|TCCNS Course||MATH 1332: Contemporary Mathematics|
|UT Austin Course||M 302: Introduction to Mathematics|
Suggested Resources and Preparation
Materials and Technology
For the instructor: either a computer/projector or a blackboard/whiteboard.
For the student: a gameboard template may be helpful, but blank paper is sufficient.
This first exploration-based lesson required no previous experience or knowledge. In a university environment where students often add the course late, can't find their lecture hall, or have been away from math courses for multiple years, it is helpful to have these "puzzle days" where students are working through content that is relevant yet not critical.
Overview and Student Objectives
Students are briefly introduced to the concept of winning or "hacking" a simple two-player game and are tasked with playing a short two-player game called Dodgeball.
Students will understand that:
- In some games, one player or the other will have an advantage, meaning if they play with a particular strategy, they can always win.
Students will be able to:
- Play the game of Dodgeball as either player.
- Decide if a particular move was a mistake.
- Articulate verbally and in writing the reason why a particular move is strategic.
Imagine a two player game. Each player gets six turns, so a game consists of 12 total moves. Player 1 and Player 2 have two different game boards. Player 1 has a six by six grid and Player 2 only has a single row of six boxes:
- Player 1 fills out one horizontal row of their game board on each turn.
- Player 2 fills out a single box on their game board on each turn.
- Who wins? After all twelve moves are made:
- If one of Player 1's horizontal rows matches Player 2's row, then Player 1 wins.
- If none of Player 1's horizontal rows matches Player 2's row, then Player 2 wins.
The puzzle: can Dodgeball be solved?
Does Player 1 have an advantage, meaning there's way for Player 1 to always win? Does Player 2 have an advantage instead?
Game theory: the art of hacking games
When you give a step-by-step recipe for always beating a particular game, you are solving the game or "hacking the game." It's possible that you've played the simpler game Tic-Tac-Toe (or Naughts and Crosses, or X's and O's) where the objective is to get three in a row horizontally, vertically, or diagonally.
Back to Dodgeball: this video walks through a sample game where Player 1 wins. Can you guess where Player 2 might have messed up?
When you guess something that you are later going to try to argue, you are making a conjecture. Make a conjecture now!
Question: Do you think that Player 1 or Player 2 can always win if they play with the right strategy? Or does it depend on some other factor?
Exploring and expressing a solution
When students work this problem in a live class, a few things usually happen:
- Some people get stuck right away and don't know how to start. Sometimes this makes them feel anxious or upset. Or sometimes they are afraid to write things down because they don't want to be wrong.
- Some people have a sense of what to do but can't express it. If your conjecture was along the lines of "Do the opposite" then you may be in this group.
- Some people get partly there but have trouble seeing the full argument. If you said something like "Player 2 because they get to take the last turn" then you may be in this group.
Importantly, it is okay to be in any of these groups! It is a normal part of problem solving. One of the things you can do to be a better problem solver is to be mindful of your own reactions. Simply acknowledging "this problem is hard and I don't know where to start" may be enough to remind you that the thing to do is start playing around on paper, even though you know that the things you are doing will probably be wrong at first. Or perhaps you would prefer to work on brainstorming questions that would help you get more clarity.
Now the grand finale: how do we find a solution and how do we write it down in a clear and complete way?
In the game of Dodgeball, which player has the advantage and can always win if they play with proper strategy?
- Player 1
- Player 2
- Neither player
You are playing Dodgeball and you are Player 2. Your opponent takes their first turn: X O O X X X. What should you choose and why?
- O because there are more X's in what they chose.
- O because the first letter they chose was X.
- X because you want to do the same thing that Player 1 does.
- X because the first letter they chose was X.
What does it mean to solve a game?
- To solve a game you determine if one player can always win the game with a certain strategy and you give step-by-step instructions for always winning.
- To solve a game you must beat the other player repeatedly.
- To solve a game you calculate the percent odds of winning.
- To solve a game you determine if a player will always win the game no matter how they play.
Player 2 should look at the nth box of the nth row on their nth turn and choose the opposite symbol. This works for n = 1, 2, 3, 4, 5, 6.
Player 2 should choose O because the first box is X.
Player 2 should choose X because the fourth box is O.
Yes to all three. As long as Player 2 puts the opposite of whatever Player 1 writes in the nth box of the nth row, the strategy will work. This is because there is one box dedicated to each row that will ensure Player 2 has something different.
- Player 1 should use O O X O O and then either symbol in the 6th spot. That way, if Player 2 uses X in the fifth spot, then Player 1 can write O O X O X O and trap Player 2. If Player 2 uses O in the fifth spot, then Player 1 can write O O X O O followed by the opposite symbol they used in turn 5, box 6.
- Player 2 only really messed up on turn 2. They matched the O that Player 1 put in box 2 of row 2. The strategy indicates that you should always choose the opposite.
Mindset question. Responses will vary.
Describe in your own words an algorithm that Player 2 can follow to win Dodgeball.
You are playing Dodgeball and you are Player 2. Player 1 writes "X O O X O O" for their first turn. What do you choose for your first box and why?
You are playing Dodgeball and you are Player 2. Player 1 writes "X O X O O O" for their fourth turn. What do you choose for your fourth box and why?
Does the Dodgeball strategy still work for Player 2 if:
- Player 1 has seven rows of seven boxes and Player 2 has seven boxes?
- Player 1 has eight rows of eight boxes and Player 2 has eight boxes?
- Player 2 has 100 rows of 100 boxes and Player 2 has 100 boxes?
Here is a game in progress where Player 2 has made a mistake along the way.
- Give Player 1 instructions for beating Player 2.
- If you were Player 2, which turns would you have done differently and why?
In this introductory lesson, in what ways were you challenged today and why?