Missing Number Puzzle | Maths Puzzle | How to solve maths puzzle easily | imran sir maths - Free Printable

Problem Analysis:

The task involves completing a table by identifying the pattern or rule governing the numbers in each row and column. The table is as follows:

\[
\begin{array}{|c|c|c|c|}
\hline
1 & 3 & 5 & 7 \\
\hline
4 & 8 & ? & 16 \\
\hline
7 & ? & 19 & 25 \\
\hline
10 & 18 & ? & 34 \\
\hline
\end{array}
\]

We need to determine the missing values marked with `?`.

---

Step-by-Step Solution:

#### Step 1: Analyze the First Row
The first row is:
\[
1, 3, 5, 7
\]
This is an arithmetic sequence with a common difference of 2:
\[
a_n = 1 + (n-1) \cdot 2
\]
where \( n \) is the position of the term in the sequence.

#### Step 2: Analyze the Second Row
The second row is:
\[
4, 8, ?, 16
\]
Let's check if there is a pattern:
- The difference between the first two terms is \( 8 - 4 = 4 \).
- The difference between the last two terms is \( 16 - ? \).

Assuming the pattern is consistent, the differences should be equal. Therefore:
\[
8 - 4 = 4 \quad \text{and} \quad 16 - ? = 4
\]
Solving for `?`:
\[
16 - ? = 4 \implies ? = 16 - 4 = 12
\]

So, the second row becomes:
\[
4, 8, 12, 16
\]

#### Step 3: Analyze the Third Row
The third row is:
\[
7, ?, 19, 25
\]
Let's check the differences:
- The difference between the third and fourth terms is \( 25 - 19 = 6 \).
- Assuming the pattern is consistent, the difference between the second and third terms should also be 6:
\[
19 - ? = 6 \implies ? = 19 - 6 = 13
\]

So, the third row becomes:
\[
7, 13, 19, 25
\]

#### Step 4: Analyze the Fourth Row
The fourth row is:
\[
10, 18, ?, 34
\]
Let's check the differences:
- The difference between the second and first terms is \( 18 - 10 = 8 \).
- The difference between the fourth and third terms is \( 34 - ? \).

Assuming the pattern is consistent, the differences should be equal. Therefore:
\[
18 - 10 = 8 \quad \text{and} \quad 34 - ? = 8
\]
Solving for `?`:
\[
34 - ? = 8 \implies ? = 34 - 8 = 26
\]

So, the fourth row becomes:
\[
10, 18, 26, 34
\]

#### Step 5: Verify Column Patterns
To ensure consistency, let's verify the columns:
1. First Column: \( 1, 4, 7, 10 \)
- This is an arithmetic sequence with a common difference of 3.
2. Second Column: \( 3, 8, 13, 18 \)
- This is an arithmetic sequence with a common difference of 5.
3. Third Column: \( 5, 12, 19, 26 \)
- This is an arithmetic sequence with a common difference of 7.
4. Fourth Column: \( 7, 16, 25, 34 \)
- This is an arithmetic sequence with a common difference of 9.

All columns follow arithmetic sequences, confirming the correctness of our solution.

---

Final Answer:

The completed table is:
\[
\begin{array}{|c|c|c|c|}
\hline
1 & 3 & 5 & 7 \\
\hline
4 & 8 & 12 & 16 \\
\hline
7 & 13 & 19 & 25 \\
\hline
10 & 18 & 26 & 34 \\
\hline
\end{array}
\]

The missing values are:
\[
\boxed{12, 13, 26}
\]