## The Mathematics Of Game Shows

Judging from his slides, it looks like Bowen Kerins produced a corker of a presentation at NCTM last week, running through the math inherent to game shows across several genres and decades. Among other revelations, he’ll help you calculate the best location for dropping your Plinko puck.

2011 April 19. Dave likes the problem:

In the past, I’ve definitely understood the idea of compelling WCYDWT problems, and been drawn to know the solution. This was almost on a new level. There was nothing as important in my life as solving the Plinko board, right now. It’s like math crack.

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. More here.

1. #### Bowen Kerins

April 17, 2011 - 8:47 pm -

Thanks, Dan. There’s a slide in there (it’s two before the one you just screen-shotted) where I realized afterward that it could have been a WCYDWT video…

Player walks to the top… drops a Plinko chip… it’s heading toward the 10000… then FREEZE right at the spot where it can either go 10000 or 0.

Probably, if I’d done that, the question of “how much is that chip worth *right now*” would come out more easily and quickly, and I wouldn’t have to explain the rules of Plinko either.

Guess we’ll have to do it again next year. Is that kosher, submitting the same NCTM talk title two years in a row? What other games make good math problems? (A friend’s been working on the “when is it best to stop and pass” logic for Press Your Luck for a long, long time.)

2. #### Steve Thomas

April 18, 2011 - 5:00 am -

Robert B. Davis came up with a game as part of the Madison project back in the 60’s. It was a Tic-Tac-Toe with a twist, you don’t tell the kids the rules. He had teachers use it as a warm-up for classes. On the board you draw a 4 x 4 grid of dots, then ask the kids for two numbers. Then you start counting from the zero on the dot in the lower left corner, the first # the child gave being the X coordinate and the second # Y coordinate. You count pointing at the dots as you go and if they give a # greater than 4 you say “sorry off the board”. The first team to get three in a row wins. You can extend it to negative numbers and expressions. The power is kids are motivated, they figure it out and they learn about cartesian coordinates without you having to tell them anything. I wrote about it in more detail here: http://mrstevesscience.blogspot.com/2009/12/taking-tic-tac-toe-to-next-level-if-you.html

3. #### Anthony Purcell

April 18, 2011 - 5:11 am -

I wish I could have seen this presentation. I’m going through the slides and there is a lot of information here! Thanks for sharing!

4. #### Dan Meyer

April 18, 2011 - 8:11 am -

Bowen: Player walks to the top… drops a Plinko chip… it’s heading toward the 10000… then FREEZE right at the spot where it can either go 10000 or 0. Probably, if I’d done that, the question of “how much is that chip worth *right now*” would come out more easily and quickly, and I wouldn’t have to explain the rules of Plinko either.

“How much is that chip worth right now?” is a question that might occur naturally in the mind of a math teacher. In the mind of a math student, though? I have a hard time seeing any question being more compelling or occurring more naturally than “where should I drop the puck?”

Right?

The idea that a puck might be worth more or less in an instant as the player moves it back and forth across the top is awesome and a necessary revelation in the course of the problem. But the best lever for that revelation is probably the question that’s inside the mind of the player at that exact moment.

Bowen: Guess we’ll have to do it again next year. Is that kosher, submitting the same NCTM talk title two years in a row?

Beats me. My mentor, Allan Bellman, says no. At minimum, alternate yearly, he says. I just applied to CMC-South, though, and the guidelines said something like “If you’ve given the same presentation two years in a row, it’s time to probably move on.” Me, I tend to do an evolution of the same shtick year after year. It’s bizarre to look back at my presentation from 2007 which is like a Cro-Magnon version of what I did at TEDxNYED in 2010. So it goes.

5. #### Dave

April 18, 2011 - 8:37 am -

The graphic alone was so compelling that I stopped everything I was doing to solve it myself. I printed it and filled in an additional row and a half…but I was so impatient that I switched to Excel (had to alternate rows of formulas).

In the past, I’ve definitely understood the idea of compelling WCYDWT problems, and been drawn to know the solution. This was almost on a new level. There was nothing as important in my life as solving the Plinko board, right now. It’s like math crack.

6. #### MathZombie

April 18, 2011 - 9:04 am -

I saw this session. It was fantastic. What I liked is that I could fit these problems/discussions in whenever we have a spare day/time. Fitting the commercial breaks in and prizes was also great. Thanks for s good session Bowen.

