What is the least number of unit squares that you need to remove from a 5 by 3 checker board in order to make it impossible for anyone to put an "X" in 3 remaining squares to make a connected vertical, horizontal, or diagonal set of 3 (a set that looks like a "win" in a standard Tic Tac Toe game)?
R will be removed square, O will be remaining
ROROR
OOROO
ORORO
To find the least number of unit squares that you need to remove from a 5 by 3 checkerboard, consider the following steps:
Step 1: Calculate the total number of unit squares on the checkerboard.
In a 5 by 3 checkerboard, there are 5 columns and 3 rows, so the total number of unit squares is 5 * 3 = 15.
Step 2: Determine the minimum number of squares needed to form a "win".
In a standard Tic Tac Toe game, a win consists of three connected squares in a row, column, or diagonal. Since you want to make it impossible to form a win, you need to remove enough squares to ensure that there are no possible sets of 3 squares that are connected.
Step 3: Find the maximum number of connected squares that can be formed.
To find the maximum number of connected squares that can be formed, consider the worst-case scenario where all squares are connected. In a 5 by 3 checkerboard, the largest possible win set can be formed by connecting three squares horizontally across any row, vertically along any column, or diagonally from one corner to the opposite corner.
Step 4: Calculate the minimum number of squares to remove.
Since the worst-case scenario has three squares in a win set, you need to remove at least 3 squares to make it impossible to form a win.
Therefore, the least number of unit squares that you need to remove from a 5 by 3 checkerboard is 3.
To find the least number of unit squares that you need to remove from a 5 by 3 checkerboard, we can follow these steps:
1. Draw a 5 by 3 checkerboard grid on a piece of paper or visualize it in your mind.
2. Since we need to make it impossible for any player to make a connected set of three "X"s, let's consider the worst-case scenario where one player has the opportunity to connect three "X"s in any direction.
3. Start by imagining all the possible winning lines on the checkerboard. In this case, we are looking for connected vertical, horizontal, or diagonal sets of three squares.
4. Consider the following winning lines on the checkerboard:
- Three horizontal lines (each with three squares)
- Three vertical lines (each with three squares)
- Two diagonal lines (each with three squares)
5. Calculate the total number of squares covered by these winning lines:
- 3 (horizontal lines) × 3 (squares per line) = 9
- 3 (vertical lines) × 3 (squares per line) = 9
- 2 (diagonal lines) × 3 (squares per line) = 6
6. Add up the total number of squares covered by all the winning lines: 9 + 9 + 6 = 24.
7. Since the checkerboard has 5 rows × 3 columns = 15 squares, subtract the total number of squares covered by winning lines from the total number of squares on the checkerboard: 15 - 24 = -9.
8. We get a negative value (-9) because it is not possible to remove a negative number of squares. In this case, we cannot remove enough squares to make it impossible for any player to make a connected set of three "X"s.
Therefore, the answer to the given question is that it is not possible to remove a certain number of unit squares from a 5 by 3 checkerboard to make it impossible for anyone to form a connected set of three "X"s in a vertical, horizontal, or diagonal direction.