Number Pyramids | wild.maths.org - Free Printable

To solve the problem, we need to determine the missing numbers in the triangular arrangement based on a logical pattern. Let's analyze the given structure step by step.

Step 1: Understand the Structure

The triangle has three levels:
- Top level: One number (38).
- Middle level: Three numbers (9, ?, 21).
- Bottom level: Four numbers (7, ?, 1, ?).

We need to find the missing numbers in the middle and bottom levels.

Step 2: Identify the Pattern

Triangles like this often follow a specific rule where each number in a higher level is derived from the numbers directly below it. Common patterns include:
1. Summing two adjacent numbers.
2. Subtracting one number from another.
3. Applying a mathematical operation (e.g., multiplication, division).

Let's test these possibilities.

#### Hypothesis 1: Summing Two Adjacent Numbers
If each number in a higher level is the sum of the two numbers directly below it, we can test this hypothesis:

- For the top number (38):
- It should be the sum of the two numbers in the middle level: \( ? + 21 \).
- For the middle level:
- The left number (9) should be the sum of the two numbers directly below it: \( 7 + ? \).
- The right number (21) should be the sum of the two numbers directly below it: \( ? + 1 \).

Let's denote the missing numbers as follows:
- Bottom level: \( 7, x_1, 1, x_2 \).
- Middle level: \( 9, x_3, 21 \).

#### Step 3: Solve for the Missing Numbers
1. Find \( x_1 \) (bottom level, second number):
- The middle-level number 9 is the sum of 7 and \( x_1 \):
\[
9 = 7 + x_1 \implies x_1 = 9 - 7 = 2
\]

2. Find \( x_2 \) (bottom level, fourth number):
- The middle-level number 21 is the sum of \( x_2 \) and 1:
\[
21 = x_2 + 1 \implies x_2 = 21 - 1 = 20
\]

3. Find \( x_3 \) (middle level, second number):
- The top-level number 38 is the sum of \( x_3 \) and 21:
\[
38 = x_3 + 21 \implies x_3 = 38 - 21 = 17
\]

Step 4: Verify the Solution

Let's verify the solution by reconstructing the triangle:
- Bottom level: \( 7, 2, 1, 20 \).
- Middle level: \( 9, 17, 21 \).
- Top level: \( 38 \).

- Check the sums:
- Middle level (left): \( 7 + 2 = 9 \) (correct).
- Middle level (right): \( 20 + 1 = 21 \) (correct).
- Top level: \( 17 + 21 = 38 \) (correct).

All checks are consistent with the summing pattern.

Final Answer

The missing numbers are:
- Bottom level: \( 2 \) and \( 20 \).
- Middle level: \( 17 \).

Thus, the complete triangle is:
\[
\begin{array}{cccccc}
& & 38 & & \\
& 17 & & 21 & \\
9 & & 2 & & 1 & 20 \\
\end{array}
\]

The final answer is:
\[
\boxed{17, 2, 20}
\]