7. #### Karim

April 18, 2011 - 5:29 pm -

“But the best lever for that revelation is probably the question that’s inside the mind of the player at that exact moment.”

If a student comes up with the question on her own, great. But I think a prohibition on exogenous questions is a bit too strict. Isn’t part of a teacher’s job to sometimes ask questions or pose scenarios that wouldn’t have been asked otherwise, either to keep the train moving or to upgrade it to another track altogether?

Dan, this isn’t too different from our conversation re: whether expected value is too abstract. I agree that students may not come up with the concept on their own, just as I agree that they might not come up with the Pythagorean Theorem on their own. Insofar as mathematics is both a tool for solving problems as well as a history of human thought, though, I think it’s perfectly fine if teachers at times say, “We as a species have confronted this question before, and here’s how we approached it.”

I appreciate the prioritization of authentic student-led inquiry. At the same time, I also think that “Oh, I hadn’t thought of that” can lead to great things.

8. #### Jason Dyer

April 18, 2011 - 6:14 pm -

I think the “how much is it worth at this moment?” question given straight might be a little odd to the student, in a confusing-math-is-hard way. Perhaps more intuitive would be a phrasing like “if you could pay money to play with the chip right at the spot it is now rather than at the top, how much would you be willing to spend?” Or even better, set up a dirt simple version of the game with only the left-right choice between 0 and 10000, and then ask what they’d be willing to spend to play the game.

9. #### Dan Meyer

April 18, 2011 - 6:58 pm -

Karim: If a student comes up with the question on her own, great. But I think a prohibition on exogenous questions is a bit too strict. Isn’t part of a teacher’s job to sometimes ask questions or pose scenarios that wouldn’t have been asked otherwise, either to keep the train moving or to upgrade it to another track altogether?

No one’s saying you can’t start a lesson by ignoring the question everyone is thinking and asking another. It just seems like an unforced error to me.

Assume both lessons get to the other side of Expected Value Mountain. One route follows the direction of student momentum towards the question “where would you drop the Plinko punk?” The other forces the student body to dial down or delay their momentum and follow a direction of the teacher’s choosing.

Both routes lead to expected value. I expect that one of those routes will be more accessible to more students than the other.

10. #### Bowen Kerins

April 18, 2011 - 8:16 pm -

While I think the clear question to answer is “where should I drop the Plinko puck” (“drop the Plinko punk” is a line for Dirty Harry), I think it’s too wide a gulp to work right away. It’s a tough question with no obvious means of solution and several potential dead ends. (The best kind.) I think it would be good to let students loose on the big question for 5 or 10 minutes or so before refocusing them on a simpler question.

The simpler question — how much is a Plinko chip (attend to precision!) worth teetering between 0 and 10k — is therefore a little forced. But it’s almost necessary to ask, then answer, that question to realize it can be used to answer the big one. And it’s also necessary to spend time on the big question to realize how difficult it is without a direction of focus.

The order of game presentation was deliberate, too: Deal Or No Deal comes first in the talk, because all its mathematics is about expected value of an incomplete game. Then when “0 or 10k = 5k” concept comes up in Plinko, it’s not foreign since it’s the same analysis one would do on a “10k or 400k = 205k” in Deal.

I haven’t tried this with students, so someone get out there and try this! One big pedagogical question is how, or if, you make students complete the board’s numbers: I say do it. Kids will figure out any shortcuts, such as the fact that they only need to fill in half the board at each step, then be able to explain why that has to be true. And then it’s they who construct the eventual result, instead of it being handed to them. (I’ll admit to violating this in the NCTM talk, but I don’t think there was any way around it in the time frame, and I’d rather show and play another game than have teachers crunching numbers for 10 minutes.)

Dan, I apologize for giving a contestant a selection of fine cheeses, when obviously I should have given them to you for further microwave experiments. They were wedges, though…

11. #### Dan Meyer

April 18, 2011 - 8:45 pm -

Bowen: While I think the clear question to answer is “where should I drop the Plinko puck” (“drop the Plinko punk” is a line for Dirty Harry), I think it’s too wide a gulp to work right away. It’s a tough question with no obvious means of solution and several potential dead ends.

