(Spring 2026) This is a story about how a planned course changed course based upon student curiosity and excitement. What began as a course called “Problem Solving with Alice” ended up primarily questioning whether there is a relationship between counterexamples and proof.

Here’s the original course advertisement: Martin Gardner states, “No other books written for children are more in need of explication than the Alice books.” Yet many mathematicians love them. Why? We will look at some of the math problems related to the book “Alice in Wonderland,” many of which are based in logic. Our primary text will be Gardner’s “The Annotated Alice: The Definitive Edition.” We’ll then connect those logic problems to contemporary mathematics by solving related problems that were presented at the Math Circles sessions at the Joint Meetings of Mathematics from 2011-2025. We will discuss and use the problem-solving methods in “How to Solve It” (Polya) and “Avoid Hard Work!” (Droujkova, Tanton, and McManaman).

We did cover these topics but then joyfully digressed, while still keeping everything connected to Alice and Gardner.

Here’s what happened, as I described in weekly parent emails:

Session 1 (5/13/26)

“Name one thing you know about Alice.” I asked students to use the “mutual invitation”  discussion principle, in other words to say their answer (or pass) and then invite the next person to answer. I also asked them to not say a second thing until everyone else had a chance to speak once. This set the stage for our Math Circle as a conversation with everyone involved. (We did not use this discussion convention for the rest of the class.)

A student asked whether it’s okay to use a calculator in this course and in general. I said that using one is fine for repeated calculations that you already know how to do by hand. The students seemed to agree that this is reasonable.

“Do you think multiplication is boring?” Students refined my question to “Do you think the way multiplication is taught is boring?” Lewis Carroll, the pen name of mathematician/author/educator Charles Dodgson, found multiplication fascinating but its instructional methods boring. Then I asked “What’s the explanation of this quote from Alice: ‘Let me see, four times five is twelve, and four times six is thirteen, and four times seven is – oh dear. I shall never get to twenty at that rate’” (The Annotated Alice, p23). Students debated  this and came up with some explanations involving a pattern. They also formed a conjecture about how algebra/an equation might be used. See the attached photo for student work. We kept a running list of student questions.

We then did some set theory work by standing in rope circles. This introduced the concepts of elements in a set, the cardinality of sets (how many elements are in the set), and the empty/null set. The empty set was of particular interest to Carroll/Dodgson, as was the general concept of contradictions. Students then worked on some quotes from Alice that connects these two topic (p143).

“Will the rope circles would overlap?” asked the students. I explained that the Carrollian set theory we were doing today did not involve the overlap/intersection of sets, so we would not be using/making Venn Diagrams. I just wanted to use the term “Venn Diagrams” aloud to see whether it generated a reaction. It didn’t. But now the students have heard it, in case it comes up again.

Of particular interest to the students is the question “Is a blank map actually a map?” (p143). A heated student discussion led to the important mathematical skill of defining the terms in a question/problem. What exactly is a map? Can we agree on the definition? What actually qualifies as a definition?

“Maps are a human concept, they don’t just exist without humans creating them,” argued one student. I mentioned that in the field of math philosophy, mathematicians discuss whether math itself is a human concept. A few eyebrows rose, then we returned to the problem.

“Is a blank piece of paper a work of art if the person handing it to you declares that it is?” argued another participant. This student was using the problem-solving skill of asking a related question to help with the first. We then did some more of these Carroll/Alice questions that Gardner – the writer of the annotations in The Annotated Alice  –  refers to as “logical nonsense” (p143). Students helped to turn these quotes into questions that could be analyzed mathematically. All of the students argued their cases with great emotion. They cared so much about conveying their reasoning.

I showed students the books Avoid Hard Work and How to Solve It, two books about problem-solving techniques. I asked students what problem solving techniques they used, then connected some from the book to the techniques students had been using to answer the above questions.

We ended with a riddle posited by the March Hare (a character in Alice): “Why is a raven like a writing-desk?”

Session 2 (5/20/26)

We continued last week’s discussion topics of whether multiplication and/or its instruction is boring, and of Carroll/Dodgson’s take on this. Students role-played. One student played Dodgson (with a British accent); another played Dodgson’s student. The group ended up doing a multiplication problem with the surprising product of 444,444,444.

“When are we going to see one of his magic tricks?” said someone. “This IS one of his tricks,” I said. “More like corporal punishment,” said someone else. “Or just “punishment,” added yet another student. Or “math punishment” at least, said another. It was exciting that our students were participating actively and with curiousity the whole time and figured out how the trick works. I promised a more impressive magic trick next week. Here’s a link to the full exercise: https://fractad.wordpress.com/2024/12/30/lewis-carrolls-other-life/

Then we used Gardner’s annotations to Alice (pp70-71) to explore last week’s riddle (desk and raven). There was some interest in the work of famed puzzle designer Sam Lloyd, referenced in the annotations.

