One potato, two potato, three potato, four — five potato, six potato, seven potato, more was a favourite children’s game when Hilary Armstrong was growing up in the 1950s. Just where the blame lies for her apparent inability to count any higher will be a moot point in her next challenge: persuading Labour MPs to vote for the prime minister’s education reforms.

The chief whip’s disastrous mathematical calculations last week not only saw backbench Labour rebellions take Tony Blair to his second Commons defeat but also — unbelievably — to his third, just a few hours later, by a single vote: his own. He had gone home early because Armstrong (dubbed Squawker, with a voice like a howler monkey that has just hit its thumb with a hammer) had assured him his presence was not necessary.

David Cameron, the Tory leader, used to taking a barracking from Armstrong who has a schoolmarmish habit of wagging a finger at him during prime minister’s questions, revelled in taking his revenge. “I’ve noticed she’s a little quieter than normal” he quipped. “She’s the first chief whip to put the prime minister in the frame for losing a key vote. It’s an interesting career move, to say the least.”

Miscalculating how many of her own party would vote for amendments to the government’s draconian “religious hatred” bill was a slip-up richly relished by the rank-and-file, who loathe the government’s bully-girl-in-chief as much as the Tories.

Those who resent her are hoping that Blair might finally realise that he has over-promoted a dozy sycophant who sees the chief whip’s job as less about jollying along — or threatening — the troops, than issuing orders down the chain of command and “telling Tony what he wants to hear”.

Since she was elevated in 2001 from local government minister to chief whip with cabinet rank but no previous experience in the whips’ office, Armstrong has forced through government bills against threatened backbench rebellions on foundation hospitals, university tuition fees and the war with Iraq.

The prime minister’s first defeat came late last year when Armstrong’s best half nelsons failed to persuade enough Labour MPs to back Charles Clarke’s measure to detain terrorist suspects without trial for 90 days.

Prior to that, the closest she had come to a setback was a run-in with the late Robin Cook in the summer of 2001. The whips’ main methods in enforcing their writ and getting unpopular government measures past their own party without revolt are a carrot-and-stick mix of dangled blandishments and threatened punishments.

But Armstrong has form on bodging the job. In July that year, Downing Street tried to take out two troublemakers. Gwyneth Dunwoody, the rumbustious left-wing member for Crewe and Nantwich and longest serving female MP, was rung up by Armstrong at 11.20am and told that an announcement that she was being sacked as chairwoman of the select committee on transport was being made 10 minutes later. Simultaneously it was revealed that Donald Anderson, another thorn in the sides of the whips, would be removed as chair of the foreign affairs committee. The pair were saved only when Cook, leader of the House, forced a vote in which 100 MPs rebelled; a disastrous situation for a whip to have brought upon herself.

Cook considered Armstrong “a bit thick” while she complained that he “patronised” her. Despite an avowed pride in her northern roots, Armstrong has shown evidence of a mighty chip on her shoulder.

Take the time, for instance, that she hauled in Paul Marsden, MP for Shrewsbury and Atcham, over his refusal to support military action in Afghanistan. She first accused him of absenteeism, which he defended by saying that his wife had been seriously ill and that he had been campaigning in the general election. Then she demanded that he consult her before making any statements to the press. When he refused, Armstrong complained that “the trouble with people like you is you are so clever with words that us up north can’t answer back”. Marsden retorted that he came from Cheshire.

His disobedience ended in a scuffle — in which Armstrong was not involved but several of her male junior whips were — in a Commons bar at 3am. Her tactics had proved so persuasive that Marsden defected to the Liberal Democrats.

At Armstrong’s 2003 summer party for the whips’ office, just months after the invasion of Iraq, she congratulated her shock troops on their performance in difficult times. She added that recently re-reading her father’s diaries when he was a whip in Harold Wilson’s government had made her realise that a whip’s lot is not a happy one.