Four Fours/Table of 4s

One 4

Believe it or not, the rules allow you to do quite a bit with a single 4. The rules say you may combine 4s with any mathematical symbol except numbers. Thus, in addition to 4 alone, we also have:

$ 1 = i^{4} $

$ 2 = \sqrt{4} $

$ 24 = 4! $

Two 4s

$ 1 = \dfrac{4}{4} $

$ 2 = 4 - \sqrt{4} $

$ 2 = \dfrac{4}{\sqrt{4}} $

$ 4 = \sqrt{4} \times \sqrt{4} $

$ 4 = \sqrt{4} + \sqrt{4} $

$ 5 = 4 + i^{4} $

$ 6 = 4 + \sqrt{4} $

$ 8 = 4+4 $

$ 8 = 4 \sqrt{4} $

$ 16 = 4 \times 4 $

$ 16 = 4^{\sqrt{4}} $

$ 256 = 4^4 $

Three 4s

These lists blow up pretty fast... as you can see, focusing on using a smaller number of 4s can force you to be creative. This makes it possible to combine 4 4's beyond the integers from 1 to 20, and keep on going...

$ 2 = \dfrac{4+4}{4} $

$ 2 = \sqrt{4} \times \left( \dfrac{4}{4} \right) $

$ 3 = \dfrac{ \ln{(4+4)} }{ \ln{\sqrt{4}} } $

$ 4 = 4 \times \left( \dfrac{4}{4} \right) $

$ 4 = 4 + 4 - 4 $

$ 5 = 4 + \dfrac{4}{4} $

$ 8 = 4 \times \left( \dfrac{4}{\sqrt{4}} \right) $

$ 10 = \dfrac{ \ln{(4^4 \times 4)} }{ \ln{\sqrt{4}} } $

$ 12 = 4+4+4 $

$ 18 = 4 \times 4 + \sqrt{4} $

$ 20 = 4 \times 4 + 4 $

$ 32 = 4(4+4) $

$ 64 = \dfrac{4^4}{4} $

$ 252 = 4^4 - 4 $

$ 254 = 4^4 - \sqrt{4} $

$ 258 = 4^4 + \sqrt{4} $

$ 260 = 4^4 + 4 $

$ 1,024 = 4^4 \times 4 $

$ 4,096 = (4+4)^{4} $

$ 65,536 = (4 \times 4)^{4} $