually won. By contrast, Prime Minister Boris Johnson led his Conservative Party to victory via a far more economical average of 38,265 votes for each of its 365 seats — a roughly tenfold difference in the parties’ ability to translate votes cast into seats won.

The Conservatives’ triumph and the Liberal Democrats’ disaster were both the result, in large part, of a factor that is rarely discussed but crucial for understanding the country’s political chaos: Britain, like the United States, operates a “first past the post” electoral system, in which parliamentary seats are awarded to the candidate who wins the most votes in each individual race, rather than by proportion of the total national vote.

Brexit, which has polarized Britain around a new political divide since the 2016 referendum in which the country narrowly voted to leave the European Union, has thrown into sharp relief the ways that first-past-the-post systems can skew political outcomes.

And so 336,038 also serves as an epitaph for the political career of Jo Swinson, the dynamic 39-year-old who was the leader of the Liberal Democrats until she lost her seat on Thursday. Just months ago she seemed triumphant, her party surging in the polls. But Thursday’s election put an end to those hopes.

The first-past-the-post system works well within a two-party system, but not where there are multiple parties. For Britain, that generally wasn’t a problem until Brexit fractured the long-stable coalitions of its two major political parties, creating an opening for challengers like the Liberal Democrats and Brexit Party.

“When you don’t have two parties, the first-past-the-post system is really bad at translating voter beliefs into seats,” said Sara Hobolt, a political scientist at the London School of Economics.

That weakness was on display in the most recent election, in which the roughly half of the electorate who oppose leaving the European Union found that their votes had only a fraction of the power that votes for the pro-Brexit Conservatives did.

Things looked very different last September, when Ms. Swinson took the stage at her party’s conference in Bournemouth, an old-fashioned resort town on the south coast of England. To rapturous applause, she promised a future that much of the country had hoped for since the 2016 referendum: If her party won power, she would stop Brexit.

In a different political system, that might have been her moment. “There is a pattern to how small parties break through,” said Dr. Hobolt, a co-author of a coming book about challenger parties in Europe. “They find an issue that cuts across a mainstream party coalition and exploit it as a wedge.”

Across much of Europe, the same issues at the heart of the Brexit debate, such as immigration and membership in the European Union, have been just such a wedge for small parties. In countries with proportional representation, the result has been a major party realignment: instead of two major parties, on the center left and center right, many countries now have four parties divided along both social and economic lines.

In Germany, for instance, the Green Party on the far left and the anti-immigration Alternative for Germany on the far right have drawn support away from the center-left and center-right parties that have traditionally dominated politics. Though that means no party wins an outright majority, coalitions and compromise offer a way to reflect voter beliefs with relative accuracy.

If Britain had a proportional system, the pro-Remain parties could have formed a coalition with a majority in Parliament. The Liberal Democrats, the Scottish National Party, the Greens and Labour, which all promised to stop Brexit directly or hold a new referendum offering that as an option, won over 50 percent of the votes between them.

But under first past the post, things played out very differently. Instead of giving Remain voters the option of a powerful coalition government, the increased popularity of the Liberal Democrats and other small parties split the pro-Remain electorate, ultimately helping to hand victory to Mr. Johnson’s Brexiteers. For example, in Kensington and Wimbledon, wealthy districts of London that voted to remain in the 2016 referendum, Conservative candidates scraped to victory with less than 40 percent of the vote after Remain voters divided between Labour and the Liberal Democrats.

The European Parliament elections in May, by contrast, used a proportional system. Although the results are not directly comparable — voters tend to treat European elections as more of an expressive choice than a practical one — they did offer a hint of what a different system might bring. The British electorate was much more atomized, with the Brexit Party in first place with 30 percent of the vote, the Liberal Democrats second with just under 20 percent and then the Greens, Labour and the Conservatives clustered around 10 percent each.

But in first past the post, casting a vote outside of the two largest parties is a risky move. “Two-party systems are particularly problematic when you have the kind of crosscutting dimensions you have now,” Dr. Hobolt said. And, she added, “it’s really hard to break through as a third party in this kind of system.” Reflecting the will of the people can be political suicide.

Which brings us to another mystery: Why weren’t Ms. Swinson and her party able to exert more influence over Labour’s Brexit platform, as the far-right Brexit party did over Mr. Johnson’s Conservatives?

To stave off the Brexit Party threat, Mr. Johnson made support for Brexit by any means necessary a litmus test for Conservative politicians, going so far as to expel 21 legislators for voting to block a no-deal Brexit.

But pressure from the Liberal Democrats did not have a similar effect on the Labour party. Jeremy Corbyn, the Labour leader, grudgingly agreed to hold a new referendum, but avoided focusing on the Brexit issue, instead emphasizing his party’s economic platform and commitment to expanding the welfare state.

“That’s a geography story,” said Simon Hix, a political scientist at the London School of Economics who studies European politics. Brexit had divided the country along geographic as well as political lines: Large, cosmopolitan cities like London voted heavily to remain, while rural areas and postindustrial towns that have seen little benefit from globalization, including many traditional Labour strongholds, voted to leave.

The result was that Conservative Remainers were concentrated in a smaller number of wealthy areas, mostly in London, leaving the Conservatives with relatively few seats to lose from alienating them. Labour Leave voters, on the other hand, were more spread out — putting many more Labour seats at risk.

Mr. Corbyn tried to “have his cake and eat it,” Dr. Hix said, by prioritizing economic messages instead. But in an election where Brexit was the most salient issue, that strategy proved disastrously ineffective. Mr. Johnson’s promise to “get Brexit done” attracted leave voters in traditional Labour strongholds, winning those districts by often-narrow margins. Mr. Corbyn, meanwhile, failed to pick up areas like Wimbledon and Kensington where many wealthy Leave voters could not stomach his far-left economic policies and cast ballots for the Liberal Democrats.

That may offer some lessons for the United States, which shares first past the post. “When you have all these liberal cosmopolitan voters piled up in urban areas, the left are winning those seats by massive margins,” Dr. Hix said. “But the right can win many more seats with smaller margins.” That helps to explain why immigration, transgender rights and other social issues, which are effective wedge issues in rural and postindustrial areas, have become so prominent in American politics.

The United States does not have small party challengers akin to the Brexit Party or the Liberal Democrats. But its primary system offers an opportunity for populist challengers within the major parties to exploit wedge issues. That strategy propelled Donald Trump to victory in the 2016 Republican primary.

“People say that the good thing about first-past-the-post systems is you don’t get radical-right parties,” Dr. Hobolt said. “But there’s a danger that the radical right wing of a party takes over.”

The Interpreter is a column by Max Fisher and Amanda Taub exploring the ideas and context behind major world events. Follow them on Twitter @Max_Fisher and @amandataub.