**Problem 5.** Can tic-tac-toe be completed (5 ones and 4 zeros in $A$) so that $\text{rank}(A) = 2$ but neither side passed up a winning move?In this problem, we are asked to consider a tic-tac-toe game that is filled with five '1's and four '0's such that the matrix representation $A$ has a rank of 2. Additionally, it's specified that neither player missed an opportunity to make a winning move throughout the game. This implies the game must end in a draw. The challenge is to determine if such a scenario is possible.

Answered: Problem 5. Can tic-tac-toe be completed…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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**Problem 5.** Can tic-tac-toe be completed (5 ones and 4 zeros in $A$) so that $\text{rank}(A) = 2$ but neither side passed up a winning move?

In this problem, we are asked to consider a tic-tac-toe game that is filled with five '1's and four '0's such that the matrix representation $A$ has a rank of 2. Additionally, it's specified that neither player missed an opportunity to make a winning move throughout the game. This implies the game must end in a draw. The challenge is to determine if such a scenario is possible.

Expert Solution

Step 1: Introduction

To answer this question, we need to understand what the rank of a matrix represents in the context of tic-tac-toe.

In tic-tac-toe, we can represent the game board as a $3 cross times 3$ matrix A, where each entry is either a $1$ (representing a move by player X) or a $0$ (representing a move by player O). The rank of this matrix represents the dimension of the subspace of the vector space spanned by the rows (or columns) of the matrix. In other words, the rank of A is the maximum number of rows (or columns) that are linearly independent.