(January 15, 2013) As students trickled in, the kids immediately resumed debate over one of last week’s question, “What happens if an irresistible cannonball hits an immovable post?”^{1} The new kids, H and L, were now clued in, but the question was not resolved. Once everyone had arrived I introduced 2 puppets who came to help them with our questions: Waggy (a fox) and Penelope (a pig). The kids were very excited to find out why puppets had come to Math Circle.

I explained that Waggy and Penelope were from an island where every person is either a knight, who always tells the truth, or a liar, who always lies. First, the kids insisted that I clarify that Waggy and Penelope were just acting, then they asked, “Which one is which?”

“That’s what *I *was about to ask *you*,” I said. “Listen to what they say and see if you can tell.”

“Jack is taller than John, and John is taller than Jack,”^{1} said Penelope. This statement set off a huge debate.

“What if the second part of the statement refers to a different Jack and John?” challenged M. So Penelope refined the statement to specify that Jack and John remained constant throughout the statement.

“What if the second part of the statement occurred after time passed and John grew taller?” challenged L. So Penelope refined the statement again. (In retrospect, this would have been a good time to discuss the meaning of the conjunction “and.”)

“What if they are the same height?” challenged D. This challenge was harder to answer. The kids (totally independently, without me) debated whether this could be true. Actually, many were trying to convince D that this could not be true. They even demonstrated by asking kids of different heights to stand up. Finally, with a couple of final questions from me, everyone was convinced that Penelope is a Liar because the “Jack and John” statement was not true. I asked why it was not true. This was a tough question. The kids offered various restatements of their prior explanations. I was trying, unsuccessfully, to elicit a generalization. Either I was unable to phrase the question well, or the kids were not able to generalize at this point. Kids now started to doubt their conclusion.

“We have to make sure or else we’d be badly mistaken,” said M, concerned. I suggested that we shelve this for now and move on to Waggy’s statement.

“Charles Dogson and Lewis Carroll are the same person,” he said. After some quick clarifying questions, the kids agreed that Waggy is a Knight. Next, I put some statements on the board and asked the children to determine who said each, Penelope (the Liar) or Waggy (the Knight):

1. Shane Victorino is on the Phillies

“Knight!” announced C immediately.

“I’ve heard of him!” said D. “I think he *is* on the Phillies.”

“He got moved to the Red Sox,” said V, who was holding his hand high in the air. At this point, most of the kids were standing up.

2. Percy Jackson’s father is Poseidon

Upon hearing this statement, the rest of the class was standing, and many kids were jumping up and down and shouting. It seems that they were familiar with this character, and once I specified which Percy Jackson I meant, the group agreed that Waggy said this.

3. Percy Jackson is a real person

It turns out that not everyone was familiar with Percy. I told those who weren’t that he is the main character in a novel. After we briefly discussed the meanings of the terms “novel” and “fiction,” the kids easily attributed statement 3 to Penelope. I kept a tally of statement numbers and puppets on the board. I then expressed some confusion over attributing statement 2 to Waggy, since it turns out that Percy Jackson isn’t real. Immediately, E instructed me to write “in the novel” at the end of statement 2 to make it true for sure.

4. An irresistible cannon hit an immovable post

Yes, this one again. Student debate resumed as though I weren’t in the room. I considered leaving for a water break, but wanted to hear the conversation. “Maybe there is no answer,” suggested E. M proposed giving up. I asked H, who had been quiet, whether I should give a hint.

“Yes,” she said.

“Here’s the hint,” I said. “The same person who published the irresistible cannonball question, Raymond Smullyan, also published the knights and liars game, and both of those are in this book.” (I held up the book.) An enthusiastic discussion ensued. Kids questioned the quality of the hint, whether a statement in general could be a hint, and whether the hint was even true. Waggy repeated the hint, convincing folks of its veracity. Finally, the kids’ discussion moved to the exact wording of the question. I reread it. V honed in on that pesky little word “if.” The kids then realized that the question might have an impossible solution and that the question-asker might be a liar.

“Penelope,” the kids chorused excitedly, announcing that statement 4 is attributable to the liar.

5. All red objects have color

6. If an object has color, then it is red.

7. If an object is not red, then it has no color

Each statement (5, 6, and 7) provoked serious debate. Kids were still standing up, and again talking to each other, not to me. (Conversation that excludes the “leader” is what a Math Circle is all about.) Statement 7 was especially troubling. E agreed that it makes sense to negate both “red” and “color” to get a logical statement, but the statement is demonstrably false. Finally, all the statements were evaluated and tallied. Waggy had 2 and Penelope had 5. If this were a contest, Penelope won.

I gave the kids 3 more statements, and asked which of them both Waggy and Penelope could say at the same time:

(a) I won.

(b) I lost.

(c) One of us won and one of us lost.

We discussed, and got confused. “This is confusing,” I said, and both D and E reiterated. Most of the kids worked together, though, and figured it out. A few kids had stopped talking and started drawing by this last question, but they were still listening. Today had been another solid hour of hard thinking.

This Math Circle opened up a delightful can of worms (statement? true? negate? contradict? converse? logical?). Next time, we’ll continue to play with these worms, in hopes of moving closer to a solution to our Dodgson problem (“All Puddings are Nice”). But expect a short break from the non-stop hard thinking; we’ll have a math history interlude.

Thanks to Beth for the photos and to Rachel for the (as always) detailed note-taking.

Rodi

^{1} Raymond Smullyan, What is the Name of this Book, p8