MAY’S INSOLUBLE BORDER WOES.

Before geography morphed into the study of how capitalism screws up the world, it concerned itself with problems of topography, terrain and, yes, maps. The classic map problem was: What’s the smallest number of colours guaranteeing no contiguous countries get coloured the same? The answer turns out to be four: You can do it with just four different colours. For the world as we know it, geographers solved this problem centuries past. To prove it for any and all possible worlds required the intervention of mathematicians, who accomplished the proof in 1976.

Perhaps mathematicians could now turn their attention to Brexit, where poor Theresa May (who didn’t originally support Brexit, remember), is seized of a puzzle even more intractable than the map-colour problem. Two regions formerly united — the United Kingdom and the European Union — agree to disunite. The U.K. wants its own domestic policies. That implies that the border between the U.K. and the EU must interfere in a meaningful way with the current free movement of labour, capital, goods and services, for, as the geo-philosopher and punctuation artist Donald J. Trump has tweeted, “If you don’t have Borders, you don’t have a Country.”

Geography ordains that the only land border between the U.K. and the EU is on the island of Ireland, whose northeast one-sixth is a province of the U.K. Its other five-sixths, the Republic of Ireland, is and will remain part of the EU. But an agreement arrived at on Good Friday of 1998, after four decades of sectarian violence, requires that the Irish border involve nothing more than a sign indicating that speed limits are switching from kilometres per hour to miles per hour or vice versa. In the negotiations leading to the recent EU-U.K. Brexit agreement, the EU took the position that this “soft” border was a red line, as it were. The U.K. agrees there should be no hard border, so it has been looking for alternatives.

The most obvious is, in effect, to put the border in the Irish Sea. Northern Ireland accounts for fewer than two million of the U.K.’S 66 million people. If for customs, excise and regulatory matters, everything moving between Northern Ireland and Great Britain received the same screening as items moving between the U.K. and EU members, that might seem a small price for U.K. sovereignty.

But the North wants none of it. Its 10 Democratic Unionist Party (DUP) MPS in the British House of Commons fear that even modest practical separation will eat away at the ties binding it to the U.K. The North’s seven Sinn Fein MPS certainly don’t oppose weaker ties with the U.K. but as, on principle, they don’t take their seats in Westminster, their views aren’t as crucial as the DUP’S — which are very crucial indeed, given that it holds the balance of power in Parliament and is keeping Prime Minister Theresa May in power.

With all the constraints enmeshing her, May must feel her power is overrated. Those who have always been unhappy with “Britain in Ireland” are delighted the problem of the Irish border is causing her so much trouble.

In sum: A deal requires a border. But you can’t put a border in Ireland. And you can’t put a border east of Ireland. So what do you do? The fudge — and it’s clearly a fudge — is that you keep negotiating with the EU during a two-year “implementation” in hopes that mathematicians and trade bureaucrats will find some way to square this circle. And you agree that, if in the end the circle remains unsquared, you will continue to follow EU rules and impose EU tariffs on imports from non-eu countries and you will keep doing so until you and the EU agree to replace it with something different. This “EU indefinitely” is called “the backstop” and among Brexiteers is very unpopular. Why Brexit in the first place if you keep EU rules and tariffs, supposedly temporarily but possibly forever?

Can the circle be squared? Full reciprocity would help. The EU and the U.K. could recognize each other’s rules and standards. If a fridge is good enough for sale in the EU, consider it good enough for the U.K., and vice versa. Very few people die in either jurisdiction from faulty fridges.

Unfortunately, that way out of the backstop leaves the problem of the backdoor. Suppose a Brexited U.K. makes a free-trade deal with the U.S. but the EU doesn’t. Without real, functioning borders between the U.K. and the EU, how does the EU prevent the U.K. from becoming a backdoor through which U.S. goods sneak into Europe without paying EU tariffs?

In the end, the four-colour theorem was proved using computers. Maybe some clever computer app will allow governments to track and tax goods wherever they end up. Until then, pity poor Theresa May.

WHY BREXIT IN THE FIRST PLACE IF YOU KEEP EU RULES AND TARIFFS?