The point of “where should I drop the Plinko chip?” is that everyone has an opinion. You get everybody in the class to put a guess down on paper. You create a quick histogram above the Plinko board in the video you’re projecting on the whiteboard. You get two opposing students to explain their rationales.

Then go wherever you want with it. I may be overlooking a huge hole in the ground I’ve carved out for myself, but I’m fairly sure that if you want the start of the problem to be generative and accessible for as many students as possible, you start with the most perplexing question.

12. #### Matthew Bardoe

April 19, 2011 - 10:06 am -

Another good game, at least there was time it was my math crack is Risk. What are the chances that a 10 army offense is going to defeat a 5 army defense. That one had me going for hours. Then what is the expected value of the armies that remain. What are the probabilities of all the possible endings. These questions are very similar to the ones that I am asking now about Plinko.

13. #### xander

May 16, 2011 - 4:05 pm -

Sorry to come the party a little late—I just found this website today.

It seems to me that Meyer’s question “Where should I drop the puck?” and Bowen’s question “What is the puck worth right now when it is teetering between 0 and 10K?” are really the same question in a simplified context.

Personally, if I were confronted with a Plinko board, my first idea would be to simplify the problem as much as possible. What if there is only one place to drop the puck, and only two possible prizes? This model might get me thinking about the “value” of the puck when it is in any particular position (the question asked on slide 35).

Unfortunately, that doesn’t really get me to the “interesting” question, which is “where should I drop the damn puck?” So let me make the model a bit more complicated. Instead of having only one place to drop the puck, maybe I have 8 positions, and 9 possible prizes in the row just below. Then I can ask the question, “where should I drop the puck?”

Of course, as I work out the solution to this problem, I am generating the expected value of the puck just before it finally reaches some prize—working backward as Bowen suggests, and finding “instantaneous” values for the puck. On the other hand, I am also answering my main motivating question, which is the one that Meyer suggests, only in a much more restricted scenario.

Am I missing something important?

14. #### Bowen Kerins

May 16, 2011 - 7:36 pm -

It’s tough, because the things you add or subtract from the problem need to feel natural. I like the concept of a much shorter board as a natural change: a full board with only 1 or 2 rows of drop feels more natural than limiting the game to one drop zone for two prizes — and partially because of what you said: it doesn’t lead to the same interesting question that the original game has.

In doing this with students, I have usually approached this problem while in a unit on combinatorics, so I simplify the game in a different way — there is one entrance and 9 prizes, with 8 rows in between. Then we calculate the expected value of the puck in several ways, including the use of Pascal’s Triangle and combination counts. That’s still a pretty bitter simplification, and also doesn’t help kids with the “where do I drop the puck” question. But I start there, make sure that understanding is solid, then expand the problem back to its origins. (This is also the way the problem is presented in our Precalculus book, where it sits in a lesson on expected value.) If I were teaching this year I would first expand the problem back to a one-row or two-row drop with a full prize bank, then work back to the full board.

It is possible to solve the full Plinko problem using Pascal’s Triangle, but I was saving that method for next year’s talk. The difficulty lies in the board’s edges — Pascal never has a Plinko chip hit the side wall…

15. #### xander

May 16, 2011 - 7:43 pm -

Oddly enough, as soon as I finished posting, I started playing with a Pascal-like triangle. It wasn’t making any sense until I stopped reducing to lowest terms. ;)

As another thought, is it bad that my next impulse was to attempt to create an infinitely high Plinko board, and model the entire thing with a 17 state Markov chain?

16. #### Bowen Kerins

May 16, 2011 - 7:51 pm -

Nope, I did that to determine whether or not, in the long run, it would matter where you dropped the chip on a huge board. (I’m being less helpful, so I’m not telling.) Plinko’s not the only Price is Right game where a Markov chain is very helpful in analysis.

Trying to do it “like Pascal” runs into issues in terms of whether you use probability or frequency count. An article once appeared in Mathematics Teacher about Plinko and it got the wrong answers because it was doing a frequency count, then at the end computing the probability as (frequency) / (total outcomes). This would normally be right, but only when the outcomes are all equally likely. And for Plinko, they’re not, because a puck off the wall has only one option… at twice the probability of the usual options.