Nikki Haley lost badly in Iowa and New Hampshire. In Nevada on Tuesday, she ran unopposed in an irrelevant primary and finished behind “None of These Candidates.” But she is still trying to broaden voters’ imagination of what’s possible, both for her prospects in the 2024 Republican presidential primary, and for the nation.

She has money in the bank, a strategy to expand the Republican base and a new unabashedly adversarial stance toward former President Donald Trump.

Yet she is coming up against tough electoral math, and a Republican base that seems determined to nominate Trump for the third time.

Mike Noble, a pollster who works in Arizona and Nevada, noted that about seven in 10 Republicans nationally say they don’t believe that Trump actually lost in 2020.

To those voters, Trump is “a two-time winner now going for a three-peat,” Noble said. “Basically you’re holding out for a Hail Mary.”

To pull out an unlikely victory, Haley, the former South Carolina governor and United Nations ambassador, must secure the support of 1,215 G.O.P. delegates — a count that, as of now, stands at 33 for Trump and 17 for Haley.