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 #1
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Here's how to find the number of ways to arrange the numbers 1 through 9 in the grid:

 

Step 1: Analyze the limitations:

 

1 and 2 cannot be placed next to each other. This means they must be in diagonally opposite corners.

 

Rotations and reflections are considered the same. So, we only need to find solutions for one orientation and multiply by the number of possible rotations/reflections.

 

Step 2: Place 1 and 2:

 

Choose two diagonally opposite corners for 1 and 2. There are 4 corner positions, so there are 4 choices.

 

Step 3: Place the remaining numbers:

 

Center square: Since 1 and 2 are fixed, place any number except 1, 2, 5, or 8 in the center square (these are diagonally adjacent to 1 or 2). This leaves 5 options.

 

Squares adjacent to 1 and 2: Place numbers diagonally adjacent to 1 and 2. Each square has 2 options. Since there are 4 such squares, there are 2^4 = 16 possibilities.

 

Remaining squares: Fill the remaining squares with any of the remaining numbers (3, 4, 5, 6, 7, 8), ensuring no violations occur. There are 6 options for each square, giving 6^4 = 1296 possibilities.

 

Step 4: Account for rotations/reflections:

 

The chosen orientation has 8 rotations and 4 reflections (including the original). Therefore, there are 8 + 4 = 12 possibilities.

 

Step 5: Calculate the total number of arrangements:

 

Multiply the choices at each step: 4 * 5 * 16 * 1296 * 12 = 1,032,192

 

Therefore, there are 1,032,192 possible arrangements of the numbers 1 through 9 in the grid under the given conditions.

25.02.2024
 #1
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Here's how we can find the number of possible ways to assign values to A, B, C, and D:

 

Constraints:

 

A, B, C, and D are distinct integers from 0 to 9.

 

A + B = C

 

A - B = D

 

Solving for B:

 

From the second equation, we can express B as B = A - D.

 

Substituting B in the first equation:

 

A + (A - D) = C

 

Combine like terms: 2A - D = C

 

Relating C and D:We now have two equations with two unknowns (A and D). However, we can't directly solve for A or D because both equations involve them.

 

Instead, we can try to relate C and D. Notice that the difference between the two equations is simply D: (2A - D) - (A - D) = C - D.

 

This simplifies to A = C - D.

 

Finding possible values:

 

Now we have two independent equations:

 

A = C - D

 

2A - D = C

 

We can substitute the first equation into the second equation to eliminate A:

 

2(C - D) - D = C

 

Expand and rearrange: C - 3D = 0

 

Therefore, C = 3D (since C and D are distinct, D cannot be 0).

 

Counting possibilities:

 

We know that C = 3D and 0 <= C, D <= 9.

 

The possible values for D are 1, 3 (since C and D are distinct).

 

For each value of D, there are 9 possible values for C (since 0 <= C <= 9).

 

Therefore, there are 2 * 9 = 18 possible combinations of C and D.

 

Considering A and B:

 

For each valid combination of C and D, there is only one possible value for A (A = C - D).

 

The value of B is then uniquely determined by B = A - D.

 

Final answer:

 

There are 18 possible combinations of C and D, and each combination leads to a unique solution for A, B, C, and D.

 

Therefore, there are a total of 18 possible ways to assign values to A, B, C, and D.

 

So, there are 18​ possible ways to assign values to A, B, C, and D.

25.02.2024