The students in Mrs. Dooley’s 5^{th} grade math class are scratching their heads. According to Mrs. Dooley, someone has challenged the great Achilles to a foot race. But what makes this particularly puzzling is that the challenge is coming from a tortoise.

You see, Achilles is the fastest runner in all of Greece while tortoises are, well, tortoises. So this can’t end well for our Tortoise.

Fortunately, though, Achilles is a generous man. He agrees to give Tortoise a head start. Achilles will begin the race at the Start but Tortoise will begin half way to the finish line.

Still, Tortoise has a problem. As soon as the race begins, ESPN measures the speed of both runners and Achilles is running 4 times as fast as Tortoise. Remember, though, Tortoise has a head start. Achilles must cover twice as much ground as Tortoise; but then again, Achilles is running 4 times as fast.

Mrs. Dooley asks the class, “Do you think Achilles can catch up to Tortoise and win the race?”

All the students seem to nod their heads.

“Does anyone disagree with that?”, Mrs. Dooley asks.

Slowly, a small boy hiding in the last seat in the last row raises his hand. The class groans impolitely.

Zeno has a reputation for coming up with contrary answers and weird arguments. One time the banter between Mrs. Dooley and Zeno became so intense that the class was two minutes late for recess. No one has forgiven Zeno for *that*.

But Mrs. Dooley is always polite. “How do you see it, Zeno?”

“Well,” the boy began, “Achilles can’t win the race unless he can catch up to Tortoise. He must catch Tortoise before he can pass him. Right?”

“Ok,” Mrs. Dooley agreed, warily.

“But Achilles can never catch up to Tortoise!”, Zeno asserted confidently, as though their argument was over and the matter settled.

“Why not?” Mrs. Dooley ventured.

Zeno seemed a bit annoyed. After all, Mrs. Dooley was supposed to be *his* teacher and now he was going to have to teach *her*.

“Before Achilles can pass Tortoise, we’ve agreed that he has to catch up to Tortoise, right?”

“Yes.”

“So, the first thing Achilles has to do is get to the half way marker where Tortoise began the race,” Zeno explained.

“Ok,” Mrs. Dooley replied, still not understanding where the argument was headed.

Zeno, by now a little bit exasperated, continued, “Well, by the time Achilles gets to the half way mark, Tortoise isn’t there anymore, is he?”

“No, I don’t suppose he is.”

“So then, Achilles has to get to where Tortoise is *now*, doesn’t he?”

“Right.”

“But when Achilles gets there, Tortoise has moved on again.”

“Of course.”

“Well, Mrs. Dooley,” Zeno concluded, trying to be as respectful as possible, “Don’t you see that this same process will repeat over and over again? No matter how many times Achilles gets to where Tortoise *was*, Tortoise won’t be there anymore.”

“So, Zeno,” asked an exasperated Mrs. Dooley, “Who wins the race?”

“Easy, Mrs. Dooley,” Zeno replies. “Achilles can never catch up to Tortoise, and if Achilles can never catch up to Tortoise, then Achilles can’t win, or even tie, the race. Eventually Tortoise, slow as he is, will cross the finish line and when he does, he will win the race.”

Mercifully, at that moment the bell rang for lunch.