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Problem Analysis:

The image shows a grid of numbers with some entries marked with asterisks (`*`). The task appears to involve identifying patterns or rules governing the placement of these asterisks and filling in the missing values.

Observations:

1. Grid Structure: The grid is composed of rows and columns, with numbers ranging from 55 to 90.
2. Asterisks (`*`): Some cells contain asterisks instead of numbers. These appear to follow a specific pattern.
3. Number Sequence: The numbers in each row increase sequentially from left to right, and the rows decrease sequentially from top to bottom.

Step-by-Step Solution:

#### Step 1: Identify the Pattern for Asterisks
Let's examine the positions of the asterisks in the grid:
- Row 1: `71`, `72`, `73`, `83`, `84`, `87`, `90`
- Row 2: `69`, `70`, `82`, `85`, `88`, `91`
- Row 3: `68`, `74`, `80`, `86`, `94`
- Row 4: `67`, `75`, `89`, `95`, `99`
- Row 5: `66`, `76`, `78`, `88`, `96`, `100`
- Row 6: `64`, `65`, `77`, `88`, `97`, `101`
- Row 7: `63`, `62`, `79`, `88`, `98`, `102`
- Row 8: `60`, `69`, `78`, `88`, `98`, `103`
- Row 9: `57`, `51`, `69`, `70`, `88`, `98`, `104`
- Row 10: `56`, `54`, `66`, `67`, `88`, `98`, `105`
- Row 11: `55`, `65`, `67`, `88`, `98`, `106`

From this, we observe that:
- The number `88` appears frequently in the grid, often replacing other numbers with an asterisk.
- Other numbers like `98`, `100`, `101`, etc., also appear as replacements.

#### Step 2: Determine the Replacement Rule
Upon closer inspection:
- The number `88` replaces numbers in a specific sequence.
- The replacement seems to occur when a number reaches a certain threshold or follows a modular pattern.

#### Step 3: Fill in Missing Values
We need to determine the rule for replacing numbers with `88` and fill in the missing values.

##### Rule Identification:
- Numbers are replaced by `88` if they satisfy a specific condition. For example:
- In Row 1, `83`, `84`, `87`, `90` are replaced.
- In Row 2, `82`, `85`, `88`, `91` are replaced.
- This suggests a pattern where numbers greater than or equal to `88` are replaced by `88`.

##### Apply the Rule:
Using the identified rule, we can fill in the missing values:
- Replace any number ≥ `88` with `88`.

#### Final Grid:
After applying the rule, the grid becomes:
```
71 72 73 88 88 88 88
69 70 88 88 88 88
68 74 88 88 88
67 75 88 88 88
66 76 78 88 88 88
64 65 77 88 88 88
63 62 79 88 88 88
60 69 78 88 88 88
57 51 69 70 88 88 88
56 54 66 67 88 88 88
55 65 67 88 88 88
```

Final Answer:

The completed grid is:
\[
\boxed{
\begin{array}{ccccccc}
71 & 72 & 73 & 88 & 88 & 88 & 88 \\
69 & 70 & 88 & 88 & 88 & 88 \\
68 & 74 & 88 & 88 & 88 \\
67 & 75 & 88 & 88 & 88 \\
66 & 76 & 78 & 88 & 88 & 88 \\
64 & 65 & 77 & 88 & 88 & 88 \\
63 & 62 & 79 & 88 & 88 & 88 \\
60 & 69 & 78 & 88 & 88 & 88 \\
57 & 51 & 69 & 70 & 88 & 88 & 88 \\
56 & 54 & 66 & 67 & 88 & 88 & 88 \\
55 & 65 & 67 & 88 & 88 & 88 \\
\end{array}
}
\]