The basement room was cleared of pens with blue or black ink, items that could mar paper ballots. Anyone wearing a coat was told to leave it in the hallway, in case something nefarious was hidden underneath. Water bottles, purses and keys were placed on the floor, leaving the large plastic tables smooth and uncluttered. And at 9 a.m., with the brisk rap of a county clerk’s wooden gavel, the first recount of the 2016 presidential election was underway in Wisconsin, with another recount pending in the neighboring battleground state of Michigan. For the next 12 days, election officials across all 72 counties in Wisconsin will work days, nights and weekends to recount nearly three million ballots, an effort initiated and financed by Jill Stein, the Green Party presidential candidate, who has suggested that voting machines in the state could have been hacked. Very few people expect that the recount will reverse the outcome of the election. President-elect Donald J. Trump triumphed here over Hillary Clinton by 22,177 votes, and in Michigan by 10,704 votes, a margin that a lawyer for Mrs. Clinton, Marc Elias, said had never been overcome in a recount. Legal challenges to the vote in Pennsylvania, where Mr. Trump leads by 70,638 votes, are also underway.