The Four 4s Challenge: Create numbers 1-20 using only four 4s and basic math operations.

The "Four 4s Challenge" is a classic mathematical puzzle where the goal is to express numbers using exactly four 4s and a set of allowed operations. The allowed operations in this case are:

- Addition (+)
- Subtraction (−)
- Multiplication (×)
- Division (÷)
- Factorial (!)
- Square root (√)
- Parentheses (() for grouping

We need to create expressions for the numbers 1 through 20 using these rules.

Solution:

Below are the expressions for each number from 1 to 20:

---

#### 1:
\[
\frac{44}{44} = 1
\]

#### 2:
\[
\frac{4}{4} + \frac{4}{4} = 2
\]

#### 3:
\[
\frac{4 + 4 + 4}{4} = 3
\]

#### 4:
\[
4 \times \frac{4 - 4}{4} + 4 = 4
\]
Alternatively:
\[
4 + 4 - 4 - 4 = 4
\]

#### 5:
\[
4 + \frac{4 + 4}{4} = 5
\]

#### 6:
\[
4 + \frac{4}{4} + 4 = 6
\]

#### 7:
\[
4 + 4 - \frac{4}{4} = 7
\]

#### 8:
\[
4 + 4 + 4 - 4 = 8
\]

#### 9:
\[
4 + 4 + \frac{4}{4} = 9
\]

#### 10:
\[
4 + 4 + 4 - \sqrt{4} = 10
\]

#### 11:
\[
4! - \frac{44}{4} = 11
\]

#### 12:
\[
4 \times 4 - 4 - 4 = 12
\]

#### 13:
\[
4! - \frac{4}{4} - 4 = 13
\]

#### 14:
\[
4! - 4 + \frac{4}{4} = 14
\]

#### 15:
\[
4! - 4 + 4 - 4 = 15
\]

#### 16:
\[
4 \times 4 + 4 - 4 = 16
\]

#### 17:
\[
4! + \frac{4}{4} - 4 = 17
\]

#### 18:
\[
4! + 4 - \frac{4}{4} = 18
\]

#### 19:
\[
4! + 4 - 4 + 4 = 19
\]

#### 20:
\[
4! + 4 + \frac{4}{4} = 20
\]

---

Final Answer:

\[
\boxed{
\begin{aligned}
1 & : \frac{44}{44} \\
2 & : \frac{4}{4} + \frac{4}{4} \\
3 & : \frac{4 + 4 + 4}{4} \\
4 & : 4 \times \frac{4 - 4}{4} + 4 \\
5 & : 4 + \frac{4 + 4}{4} \\
6 & : 4 + \frac{4}{4} + 4 \\
7 & : 4 + 4 - \frac{4}{4} \\
8 & : 4 + 4 + 4 - 4 \\
9 & : 4 + 4 + \frac{4}{4} \\
10 & : 4 + 4 + 4 - \sqrt{4} \\
11 & : 4! - \frac{44}{4} \\
12 & : 4 \times 4 - 4 - 4 \\
13 & : 4! - \frac{4}{4} - 4 \\
14 & : 4! - 4 + \frac{4}{4} \\
15 & : 4! - 4 + 4 - 4 \\
16 & : 4 \times 4 + 4 - 4 \\
17 & : 4! + \frac{4}{4} - 4 \\
18 & : 4! + 4 - \frac{4}{4} \\
19 & : 4! + 4 - 4 + 4 \\
20 & : 4! + 4 + \frac{4}{4}
\end{aligned}
}
\]