The game of #TicTacToe is well known and has been played for ages. It’s a very simple game – the first person to get 3 in a row wins. But have you ever thought about the math underlying this simple game? This is what we will explore in our story #18.
Combinatorial Game Theory – is used to analyse the possible outcomes of tic-tac-toe.
A tic-tac-toe board is a 3x3 grid. If we were to fill in each of the nine spaces with a unique value (like 1, 2, 3, 4, 5, 6, 7, 8, 9), then we would have nine choices for filling the first slot, eight choices for filling the second slot, and so on.
Counting this way, we’d have 9*8*7*6*5*4*3*2*1 = 9! = 362,880 ways to fill the board.
But, in tic-tac-toe, we fill the board with X, O, or leave it blank. Thus, the total number of ways to fill the 3×3 grid is 3^9 = 19,683 – as every square will either be a X, O or blank.
The number of different combinations of filling the grid with 5 Xs and 4 Os is 9 choose 5 = 126. So, we get 126 (3×3) grids for the game tic-tac-toe.
Wait, that's not it. In these 126 boards- there few boards that when rotated and reflected are the same.
Thus, there are 23 unique boards which, when rotated and reflected, end up giving us all 126 boards.
Uff, so much math to get us 23 unique boards isn’t it?!
Sources:
<1> “The Mathematics of Tic-Tac-Toe” by Leslie N. Gruis in the Association of Old Crows.
<2> “Game Theory and Tic Tac Toe” by Intermathematics.
<3> Maths Problem: Complete Noughts and Crosses (Burnside's Lemma) by ‘singingbanana’ in YouTube.
<4> Google images: tic-tac-toe.
#gametheory #tictactoe #math #mathstories #mathnmovies #mathngames #mathnbooks #story18
Combinatorial Game Theory – is used to analyse the possible outcomes of tic-tac-toe.
A tic-tac-toe board is a 3x3 grid. If we were to fill in each of the nine spaces with a unique value (like 1, 2, 3, 4, 5, 6, 7, 8, 9), then we would have nine choices for filling the first slot, eight choices for filling the second slot, and so on.
Counting this way, we’d have 9*8*7*6*5*4*3*2*1 = 9! = 362,880 ways to fill the board.
But, in tic-tac-toe, we fill the board with X, O, or leave it blank. Thus, the total number of ways to fill the 3×3 grid is 3^9 = 19,683 – as every square will either be a X, O or blank.
The number of different combinations of filling the grid with 5 Xs and 4 Os is 9 choose 5 = 126. So, we get 126 (3×3) grids for the game tic-tac-toe.
Wait, that's not it. In these 126 boards- there few boards that when rotated and reflected are the same.
Thus, there are 23 unique boards which, when rotated and reflected, end up giving us all 126 boards.
Uff, so much math to get us 23 unique boards isn’t it?!
Sources:
<1> “The Mathematics of Tic-Tac-Toe” by Leslie N. Gruis in the Association of Old Crows.
<2> “Game Theory and Tic Tac Toe” by Intermathematics.
<3> Maths Problem: Complete Noughts and Crosses (Burnside's Lemma) by ‘singingbanana’ in YouTube.
<4> Google images: tic-tac-toe.
#gametheory #tictactoe #math #mathstories #mathnmovies #mathngames #mathnbooks #story18
