## THE BUTTERED TOAST PROBLEM

(2/15/2022) After last week, I realized that I have gotten away from an old, important practice of mine in facilitating Math Circles: listing assumptions, questions, and conjectures on the board/slide. My goal for today was to recover this. Since O was absent last week, I had the other students explain this problem to him as I acted as secretary on the slides. (We hadn’t finished solving the problem last week, but the students did finish writing it.) Also, I wanted to get O’s input on the problem before getting into answering it.

*The problem: You’re locked up in the basement of Leshy’s cabin (but not in a bad way). You have 5 arcade tickets that you can cash in for prizes or cash once you get out. You may play a game that costs 1 ticket to play. If you win, you earn 3 tickets. If you lose, you get nothing. You have a 50% chance of winning. Should you play?*

Me: Do you have any questions, O?

O: Yes! I want to know what the game is.

Me: We never stated that last week. I had something in mind last week, but I never mentioned it. I didn’t think it would work with the group’s stated assumption that the problem takes place in a basement.

Students: What were you thinking?

Me: Don’t you prefer to choose an assumption that makes sense about what the game is? (Last week, the students stated assumptions, thereby creating some of the premises of the game.)

Students: No. (Huh! Usually the students want to write the problem. I like to think that they didn’t want to write the problem – specifically the context – this time because they wanted to get on with the math, but I’m not sure. I wish I had asked.)

Me: Okay. I was thinking that the game is this: you stand on a high platform (as high as a bungee jump) and drop a piece of buttered toast off it. If it lands butter-side-up you win, and butter-side-down you lose. Assume you have a 50-50 chance of either side. (I expected arguments about this probability and had even read up on it, but no one questioned it.)

K: Let’s just use that. (Everyone agreed, and I edited the problem to state that Leshy’s basement has a very high ceiling.)

Students were still using their gut feelings (math intuition plus emotion) to answer the question. I asked “What would the outcome be if you played the game one time? Two times? Three times?”

I took periodic polls to see if the group had come to a consensus about the answer to the problem (should you play?). Eventually they all said yes, after simulating playing 5 times. Some of them doubted that they would play if they had only one chance to play. I chose not to encourage more work on this problem because the students really, really wanted to flip their coins, which I had asked them to bring. I did ask them to name the problem – something that normally students are excited to do, but not this time. K suggested calling it “The Buttered Toast Problem” and everyone concurred.

## THE LAW OF LARGE NUMBERS

Before starting the coin tossing, I wanted to talk about it for a moment, but the students just wanted to start flipping. Finally I was able to get in my questions:

*What is the probability that you get tails if you flip once?**If I just flipped a coin 10 times and got tails each time, what’s the probability of getting tails on the 11*^{th}flip?

The students said that the probability of tails is 50% if some very specific requirements are met (see below) – I also introduced the idea of a “fair coin.” At first the students said there were pretty low chances of getting tails on the 11^{th} toss after 10 tails, but then realized it would be 50% because of the probability of independent events. Now time to toss!

I asked them to first flip their coin one time and record (on a Google sheet) how many tails they got. Then to do it twice, then three times, etc. “What does tails mean?” asked Z. The others explained. Then the flipping and recording began. “What number should we go up to?” asked O. I suggested 12. After everyone had done trials up to 12 flips, we looked at the data.

The students did notice that with the smaller number of tosses, there was a lot of variability in the percent of tails results, but that it got closer to 50% with more tosses (the Law of Large Numbers).

“What I’m interested in,” said K, “is what the total is.” I totaled up the column with the sum formula so that we could immediately see that 312 tosses produced 150 tails – 48%. We were all impressed with the math.

N also pointed out how impressive spreadsheets are at doing calculations – almost magical! Some of the students had never seen spreadsheets in action before. At one point when the students were doing the experiment and recording their raw data, N had typed a number over the percent formula in the row and said “I broke the spreadsheet!” I fixed it in less than a second, again exciting to students.

I asked for predictions about how many tails you would get if you tossed the coin a very large number of times. Students suggested the thought experiment of tossing it 1,000 times, 50,000,000 times, and 2,020 times. They predicted 50% tails each time based upon the Law of Large Numbers.

Me: How does this relate to the Buttered Toast Problem?

Students: The more you do it, the more stable the results become.

K: There’s a name or rule for that.

Me: Yes, the Law of Large Numbers.

## DIMINISHING MARGINAL UTILITY OF MONEY

Me: Would you eat mud for a million dollars?

Students: How much mud? Would it kill you?

Me: A pie pan full of mud, and it would not kill you.

Everyone but N said yes. N said he would not eat mud for any amount of money.

Me: Would you eat it for a *billion* dollars?

Everyone but N said yes. N said he would not eat mud for any amount of money.

Me: Would you eat it for a *trillion* dollars?

Everyone but N said yes. N said he would not eat mud for any amount of money.

Me: What if you already had a billion dollars. Would you eat the mud for a million dollars?

K: That depends. Do I have a source of income?

Me: Yes.

K: What is my salary?

Me: A million a day.

Now everyone said no, they would not eat the mud.

Me: But a few minutes ago you said you would eat it for a million!?

The students explained why they wouldn’t, which I told them is called the “diminishing marginal utility of money” in economics. (In other words, context matters – a key idea in the mathematical discipline of Category Theory, which I did not talk about.)

## DECISION THEORY VERSUS GAME THEORY

Me: How is the Buttered Toast Problem different from the Pirate Problem, the Prisoner’s Dilemma, and the Triple Dominance Game?

Students: Random chance. It’s about what you would do, not about figuring out what some made-up people would do

Me: In Decision Theory, one actor is making a choice. In Game Theory, multiple actors whose choices affect others.

## FUNCTIONS WITHOUT NUMBERS

I used James Tanton’s material in his book about function theory to present functions without numbers, which the class had requested last week. I presented some function machines (input and outputs) and the students, as usual, were supposed to figure out the rule. We ended up delving into formal logic as well as function theory.

My original goal in doing functions without words was to get into the concept of the actual definition of a function beyond the idea that it’s a rule. That there is only one output for each input. That they can be mapped. That there is a notation system. That some functions are not well-defined. Again, time to let go of my expectations! I feel so privileged to be able to teach in a way that allows me to follow the students’ curiosity for some very deep learning.

## PEDAGOGY

Buttered Toast: My original goal for the Buttered Toast Problem was for the students to lead themselves to discovering the mathematical (probability) concept of expected value. These students are so interested in the application of math to the behavioral sciences that I thought we could progress into a lot of different things with expected value. Didn’t happen. (Had we delved into “expected value,” the answer to “should you play?” would be mathematically yes.) At first, I couldn’t figure out why these students weren’t excited by this problem; I had done a similar problem with older students – ages 13-15 – and they had come up with this concept and solution.

These students are younger, ages 11-13, and haven’t had the same math experience, much less work with percents and probabilities compared to that older group from the past. OTOH, this current group has been doing game theory for six weeks now, which the other group had never seen, so of course the math will go differently! (These students, unlike the older ones a few years ago, are using the language of game/decision theory, i.e. decision factors, rational actors, etc., naturally and automatically. Note to self: I need to let go of my expectations more.

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