Today’s reshuffle stories suggest that Theresa May will dismiss or demote Justine Greening and Andrea Leadsom…while also wanting more women in senior posts.

If so, the Prime Minister will find it difficult to reconcile promoting more women to Cabinet with finding enough women to promote, at least from high-ranking levels within the Government.

Let’s start at the top. If we discount the Prime Minister herself and the Leader of the Lords, the Cabinet has four full female members: Amber Rudd, Greening, Karen Bradley and Penny Mordaunt.  Leadsom and Liz Truss are entitled to attend it.

But immediately below Cabinet itself – that’s to say, at Minister of State rank – we estimate that out of 23 such ministers there are only four women: Anne Milton, Claire Perry, and the recently-promoted Sarah Newton. Caroline Nokes has a full Ministerial title.

May could well promote the first two.  (It would be surprising were Newton to make Cabinet this time round, simply because she has only recently been moved up a rank.)  Milton is our recommendation for Chief Whip. There are claims she will be made Health Secretary.

But readers will grasp Downing Street’s problem. Just as the houses that Britain needs can’t be built fast enough if there are shortages of material and labour, so women Cabinet Ministers can’t be appointed speedily enough if too few women are coming through the ranks.

The Prime Minister could of course seek to promote to Cabinet from the Under-Secretarial ranks.  This is unusual but not unknown: Truss, for example, made the leap during the Cameron premiership.

We count eight female Under-Secretaries out of a total of 25: Victoria Atkins, Harriett Baldwin, Margot James, Jackie Doyle-Price, Caroline Dinenage, Chloe Smith, Tracey Crouch, Therese Coffey.

One could perhaps see, say, Baldwin or James making the top table this time round. Or for May to look to the Whips Office and Esther McVey.

But it is hard to imagine talented women MPs with no ministerial experience – Nus Ghani, Lucy Frazer, Suella Fernandes, Kemi Badenoch, Jo Churchill and so on – making it to the Cabinet at one bound. The same goes for their male equivalents.

Finally, it is worth counting up the departments without any Commons women Ministers at all.  These are: the Foreign Office, Justice, DexEU, International Trade, Transport, CLG, Scotland and Wales.

It is a statement of the obvious that women are not a minority group: they are roughly half the population.  It would therefore be sensible for May to aim to appoint at least one female Minister to each department, bar the smaller Scottish and Welsh departments.