The critical question for MPs now is what would they hate most – a no-deal Brexit or a Corbyn government?

“What would you hate most – a Corbyn government or a no-deal Brexit?”

That, I’m told, is how Downing Street is already framing the crucial question for Conservative MPs. Government whips will ram home the same message when the Commons returns from its summer break in two weeks. The answer will probably decide what happens on Brexit.

The $64,000 question is not only one for Tories who believe no deal would be an act of economic self-harm, but also for opposition MPs. Jeremy Corbyn inadvertently posed the question by urging them to support him as head of a “strictly time-limited” caretaker government if no deal opponents pass a vote of no confidence in Boris Johnson.

Jo Swinson, the Liberal Democrat leader, overshot the runway a little by dismissing Corbyn’s offer as “nonsense”. It’s easy to see why. The Lib Dems are on a roll after the local and European elections. Sarah Wollaston, the respected former Tory MP, confirmed she is joining them, as this column revealed last month. Why, with a general election looming, should the Lib Dems dilute their unique selling point by boosting the Remain credentials of Corbyn, a man they describe as a Brexiteer and Eurosceptic?

The problem was that Swinson risked breaking her own ground rule: the national interest in stopping no deal must come before “personal agendas, political games and party politics”. On that, she is right. But, initially, she got her tone wrong. Even new recruit Wollaston suggested answering the $64,000 question differently: a Corbyn government would be “the lesser of two evils”.

After criticism from Nicola Sturgeon and Caroline Lucas, Swinson softened her line by offering to meet Corbyn.

However, she was right to paraphrase Lynton B Johnson, the US president who said the first rule of politics is that its “practitioners need to be able to count”. The battle over a no-deal Brexit is a numbers game. The Commons arithmetic would be easier if MPs carve out enough parliamentary time to pass legislation instructing Boris to seek an extension of the UK’s EU membership beyond 31 October. But time will be very short, so a no-confidence vote might prove the only way to stop a no-deal scenario.

As leader of the opposition, Corbyn would be first in line if Johnson lost such a vote. It is often conveniently forgotten that Corbyn has 247 MPs, more than three times the other opposition parties combined. But he will undoubtedly have a huge problem in winning a majority to head even a short-lived government. He might need to persuade about 15 Tory MPs to back him, given there are 10 former Labour MPs who quit the party because of his leadership, and other independents, such as Lady Sylvia Hermon, a unionist in Northern Ireland, who would not support him. So far, only one Tory has answered the $64,000 question in the affirmative – Guto Bebb, the former defence minister.

The numbers could be even worse for Corbyn. The critical question is perhaps most difficult not for Tories who agonise over bringing down their own government, but for Labour MPs who are reluctant to make their own leader prime minister. If the parliamentary Labour Party held a vote of confidence in Corbyn as party leader now, he would lose it. But when they staged such a vote in 2016, it went nowhere; he was re-elected by the party membership.

Several Labour MPs felt able to stand for re-election in 2017 because they could tell people on the doorstep Corbyn had no chance of becoming prime minister. Amid the Brexit turmoil, they could not say that now. So, do they hold their nose and support Corbyn as a temporary PM, or help prevent that by abstaining if he sought a positive vote of confidence in the Commons, thus ending their own Labour career? A few pro-Leave Labour MPs might also be tempted.

While Corbyn’s intervention put the spotlight on Swinson, it could soon fall on him again. His letter to MPs in other parties did not mention including anyone from outside Labour in his temporary government. This is a card he could and should play. Why not make a big, bold offer to Ian Blackford, the SNP’s Westminster leader; Liz Saville Roberts from Plaid Cymru; Caroline Lucas, the sole Green MP; and Jo Swinson?

Yes, I know Corbyn would choke on his muesli at Swinson’s name. How could he cosy up to the enemy when he constantly attacks the Lib Dems for entering a coalition with the Tories in 2010? But it would show he is serious about stopping no deal, and is not merely trying to spread the blame to the Lib Dems if opponents fail to halt it.

A Corbyn-led cross-party government would offer him a big prize: if he acted in a sensible and statesmanlike way in his few weeks in office, it would be much harder for the Tories to portray him as a Marxist bogeyman at the ensuring election.

Ultimately, Corbyn could face his own agonising dilemma: if it becomes clear he cannot win a majority as caretaker PM, would he stand aside and vote for someone who could, such as Kenneth Clarke or Harriet Harman? Tory MPs won’t be the only ones having sleepless nights soon.