Using only four 4's, write an equation representing the counting integers from 1-20, inclusive.?

Question

You may use any of the commonly used operations except 4 factorial. You can ONLY use four 4's. No more, no less. The only digit you can use is 4.

Hint: You can use 4/4, exponents, square root of four, 44, .4, 4 to the square root of 4 power, etc, etc.
The only operation you may not use is 4! (4x3x2x1)

Good luck!

Answer 1

1 = (4/4) + 4 - 4
2 = (4/4) + (4/4)
3 = sqrt(4) + sqrt(4) - (4/4)
4 = sqrt(4) + sqrt(4) * (4/4)
5 = sqrt(4) + sqrt(4) + (4/4)
6 = 4 * (4/4) + sqrt(4)
7 = 4 + 4 - (4/4)
8 = 4 + 4 * (4/4)
9 = 4 + 4 + (4/4)
10 = 4 + 4 + 4 - sqrt(4)
11 = 44 / (sqrt(4) + sqrt(4))
12 = (44 + 4) / 4
13 = (44/4) + sqrt(4)
14 = 4 + 4 + 4 + sqrt(4)
15 = (44/4) + 4
16 = 4 * 4 * (4/4)
17 = 4 * 4 + (4/4)
18 = 4 * 4 + (4 - sqrt(4))
19 = (4 + 4 - .4) / .4
20 = 4 * (4 + (4/4))

Answer 2

1 = 4/4

2 = sqrt4

3 = 4/4 + sqrt4

4 = 4

5 = 4 + 4/4

6 = 4 + sqrt4

7 = 4 + sqrt4 + 4/4

8 = 4 + 4

9 = 4 + 4 + 4/4

10 = 4 + 4 + sqrt4

11 = 4 x sqrt4 + sqrt4 + 4/4

12 = 4 x sqrt4 + 4

I think you got the idea.

Answer 3

4/4 = 1
4+4/4 = 2
4+4+4/4 = 3
4+4+4+4/4 = 4
... picking up on a theme here?
4+4+4+4+4/4 = 5
4+4+4+4+4+4/4 = 6
4+4+4+4+4+4+4/4 = 7
...etc etc
... all you have to to is add +4 to the top line(numerator) for each +1 increment going up.


4+4
4+4/4
4+4+4/4
4
4+(4/4)
4 +(4+4)/4
4+4 -(4/4)
4+4
etc etc