There’s More to Tic Tac Toe than You Know…

We welcomed two first time CAMI members to a meeting where we once again looked deeply at something that is so familiar, we take it for granted. Greg Fein’s exploration helped us to unpack tic-tac-toe and find the math. It turns out is an ancient game with roots all over the world, with perhaps something innately human at the heart of it.

Facilitator(s): Gregory Fein
Date of Meeting: March 8, 2018
Problem: · docx · url

Launch

Greg started by putting a familiar drawing on the board and asked us what came to mind:

We responded with “tic-tac-toe, hashtag, 9 squares, right angles. parallel lines…”

Greg then focused us on our first impression and asked us to remember the rules of tic-tac-toe:

  1. X goes first in one of 9 spaces.
  2. Then O chooses a space.
  3. X and O alternate turns.
  4. The game ends when there are 3 X’s or 3 O’s in a row or when all the spaces are filled.
  5. If all the spaces are filled and there are not 3 X’s or O’s in a row, then the game is a draw.

Greg told us the game is known by other names such as:

  • Noughts and crosses (England)
  • X’ies and O’sies (Ireland)
  • Three-Men’s Morris
  • it appears in Egyptian carvings from 1300 BC

He then asked us to write a question we have about the game. We came up with the following questions:

  • Is there always a best move, given the circumstances? What is the best square to start in?
  • How many possible appearances of the board (combinations) are possible?
  • Why X first? Does X have an advantage?
  • Why is it popular? Why has it lasted so long when it almost always ends in a tie? Where is the satisfaction?
  • Why so many ties? Is there a way to avoid draws? Is there a way to always force a draw?
  • Why is it 3×3? What if it was a different size? 4 x 4? 3 x 3 x 3?
  • Why 3 in a line? What if you said 3 adjacent winds? Or 2 in a row? Or 4 adjacent?
  • How many ways to get through the game to each ending configuration?
  • Is there  a way to avoid a draw?
  • What thought process or pattern recognition do you need to be competent? What age can you process it?

TEACHER WORK

Small groups chose questions to explore. While we were working, Gregory came around to see what we were working on. After a while, he shared questions on strips of paper to prompt us to explore related questions (see attachment above).

Eric, Tim & Spencer

For example, Greg started this group off with the questions: What is the best first move for X? What is the best first move for O? 

Tim and Eric began evaluating moves by how many paths there were to getting three in a row from any given square. They concluded by going in the center, it opens up 4 paths (vertical, horizontal and the two diagonals). Going in the corners creates three paths (horizontal, vertical and one diagonal). Going in the side spaces creates two paths (horizontal and vertical).

 

 

 

 

 

 

 

 

 

After they had looked at those questions for a while, Greg shared a tic-tac-toe board in the form of a tube. We tried to play a game and thought about how new possibilities for 3 in a row were created since the left and right side edges of the board were eliminated.

Nolan and Ramon

They explored the question, How many possible appearances of the board (combinations) are possible?

They were able to determine the possible configurations for X winning after 5 moves (3 moves by X and 2 moves by O). Nolan and Ramon determined that X could win in 8 different ways.

Then they considered the following possible ending state. With X winning across the top, they considered all the possible ways that two O’s could be distributed in the other 6 boxes. You might try counting all the ways now.

How many ways did you find?

Ramon and Nolan found 15 ways:

They saw these different ways the 2 O’s could be arranged as 5 + 4 + 3 + 2 + 1. They multiplied 15 by the 8 ways that the 3 X’s could be arranged and came up with 120 ways that the game ends in 5 moves. These 120 possibilities are a small fraction of the total possibilities of how the game could end, since the game could end after 6, 7, 8, or 9 moves as well.

(By the way, in our discussion, we realized that 120 is the same as 5! (5 x 4 x 3 x 2 x 1). Coincidence?)

Mark and Deneise

Mark and Deneise explored several questions, including 3-dimensional version of tic-tac-toe and a 4×4 grid.

 

 

 

 

 

 

They spent most of their time working backwards from different boards where x wins. The goal was to figure out different sequences that lead to arrangements where x must win.

They ended with the theory that X’s best starting move is in the corner and the goal is to try to force your opponent’s move.

For example:

 

 

 

 

 

By moving to the opposite corner, X forces O’s next move to the center square.

 

 

 

 

 

 

Once O takes the center, X’s win is inevitable, because X can (1) control O’s move while (2) creating two potential wining combinations.

 

 

 

 

 

 

At the end, they started to think about the problem from O’s perspective, wondering if O (or the person who goes second) can ever win or if their best hope is to draw.

Extensions

Greg recommended the resources and games at http://geometrygames.org, which include 3-D versions of tic-tac-toe, as well as tic-tac-toe on a torus!

Just for fun, here is a clip from the end of War Games (1982) where an artificial intelligence named Joshua is taught through the futility of trying to win at tac-tac-toe that there can be no winners in thermonuclear warfare.

 

In Attendance: Deniece, Eric, Greg, Stephanie, Mark, Nolan, Spencer, Tim

Programs Represented: CUNY Start, the Manhattan EOC, ParentJobNet, CUNY LINCT, CUNY Adult Literacy and HSE Program, Brooklyn Public Library, Bard Prison Initiative, The Adult Learning Center at BMCC, District 79

Respectfully submitted by Mark and Eric


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