Zeno’s Achilles Paradox

(October 27, 2021) Draw 2 long lines. On each of them label one end START and the other end FINISH. What are the fastest and slowest runners in the world?

Done. Cheetah and snail, same responses as the School House Lane group. (After a brief debate about G’s suggestion that we use birds such as the peregrine falcon that “can fly faster than land animals can run.” Then slugs: A mentioned that “every animal has a role; scientists just found a way to use slug slime in medicine.” R claimed that spotted lanternflies, which seem to come up in every session, do not have a beneficial role. But then A reminded everyone that spotted lanternflies are invasive here but beneficial in their native habitat.) Back to the problem…

The problem: if the snail gets a certain distance head start, and the cheetah has to wait until the snail covers that distance to then cover the same distance, then the snail gets another head start as soon as the cheetah reaches first head-start distance, who will win? The problem is originally known as Zeno’s Achilles Paradox – you can see it in this video. We started to work the steps of the problem.

“What’s the point of this problem, anyway?”

An ancient person named Zeno did this problem to prove that motion is impossible.

G: “Actually, there’s also no such thing as touching. Nothing can really touch each other.” He started stroking a tree. “I’m not really touching this tree, since there are electrons racing around the protons of each atom.”

True. You’re using science to look at the underlying structure of things. Mathematicians and scientists have a similar goal: to figure out the underlying structure of things. That’s what Zeno was trying to do with this problem with the cheetah and the snail. (At this point, A also chimed in with an example of how science looks at the underlying structure of things.)

The students intensely debated the parameters of the problem:

  • How long should the race be? Should it be measured in units of time or distance? (Many opinions, finally settled on 100 since half the class agreed)
  • What units should we use? (Miles? Kilometers? Kilo-miles?) There was no consensus or majority or even plurality for this decision. I ended up using the term “units” throughout the class to set an example of the important math skill of generalizing.
  • How big of a head start should the snail get?
  • How far would the cheetah reasonably move after the snail got the head start?
  • Is it realistic that a snail could cover such a long distance? Or that a cheetah would be racing a snail?
  • How should we mark on the line where each animal is at each point? Draw the animal with chalk? Use a rock or shell? Both? (You can see in the photo that the decision was “both.”)

As the students worked the problem, it was becoming apparent to most that the cheetah might not win. This defies common sense. Several of the students, especially R, started looking for loopholes that would resolve the cognitive dissonance that this problem often produces. He said something like “Wouldn’t the cheetah have an advantage since he would be resting while the snail was doing the head starts, so the snail would be getting tireder?” This gave me another opportunity to revisit the idea of formal systems – we have to stick to Zeno’s rules in this formal system that Zeno designed. We have to stay in what Hofstadter calls M-mode (machine). We can’t use I-mode (intelligent) to jump out of the formal system. In M-mode, the snail doesn’t tire sooner than the cheetah. We talked about it as a “thought experiment.”

S: “Are we supposed to be rooting for the snail?”

Rooting isn’t the point. The point is that this problem is supposed to convince us that motion is impossible.

Everyone: “This problem is not convincing us!”

That’s okay. Zeno had three other problems he used with this one to convince us, and people have been studying them for thousands of years. We might not get convinced today.

Students: “Can you show us those other three problems?”

If we have time. (I was doubtful)

After working on it a bit more, we left it unresolved as A, G, and S thought that there’s no way cheetah can win working within the system. R asked how much time we had left. Not much. He wanted to keep looking for loopholes that would allow the cheetah to win, but also agreed with  the others that it would be nice to spend our last few minutes on another activity.

Function Machines: Redefining the Domain

Let’s do a function machine.

R asked to draw the T-chart and G asked to draw the machine.

A: “What’s the difference between the T-chart and the machine?”

The T-chart is your organizational device, the machine is what you literally put the items in the domain into. For this machine, the domain is animals, so don’t draw anything that would hurt them.

