Letting opponents choose which side to take was one of his patented pieces of debate brinkmanship. His “flourish,” according to Scott Angstreich, a former teammate, would be to crumple up a piece of paper of the side not taken. In reality, the page remaining in his other hand had both the pro and con arguments written on opposite sides.

“Nobody was better at setting traps,” said Austan D. Goolsbee, a Yale debater who became a leading economist for President Obama. He recalled Mr. Cruz’s attempts to control debates with carefully constructed arguments that always seemed to anticipate his opponents’ rebuttals.

But Mr. Goolsbee and other top debaters on the circuit who frequently beat Mr. Cruz discovered it was easy to get under his skin, especially with humor. “It would unravel him,” Mr. Goolsbee said.

In one round, Mr. Goolsbee pointed out that the story of Mr. Cruz’s father coming to America, as compelling as it sounded, was not entirely relevant to, say, the federal deficit.

“How dare you insult my father!” Mr. Cruz replied.