Question

In a computer game, each move requires pressing a button. When the button is pressed for the first time, as a move, the computer randomly chooses a cell from a 4x4 grid of sixteen cells and puts an "X" mark on that cell. When the button is pressed subsequently, the computer randomly chooses a cell from the remaining unmarked cells and puts an "X" mark on that cell. This goes on till the end of the game. The game ends when either all the cells in any one row, or all the cells in any one column, are marked with "X". What is the maximum possible number of times a player has to press the button to finish the game?

• A. 16
• B. 10
• C. 13
• D. 04
• E. 06

Hint:

When dealing with grid-based problems, consider the worst-case scenario where the conditions to finish the game are met as late as possible.

Answer 1

The Correct Option is C

To determine the maximum number of times a player must press the button to finish the game, we need to understand the conditions that end the game. The game ends when all cells in any one row or any one column are marked with "X".

Consider the following logical steps: 

1. There is a 4x4 grid consisting of 16 cells. So, there are 4 rows and 4 columns, each containing 4 cells.
2. The game ends when all the cells in a single row or column are filled. Therefore, ideally, the game can end with just \(4\) moves if a row or column is completed immediately.
3. However, the question asks for the maximum number of moves, meaning we must fill in a way that maximizes the number of moves before completing a row or column.
4. The maximum number of moves occurs when the grid is filled up to the third mark in each row and column without completing any row or column.
5. Each row and column should ideally have 3 cells filled. Since there are 4 rows and 4 columns, this gives us: \(4 \times 3 = 12\) filled cells.
6. The 13th move will complete a row or column, ending the game.

Therefore, the maximum number of times the button has to be pressed is 13.

Other options:

• 16: This option assumes that every single cell can be filled in random order, but the game ends sooner once a row or column is finished.
• 10, 4, 6: These options represent cases that end the game in fewer moves, not maximizing the length of the game.

 

Answer 2

Step 1: Understand the grid and conditions.
The grid is 4x4, and the game ends when all cells in any row or column are marked. Thus, it is possible for the game to finish early when one row or column is completely marked.
Step 2: Calculate the maximum number of moves.
The maximum number of moves occurs when no row or column is fully marked until the very last move. This gives a maximum of 13 moves before the game ends.
Final Answer: \[ \boxed{\text{(C) 13}} \]