President Barack Obama and House Speaker John Boehner offered hints of possible compromise but also traded heated rhetoric Tuesday, a frustratingly inconclusive combination that left the eight-day partial government shutdown firmly in place and the threat of an unprecedented national default drawing closer.

“There’s a crack there,” Boehner said of the impasse near the end of a day of maneuvering at the White House and the Capitol. Yet the Ohio Republican added that it was not enough to warrant optimism.