Day: October 20, 2008

Mathclub Hat Problem

One of the students in Math Club recently put his own twist on the age old hat question: Assume you have n people, each of whom has a red or green hat put on them. They each don’t know what color hat they have on. However they can look around and see everyone else’s hat.

After getting to spend some time in a room looking at everyone else and their hats (they may not communicate in any way), they are each placed in separate cells and asked to say whether they have a red hat on, a green hat on, or “pass.” Everyone wins the game if at least one person says their right hat color, and no person messes up their hat color. Everyone loses the game if everyone passes, or if anyone says the wrong hat color.

The question is: what is the strategy that those wearing the hats should come up with beforehand? And can you come up with a formula giving the probability that n people win with that strategy?

To make the problem clear, let’s examine the three person case. The possible combinations of hats are:

RRR | RRG | RGR | GRR | GGR | GRG | RGG | GGG

The best strategy we could come up with is to say: if you see two opposite colors (a red and a green), say “pass”. If you see two hat of the same color, say you’re wearing the opposite color.

So you’ll lose with RRR and GGG (everyone sees two of the same color, so everyone will say the opposite color).

But you’ll end up winning with RRG, RGR, GRR, GGR, GRG, and RGG. Let’s look at RRG to explain. The person wearing the first red hat sees a red and green hat. So that person says “pass.” The person wearing the second red hat sees a red and a green hat. So that person says “pass.” The third person wearing the green hat sees a red and a red hat, so that person says “green” and is right! So RRG is a winning combination. Similar arguments follow for the other five.

Since there are 8 possible combinations of hats, and 6 of them have a winning strategy, there are 6/8 chances that everyone will come out a winner! (That’s a whopping 75%!)

So we’ve been investigating what the strategy will be for n people wearing red and green hats. So far, we’ve done pretty well. In fact, we’ve even gotten Pascal’s Triangle involved, which is always great.

And there seems to be a consensus among the students (though no proof yet) that if you have any even number of people playing the game, say 8, you can actually get better odds of winning if you ask another person to join in (so you’d have, say, 9 people playing). That seems totally counter-intuitive, that adding an extra person to play the game with you would lead to a better chance of winning. So if they’re right, I’ll chalk this problem up to a win.

PS. We did talk about the Bloxorz problem for two weeks, but students grew bored and tired of it. I still think it’s a great problem. Maybe one year a student will want to do an independent study on it, and ask me to be the adviser to the project.

I survived Parent Night

Even though I was sick — aching and tired — I survived our Parent’s Night last week. I think it was pretty successful, even though I was foiled a few times by parents who tricked me into talking about their children. (I keep a general policy not to talk about individual kids at these events; it’s a time to share what we do in the classroom, introduce myself to parents, and to tell parents what their kids can do to be successful — and how they can help their kids be successful).  I’m still baffled on how they tricked me. I totally blame my weak immune system for my inability to steer conversation away from talking about little Jane or little Jake.

The night had one tragedy — when SmartBoard didn’t work for one of my classes. I knew this would happen; the same thing happened last year. I even told everyone I knew it would happen again. However, luckily, it happened when talking to the parents of my four student multivariable class. The parents all knew each other — these kids had been in the same classes for gosh knows how long — and so we just gathered ’round my laptop and I showed them what sorts of things go on in our class.

Some observations:

(1) Parents tend to start off the night stoic. Their faces won’t let anything through. Cracking jokes or smiling doesn’t phase them. As the night progresses, however, the parents get more laid back, and by our last class, parents have let their guard down. I swear I heard a few of them laugh in my last presentation. I’ve asked other teachers in my school if they have noticed this phenomenon, and it seems pretty universal.

(2) Parents like to introduce themselves (great). Parents like to follow that up by asking “how’s my kid doing?” (not great). First of all, as I said, I don’t like to talk about individual students. Second of all, who is your kid again? Believe me, unless you say “we’re the parents of Joe Schmo,” every time you meet me, I’m not going to know who you are.

(3) I realized I go into these nights actually expecting some gratitude from parents. And when I didn’t get it from more than a handful of parents, I felt a little slighted. Am I a bad teacher for needing those bits of affirmation? I don’t know. But I can’t help how I feel, and that’s what I felt.

(4) One point I made to almost all my parents is the basis for how I approach designing any class: I try to get them to do work which I think is just beyond the level that they think they can do. Of course, I’m not always successful with this, but I do try to push my students just past their perceived limits. Gauging their limits is tough though. I’m doing a really good job with this in Multivariable Calculus this year, but at the moment, I don’t think I’m pushing my Algebra II or Calculus classes enough.

With that, I’m going to eat an apple, and get me to bed, and hope to be ready tomorrow to embark on yet another week.