A helped G draw the machine (notice the “in” and “out” locations), then students started suggesting animals to put into it.

“Put in narwhal.”  Out comes 0.

“Put in snake.”  Out comes 0.

“Put in kangaroo.” Out comes 4.

“Put in butterfly.” Out comes 6.


After students tried about six different inputs, A figured out the rule, but did not tell the others – instead they told the others what came out from each additional input. (If you’ve been reading along over the weeks, you may realize that this is the same rule that we worked on in the School House Lane group.)

Most of the discussion of this machine was about its domain (the rule about what you’re allowed to put into a machine). I repeatedly had to make the domain more clearly defined because at first, G kept putting in animals that I had never heard of.

Let me redefine the domain: the domain is not just animals, but animals that Rodi has heard of.

“Put in a tardigrade,” said G.

I’ve heard of that but know nothing about it.

Both G and A started telling me about it, but not what I needed to know to be sure of what number comes out of the machine.

Let me redefine the domain: the domain is not just animals that Rodi has heard of, but animals that Rodi knows something about.

“But you do know something about it; we just told you,” said the students. They were correct.

Let me redefine the domain: the domain is not just animals that Rodi knows something about, but animals that Rodi knew something about before arriving at Math Circle today.

Getting that precise seemed to resolve the issue (for now, dun dun dun duuuuun – imagine suspenseful music – foreshadowing of next week’s report!).

S was very focused on conjectures; she came up with many that involved the spelling, which were not the rule I was thinking. We talked about how wrong conjectures still help with reasoning.

A: “I could suggest an animal to put in that would make it easier to figure out the rule, but that might spoil people’s fun.”

I agree – why don’t you wait until next week to do that if everyone else agrees. (They all did.)

Spoiler alert: I will tell you all in the next report what the rule is.

Map coloring, continued

At the end of the session, I gave students printouts of the map they created last week, as they had requested. (But I had cut them out into rectangles instead of circles as the students had asked – sorry!) Everyone wanted 2-3 copies to work with at home. During class G drew borders on his more clearly with sharpies, since the printouts were a bit hard to read. You may want to try this at home. Please email me if you’d like me to send an electronic file with the map.


Zeno’s Achilles Paradox

I didn’t use the word paradox when I named and described the history of this problem. I wanted to avoid the spoiler that the formal name of the problem implies. I just called it things like Zeno’s Motion-Is-Impossible Problem, Zeno’s Race Problem, etc.

Looking at Domain Algebraically

 Our discussion about domains is somewhat analogous to a process whereby I tell you that the domain of my function is “numbers.” You might say “Put in 2.” I’d say “Out comes 0.5” Then you say “Put in 11” I’d say “Out comes 0.09.”  After a few more examples, you might say “Put in 0” and I’d say “That breaks the machine.” Or, I might refine the domain and say “No, the domain is actually all numbers but 0.” If you say “Put in pi,” I’d also say “that breaks the machine because  I’m using the calculator on my phone to save time and it doesn’t let me input pi.” So we need to refine the domain to all non-zero rational numbers. And so on. Maybe you’ve guessed by now that my function is . In this process, we’re developing skill in the 5th grade Common Core Standard for algebraic thinking of “analyze patterns and relationships” and “attend to precision.”

Collaboration Ramping Up

While the students worked Zeno’s problem, they explained everything to each other – why conjectures wouldn’t work, what is the definition of something, what makes sense as the next step, etc.  Usually at least one person can answer anyone else’s conjecture/question. They continue to draw everything – nothing on the sidewalk chalk “board” from me. At this point in the course, the group is coalescing into a good workplace team (as I think it’s called in the field of project management). All they need me for really is providing the interesting math problems to explore and to watch the time, or today, hand out rocks since people’s papers were blowing away. Or sometimes come up with systems for decision making/consensus, or noticing if someone is waiting a long time to talk. Otherwise, I just sit quietly.

(students tracking their work)

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