Three converts came to Shammai with impossible requests. All three were turned away. All three then went to Hillel, who accepted every one of them. The contrast, recorded in Shabbat 31a, defines two competing visions of what it means to teach Torah.
The first gentile said he would convert if he could be taught the entire Torah while standing on one foot. Shammai drove him away with a builder's measuring rod. Hillel accepted him and said: "What is hateful to you, do not do to your fellow"—a distillation of the commandment "Love your neighbor as yourself" (Leviticus 19:18). "That is the whole Torah. The rest is commentary—go and learn it."
The second gentile wanted to convert on the condition that he accept only the Written Torah, not the Oral Torah. Shammai pushed him away. Hillel began teaching him the Hebrew alphabet. On the first day, Hillel taught him alef, bet, gimmel, dalet. On the second day, Hillel reversed the letters. The convert protested: "Yesterday you taught me differently!" Hillel replied: "You see—you are already relying on me to tell you what the letters mean. So you must also rely on me for the Oral Torah."
The third gentile wanted to convert in order to become the High Priest, having been dazzled by the priestly garments. Shammai drove him away. Hillel accepted him, then gently guided him through the relevant texts until the convert discovered on his own that even King David could not have served as High Priest—the position was reserved for descendants of Aaron. "If even a king of Israel is not eligible," the convert said, "then certainly not a mere convert like me."
Before the story, the Talmud tells of a man who bet four hundred zuz that he could make Hillel lose his temper. He interrupted Hillel's Shabbat preparations with absurd questions—why are Babylonians' heads oval, why are the eyes of Tadmor residents bleary? Hillel answered each one patiently. The man lost his money. Hillel's calm was unbreakable.