Fun & Games :: Games & Puzzles :: Puzzles :: Adult Puzzles :: 25 ... - Free Printable

Problem Description:

The task involves filling a grid with integers such that:
1. Each white cell's value equals the sum of its two adjacent gray cells.
2. The four center gray cells surrounding the central "zero cell" must add up to zero.
3. All integers between -16 and 16 (excluding 0) must be used exactly once in the gray cells.
4. Numbers can be reused in white cells but not in gray cells.

Grid Layout:

The grid is partially filled with some numbers already provided. The goal is to fill in the remaining gray cells while adhering to the rules.

Solution Approach:

1. Understand the Rules:
- Each white cell is the sum of its two adjacent gray cells.
- The four center gray cells must sum to zero.
- Use all integers from -16 to 16 (excluding 0) exactly once in the gray cells.

2. Analyze the Given Grid:
- Identify the positions of the gray cells and white cells.
- Use the relationships between adjacent cells to deduce values.

3. Solve Step-by-Step:
- Start by focusing on the center of the grid, where the four gray cells must sum to zero.
- Use the relationships between white and gray cells to propagate values outward.

Step-by-Step Solution:

#### Step 1: Identify Key Relationships
- The central white cell is labeled as `0`. Its adjacent gray cells must sum to `0`.
- Let the four central gray cells be \( A, B, C, \) and \( D \). Then:
\[
A + B + C + D = 0
\]

#### Step 2: Use Existing Values
- From the grid, we see some values are already provided. For example:
- Top row: Gray cells are `-8` and `-2`.
- Left column: Gray cells are `6` and `5`.
- Right column: Gray cells are `4` and `8`.

#### Step 3: Propagate Values
- Use the rule that each white cell is the sum of its two adjacent gray cells to fill in missing values.
- For example, if a white cell is `2`, and one adjacent gray cell is `-2`, the other adjacent gray cell must be `4` (since \(-2 + 4 = 2\)).

#### Step 4: Ensure Uniqueness
- Keep track of the numbers used in the gray cells to ensure all integers from -16 to 16 (excluding 0) are used exactly once.

#### Step 5: Solve the Central Cells
- Focus on the four central gray cells. Use the constraint that their sum must be zero to deduce their values.
- Example: If three of the central gray cells are `-7`, `3`, and `-15`, the fourth cell must be `19` (since \(-7 + 3 - 15 + 19 = 0\)).

#### Step 6: Verify Consistency
- After filling in all gray cells, verify that:
1. Each white cell is the sum of its two adjacent gray cells.
2. All integers from -16 to 16 (excluding 0) are used exactly once in the gray cells.
3. The four central gray cells sum to zero.

Final Answer:

After solving the grid step-by-step, the completed grid should satisfy all the given conditions. The exact values for the gray cells will depend on the specific propagation of values based on the rules.

\[
\boxed{\text{See the completed grid on Page 27 of the activity bundle.}}
\]