Clinton won all six.
From a mathematical standpoint, that's pretty remarkable. We could demonstrate that by explaining the probability numbers behind it. But we have a better way.
The odds that Clinton supporters would win all six of the coin tosses against Bernie Sanders supporters are pretty slim. (See the update below. There were other coin tosses that emerged today which Sanders won -- so, yes. Very slim.) Given the closeness of the race and the complexity of the caucus system, you'd be forgiven for assuming that Clinton earned her four-delegate statewide margin with six delegates earned via coin toss. But she didn't.
The Iowa caucus process is thoroughly complicated, from start to finish. What matters here is that the delegates that were won with the coin toss were not actual convention delegates but county delegates. The tally that Clinton won by four delegates was "state delegate equivalents," a calculation that estimates how many delegates to the state Democratic convention will result from each side's winning enough county delegates. There were nearly 1,700 precincts that held caucuses on Monday and in most of them, according to the blog Bleeding Heartland, four or more county delegates were identified. That's thousands more county delegates than the 1,400 delegates that will go to the state convention — where the actual delegates that attend the nominating convention in Philadelphia will be chosen.
Or put another way, it's like Clinton won six jump balls in the first game of the NCAA tournament, not that she was awarded free throws in the finals.
Which doesn't mean it's not an accomplishment. Clinton and Sanders may essentially have tied in Iowa, but in the coin tossing, Clinton achieved a remarkable victory.
Update: The initial 6-for-6 report, from the Des Moines Register missed a few Sanders coin-toss wins. (There were a lot of coin tosses!) The ratio of Clinton to Sanders wins was closer to 50-50, which is what we'd expect.
Here is video of one, which we are sharing mostly because of dude's mustache.