Finally, we played the game Bendo, which I learned at the Joint Meetings of Mathematics (JMM). Bendo is an embodied (physical math) version of the game “Zendo.” Both games involve pattern recognition, problem solving, and critical thinking. To solve the problem, students need to explore the unknown, make conjectures that will probably be wrong, and use incorrect information to come up with good information. The problems are inductive reasoning systems that builds from a small number of facts. (Zendo is a game designed by the game company Looney Labs and Bendo is a game designed by Simon Pfeil of the Angelo State University Math Circle.) Students explicitly developed the following math skills (with me reiterating the skills verbally, again and again):

• Be wrong on purpose to gather information
• Form conjecture, or pass on forming a conjecture until you gather more information
• Clarify your conjectures
• Don’t use a stated premise/definition itself as a conjecture
• Practice precision. You get a different answer depending upon whether the conjecture is “1 second” or “2 seconds.” You also get a different answer when it’s “at least” or “at most” or “exactly.”

Session 3 (5/28/26)

I performed a “magic trick, the coffee sleeve illusion, which I learned at the JMM from Tim Chartier. Students learned how to perform it and figured out the underlying math (concepts include geometric shape and parallel lines). I shared with students that Chartier reported using this illusion to teach his students to show their work, that just because you see it doesn’t mean it’s there/real.

I then demonstrated the “64 is the same as 65” chessboard illusion, in which an 8×8 and a 5×13 rectangle appear to have the same area. “What’s going on?” I asked students.

One conjecture: “It’s about the diagonals.”  

“Yes it is, but why?” (me)

“Is it airtight?” This question from another student was really the communication of a new conjecture: it might not be airtight.

“You are right, it is not airtight. But why?” (me)

 “If you add the heights not at the edge, it doesn’t quite total five.” (still another student)

“Correct, but why?” (me)

Students were not sure. I asked “Do you want a spoiler, or do you want to figure it out on your own?”

“No, no spoilers!” (at least one student)

“I’ll give a hint: you can model this problem with graph paper and scissors. For those of you who want to see the math solution, you can watch a video on youtube that I will send to your families, not to you.” (I later emailed the link home: https://www.youtube.com/watch?v=Gh3BRfXwmsE&t=7s) Then we discussed the provenance of the problem.

I said, “Before class today, I found this problem by googling ‘Chessboard math problem from Alice in Wonderland.’ I learned the problem, including the reported plot surrounding it. Then I checked The Annotated Alice and discovered that the problem is not in the original Alice books! It seems that Carroll wrote it somewhere else. Should I believe everything I read online.”

“No!!!” (students)

Then I reported that I read that, just like Chartier, Carroll/Dodgson used this illusion to teach students to not believe what they see. At least according to someone on the internet. I repeatedly used the word apocryphal, one theme of the day, throughout the session. (No one had heard that word before.)

By the way, puzzlemaker Sam Loyd came up in the group again. Turns out he (not Carroll) is the person who gets the most credit for making math chessboard problems famous.

We moved onto the famous “Wheat and Chessboard” problem – a problem with an apocryphal origin story. Most of our participants had heard some version of the problem before. None, however, had worked out the math in the “Would You Rather” version, so we did:

“Would you rather receive $1,000,000 or receive 1 grain of wheat on the first square of a chessboard, 2 grains of wheat on the second square, 4 grains of wheat on the third square, continuing this pattern until all 64 squares have wheat on them?”

Some interesting math content came up:

• very large numbers,
• estimation when dividing with numbers that big, 
• exponents,
• exponential versus linear growth,
• converting grains of wheat into dollars – as a mathematical problem of unit conversion, and
• converting grains of wheat into dollars –  as a practical problem of hiring a labor force, stacking the pile of grains onto those tiny squares, etc .

In Alice in Wonderland, Carroll refers to one of Zeno’s paradoxes, the one about Achilles and the Tortoise. Some students were familiar with Zeno’s dichotomy paradox, the one about never arriving, but no one was familiar with the details of the Achilles paradox. Students acted out the mathematical paradox to understand the reasoning. Of course, the result is counter-to-fact. “This is not realistic,” they argued.

“But can you actually disprove the logic behind Zeno’s reasoning?” I countered. We discussed why it is a paradox, and what the word paradox means. I promised students that next week I’ll share the actual reference from Alice.

We finished up with a short version of the game Mathematical Zendo, similar to Bendo from last week. The rule centered around square numbers, but the discussion ended up being about (1) the student question “can a prime number be a square number?” and (2) my question for students “can you use counterexamples alone to prove something?”  

