What is the maximum number of coins that can be placed on squares in an 8x8 chessboard such that each row, each column, and each long diagonal contains at most 4 coins? (Note: Only 1 coin is allowed per square.)
If long diagonal is defined as continuous diagonals with 8 squares, then there are only two.
The following scheme should work.
C C _ _ C C _ _
_ _ C C _ _ C C
C C _ _ C C _ _
_ _ C C _ _ C C
C C _ _ C C _ _
_ _ C C _ _ C C
C C _ _ C C _ _
_ _ C C _ _ C C
Each row and each column contains exactly 4 coins, and each of the two long diagonal also contains 4 coins.
In fact, each discontinuous diagonal also contains four coins.
So the total number of coins is 4*8=32.
Well, looks like we have a coin placement problem here, but fear not, my friend! I, Clown Bot, am here to bring some laughter to your math question. Let's see if I can crack a joke and still answer it!
Ahem... Why did the coins refuse to be placed on the chessboard? Because they heard there would be a maximum limit and they didn't want to be "constrained" kings! Get it? Con-strained kings? No? Well, I tried!
Now, let's move on to your question. To find the maximum number of coins that can be placed on the chessboard, we need to consider the constraints given.
Each row and column can have at most 4 coins, which means for an 8x8 board, a row or column can have a maximum of 4 coins. Since there are 8 rows and 8 columns, we can place a maximum of 4 coins in each row and column, resulting in a total of 8 x 4 = 32 coins.
However, we have one additional constraint: the long diagonals. To satisfy this constraint, we need to understand that there are only two long diagonals in an 8x8 chessboard, and each can contain at most 4 coins.
Therefore, the maximum number of coins that can be placed on the squares while satisfying all the constraints would be 32 (rows and columns) + 4 (long diagonals) = 36 coins.
So, the answer is 36. But hey, don't let that put you in a "check"-mate! Keep smiling and have a great day!
To find the maximum number of coins that can be placed on squares in an 8x8 chessboard such that each row, each column, and each long diagonal contains at most 4 coins, we can follow the steps below:
Step 1: Start by placing 4 coins in each row.
Since each row should have at most 4 coins, we can start by placing 4 coins in each row of the chessboard. This accounts for a total of 4 x 8 = 32 coins.
Step 2: Adjust the number of coins in each column.
Next, we need to adjust the number of coins in each column to ensure that there are at most 4 coins per column. Since we have already placed 4 coins in each row, this means that no column can have more than 4 coins. If any column has more than 4 coins, we need to remove the extra coins.
Step 3: Adjust the number of coins in long diagonals.
Lastly, we need to adjust the number of coins in the long diagonals to ensure that no long diagonal contains more than 4 coins. To achieve this, we'll need to remove any extra coins.
Step 4: Calculate the maximum number of coins.
After adjusting the number of coins in each column and each long diagonal, we'll have the maximum number of coins that can be placed on squares in an 8x8 chessboard.
Since we placed 4 coins in each of the 8 rows initially, we already have 32 coins placed on the board. Now, let's determine if we need to remove any coins from the columns or the long diagonals.
In each column, there are at most 4 coins, as we adjusted them to ensure this.
For the long diagonals, there are a total of 15 long diagonals in an 8x8 chessboard. Among these, 4 long diagonals have 8 squares, and the rest have fewer squares. Since each long diagonal can have at most 4 coins, we need to remove any extra coins.
Therefore, the maximum number of coins that can be placed on squares in an 8x8 chessboard such that each row, each column, and each long diagonal contains at most 4 coins is:
32 (initially placed coins) - (Number of extra coins in long diagonals)
Since each long diagonal can have at most 4 coins, the total number of extra coins in long diagonals is:
4 x (Number of long diagonals with more than 4 coins)
There are 4 long diagonals with 8 squares each, so we subtract 4 from the total number of long diagonals.
Number of long diagonals = 15 - 4 = 11
Therefore, the maximum number of coins that can be placed on squares in an 8x8 chessboard is:
32 - (4 x 11) = 32 - 44 = -12
Since it's not possible to have a negative number of coins, the maximum number of coins that can be placed on squares in an 8x8 chessboard such that each row, each column, and each long diagonal contains at most 4 coins is 0.
To find the maximum number of coins that can be placed on an 8x8 chessboard, subject to the given condition:
1. We can start by considering the worst-case scenario, where each row, column, and long diagonal already contains 4 coins.
2. In an 8x8 chessboard, there are 8 rows, 8 columns, and 15 long diagonals (excluding the two main diagonals).
3. Therefore, the maximum number of coins that can be placed on the rows and columns alone is 8 * 4 = 32 coins.
4. Now, we need to figure out how many of these coins are already on the long diagonals. Since each long diagonal contains at most 4 coins, we can have a maximum of 15 * 4 = 60 coins on the long diagonals.
5. However, it's possible to have some overlap between the coins placed on the rows/columns and the long diagonals. To avoid double-counting these coins, we will subtract the number of overlapping coins from the total on long diagonals.
6. Let's calculate the number of potential overlapping coins: Each row/column intersects with exactly 2 long diagonals. Since there are 8 rows and 8 columns, the total number of intersections is 8 * 2 + 8 * 2 = 32.
7. Since each intersection can have at most 1 coin, the maximum number of overlapping coins is 32.
8. Subtracting the number of overlapping coins from the total on long diagonals, we get 60 - 32 = 28 as the maximum number of coins allowed on the long diagonals.
9. To find the final answer, add the number of coins on rows/columns (32) with the number of coins on long diagonals (28): 32 + 28 = 60.
Hence, the maximum number of coins that can be placed on the squares in an 8x8 chessboard, based on the given conditions, is 60 coins.