The phrase "mathematically impossible" is one that's worth being careful with. In the real world, mathematical impossibility refers best to things like needing 101 percent of the vote in a precinct in order to win an election. Since you cannot get more votes than exist, that situation would be rightly deemed mathematically impossible.
Here is an example of something that is not mathematically impossible, from a Bloomberg Politics interview with Ben Carson aide Armstrong Williams. Williams predicts that Carson will be able to earn support from 13 percent of black voters should he make it to the general election against Hillary Clinton. (It is still mathematically possible she could lose in the primary, but it is not something mathematicians would bet against.)
"If we can capture that much of the African-American vote," Williams told Bloomberg's Kevin Cirilli, "it is mathematically impossible for her to win."
No, it isn't, for a variety of reasons. First of all, Clinton could win 100 percent of the white vote and 100 percent of the Hispanic vote, in which case Carson's 13 percent of the black vote isn't going to do him much good. Again: That's not likely. But it's not impossible.
Less pedantically, it's also not true because it appears that there's nothing magical about that 13 percent figure.
In 2012, Mitt Romney got six percent of the black vote. Four years earlier, John McCain got four percent and four years before that, George W. Bush got 11 percent. Bush's 11 percent was enough to get him across the finish line, which is maybe why Carson's team thinks 13 percent will put it away. But Bush also got 44 percent of the Hispanic vote to McCain's 31 and Romney's 27. Bush got 44 percent of the Asian vote, too -- versus 35 percent and 26 percent for the next two Republicans, respectively. In other words, unless Carson makes significant in-roads among other groups as well, that 13 percent among black voters won't necessarily do him much good.
If we contrast 2012 results to available exit polling, we can see that 13 percent of the African American vote in 2012 wouldn't have done Romney any good, either. The states with the highest population of black voters are primarily ones that vote heavily Republican: Alabama, Mississippi, Louisiana, South Carolina. Shifting the black electorate slightly more to the right in those states would only make the Republican victories there larger. In other states, like Colorado or New Hampshire, a Republican victory isn't far out of reach -- but there simply aren't a lot of black voters.
What Romney would have needed is a stronger black vote in places with closer results and larger black populations. Places like Florida, Ohio and Virginia. The next problem, though, is that shifting a small part of the population a small amount has an even smaller effect on the end result.
In 2012, Ohio's electorate was 15 percent black, according to exit polling, and President Obama won the state by three points. Black voters supported Obama by a 93-point margin, 96.5 percent for the incumbent and 3.5 percent for Romney. If you jack that Romney percentage up to 13 percent, as Williams would like to do, it shifts the overall vote to the Republicans -- by 2.7 percent. Not enough for Romney to have won.
In Virginia, the numbers are similar. Shifting the black vote to 13 percent Republican Obama with a 1.2-point victory in the state.
Only in Florida would it have made a difference. There, a 0.9-point Obama victory becomes a 1.3-point Romney one. Which would have shifted the electoral vote calculus from 332 votes for Obama to 303 votes -- still a 68 electoral college vote win for the president. No other state for which we have exit poll data becomes a Romney win under the "13-points" strategy, including Pennsylvania, Massachusetts or New York. Some close states like North Carolina and Indiana were won by Romney anyway.
What's left over are states that Obama or Romney won handily, states that have small black populations and/or states that don't have many electoral votes. If Romney had gotten the support from black voters that Armstrong says Carson can get, he'd still be sitting in his home right now instead of sitting in the White House.
Of course, it looks increasingly likely that Armstrong and Carson won't have the opportunity to put their theory to the test. Carson has dropped out of the lead in Iowa and Trump is pulling away from him nationally.
It is not mathematically impossible that he could win the nomination, mind you. But I wouldn't bet on it.