Session 4 (6/3/2026)

1. After discussing the character the Mock Turtle, students read aloud the Alice quote in which the Mock Turtle references the Tortoise (p96, The Annotated Alice). According to the annotations (which we read), the Tortoise is a reference to Achilles and the Tortoise. This is connected to Zeno’s dichotomy paradox explored in class last week. “So much of Alice is actually math!” the students and I marvelled. And we had fun with doing British accents and playing with Carroll’s puns.
2. We played the game Exploding Dots, in which students need to figure out a rule. In the game, the rule is about place value and different number bases, but students don’t know that in advance. Today, students figured out that the first rule used base 5, then they translated between base 5 and base 10, discussing place value to make sense of it. Some students were already familiar with different bases and shared some of their favorites, including 12.
   
   
   
   
3. We revisited the multiplication problem from our first class: “Let me see, four times five is twelve, and four times six is thirteen, and four times seven is – oh dear. I shall never get to twenty at that rate.” (The Annotated Alice, p23). One conjecture in the annotations is that this is a problem about different bases; the students figured out how that theory works mathematically.
   
   
   
   But: did Caroll intend that? We had a nice discussion about what and what not he may have intended and whether you can know an author’s intent. I let students know that they can find many “rules” for integer sequences using the OEIS. Then I asked “What would have been Carroll’s reaction to the theory about the quote being about the pattern with number bases?”
4. We did more Exploding Dots, this time in base 2. “Binary!” said some students, who did some converting between base 2 and base 10.
5. I read aloud the scene in which the Red Queen turns into a kitten and Alice talks about purring (The Annotated Alice, pp267-269, from the Alice in Wonderland sequel Through the Looking Glass). “What does this have to do with math?” I asked. “It’s binary!” a lot of students shouted. Then students discussed some questions I posed based upon the annotations on p269. “What is needed to transmit information? What is needed to record information? Is one-value logic sufficient?”
   
   _{^{Alice and The Red Queen (Wikipedia)}}
6. Our biggest discussion of the day was about counterexamples versus proof. Specifically:
   • “What are the limitations of counterexamples when you want to prove something?” Students brought up the example of planes in World War 2: the planes without damage are the only ones that came back. Students posited that the survivor bias is a logical flaw analogous to the limitations of using counterexamples for proof. I wasn’t aware of the survivor bias – I love that I’m always learning from students – so couldn’t quickly ask the students leading questions to determine for sure whether the two lines of logical reasoning are identical. That’s a question for another day or to discuss at home.
   • “Can you prove anything in the game Zendo?” (We played a quick round – with a student leading this time – to test whether you can. Unfortunately, we really needed time more time to use the game to make this point.)
   • “If you can’t find a counterexample, is the conjecture definitely right?”
   • “Is AI essentially playing Zendo?”
   • I shared Lance Eliot’s Forbes column Open AI’s Breakthrough on Famed Math Problem Actually Proves that Using AI to Find Counterexamples Is a Smart Strategy for Everyone. “What biases might there be in this piece of writing?” I posed the three examples in the column as questions:
     • How can you use a counterexample to disprove the car mechanic’s claim that they require a “highly specialized and expensive tool” to fix your car?
     • How can you use a counterexample to disprove a friend’s claim that they have never posted about unicorns?
     • Is it worth discarding a claim that just has one extremely-ridiculous counterexample? (We extended the fraud-detection example to an edge condition that involves dressing up as a moose and living 3 years as a moose on an Arctic island.)
     • Is the answer to the previous question different in math versus the business world? (Students were quite surprised to hear that yes, in math, you do just have to reject the claim/theory/conjecture/plan.)
     • What biases might there be in this piece of writing?
       
       ^{The King of Hearts (Wikipedia)}
7. In Alice, the King refers to “Rule Forty-two” (p120). Why do you think that the first Rule referred to in the book is #42? Students had lots of great conjectures. Then I read the annotation to them. Turns out that “the number forty-two held a special meaning for Carroll…” Then I asked each student “Do you have a favorite number? And why?”
8. After class today and last time, one student has been teaching me the mathematical finger game “Chopsticks.” Today we came up with extensions of that game. What if there are remainders when you tap with more fingers than a person has room for? Also, what if, when you tapped, you transferred your number of fingers, how would that work if zero were a position?

Then our time was up. Since this planned 6-week course happened in only 4 weeks, we never got around to talking about reversals, a huge topic in Alice. We could do another whole course just on reversals. We also did not get into doing actual proofs, another topic for a future course. Also, we never got around to explicitly talking about Skepticism, although we did discuss it in different words. My understanding is that Carroll was not a Skeptic in the modern sense of the word, but instead a Rationalist. BTW, what I’m talking about when I’m talking about Alice/Carroll is really Martin Gardner, “considered the founding father of the modern skeptical movement” (gathering4gardner.org).

For more info in general about the themes of this course, you and our student might enjoy this article: The Mad Hatter’s Secret Ingredient: Math