THE Scottish Conservatives are on manoeuvres. This week they aired a party political broadcast featuring their new leader.

The media had already been treated to the three minutes of Jackson Carlaw pottering around the goods entrance of the Scottish Parliament in the cold.

But now the electorate at large got their chance to enjoy the sight.

His party, he declared, was “going to up its game”. It was going to “work flat out” to become the government at next year’s Holyrood election.

In a related incident, he also tested out a new campaign slogan.

When the SNP won power in 2007, they pitched themselves as a breath of fresh air after eight years of lacklustre Labour-LibDem coalition.

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“It’s time,” they said. For change.

Thirteen years on, Mr Carlaw told Nicola Sturgeon at First Minister’s Questions: “Time’s up.” We’ll be hearing that one ad nauseam.

The SNP certainly doesn’t have its troubles to seek. Problems are piling up in a number of public services.

Ms Sturgeon’s leadership is also under fire from within her party.

Previously private muttering over her caution / inertia on a second referendum has become public.

Some of her MPs want her to test the limits of Holyrood’s power by legislating for a consultative referendum without Westminster consent, drawing Boris Johnson into a UK Supreme Court challenge.

SNP MP Joanna Cherry QC yesterday said the Lord Advocate should even “proactively” refer any such legislation to the Court for a ruling on its legality.

But that assumes he would sign off on the legislation in the first place.

Legal advice to ministers is a closely guarded secret, but Ms Sturgeon’s reluctance over the whole issue - she has said any such Bill could be struck down and send the Yes movement backwards - suggests her chief law officer may not be thrilled by the idea.

That row will rumble on, and divided parties don’t impress voters.

Allied to that, Ms Sturgeon faces questions over her shelf-life.

She has said she intends to carry on for now, but the questioning will only intensify as polling day approaches.

By that point, she will have been First Minister for six and a half years, and be applying to extend it to 11 and a half, longer than Tony Blair, and almost as long as Margaret Thatcher.

A vanishingly small number of politicians have either the stamina or the public support to pull that off.

It was no accident Mr Carlaw referred simply to “Sturgeon” in his broadcast, knowing the visceral response in his target audience.

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But the Scottish Tories also have their problems. And when it comes to achieving Mr Carlaw’s do-or-die goal of becoming the largest party at Holyrood, they have a very big problem indeed. It’s called arithmetic.

The party currently has 31 MSPs, seven in first-past-the-post seats and 24 through the regional list system.

Let’s say their aim is to match the 47 MSPs won by the SNP in 2007 when it became the largest party.

That means they need to hold all they have - by no means a given - and add another 16 MSPs. But where do they come from? Not from their most fertile territory, the lists.

There are 56 list seats, but no party has ever won more than half of them.

The most was the SNP’s 28 in 1999, essentially compensation for winning just seven of the 73 constituencies.

In 2007, the SNP won 26 list MSPs. So let’s be generous and say the Tories also win 26 in 2021, a gain of two.

That still leaves them 14 short of 47, if they hold their seven constituencies.

These 14 missing seats have to be wrestled away from their opponents.

This is where arithmetic really starts to trample Mr Carlaw’s dreams.

For the Scottish Tories have a truly woeful record at gaining Holyrood seats. Since 1999, they have gained a grand total of just eight across five elections and one byelection.

Worse still, they have been better at taking seats from Labour (four of the eight), than from the SNP (three).

They also took out a solitary Borders LibDem in 2007.

But in all 17 seats where they came second in 2016 - their best hopes next year - they are chasing the SNP.

There are no more Labour/Tory or LibDem/Tory marginals to pick up.

The Tories can take votes directly from the SNP, of course. In the 2017 general election, most of the 12 Tory gains followed a big drop in the SNP vote. Transfers from LibDems also helped elect Tories in Gordon and Berwickshire, Roxburgh & Selkirk, while Labour switchers were notable in Ochil & South Perthshire.

But half those gains were reversed in December’s election. Interestingly, not because people who voted Tory in 2017 rushed back to the SNP.

The Tory vote was generally down just a few points. The big factor in 2019 was Labour voters going SNP to help turf out the Tories. Would those same tactical voters U-turn and vote Tory to get the SNP out in 2021? It seems fanciful.

There are some constituency prospects for the Tories.

But even among the top ten target seats, where the SNP majority is less than 6,000, barely half look in serious contention. Roseanna Cunningham is vulnerable in Perthshire South & Kinross-shire with a majority of just 1422, as is Gordon MacDonald in formerly Tory Edinburgh Pentlands (majority 2,456), especially if the Tories exploit SNP divisions involving the local MP, one Joanna Cherry.

Their majorities below 3000, Maureen Watt in Aberdeen South & North Kincardine, Mairi Gougeon in Angus North and Mearns, and Richard Lochhead in Moray also need to be on their toes.

With the Tories putting problems in schools front and centre, Education Secretary John Swinney will feel the heat in Perthshire North (majority 3,336). He is likely to be targeted as a symbol of the SNP’s record.

But in many of these seats the absence of a large pool of Labour and LibDems votes to squeeze still gives the SNP a clear advantage.

Retiring SNP MSPs and the loss of personal votes may help the Tories here and there, but overall those 14 seat gains look well out of reach.

Unless he can discover a credible path to victory, not just endlessly complain about the Government, Mr Carlaw will be pottering about in the cold for some time yet.