Some answers to the four 4s problem.

Simple answers

\(0 = {\frac{4}{4}} \times 4 - 4\)	\(1 = {(\frac{4}{4}) ^ 4} ^ 4\)
\(2 = {\frac{4}{(4 + 4)}} \times 4\)	\(3 = \frac{(4 \times 4 - 4)}{4}\)
\(4 = (\frac{4}{4}) ^ 4 \times 4\)	\(5 = (\frac{4}{4}) ^ 4 + 4\)
\(6 = \frac{(4 + 4)}{4} + 4\)	\(7 = {4 - \frac{4}{4}} + 4\)
\(8 = {\frac{4}{4}} \times 4 + 4\)	\(9 = {\frac{4}{4} + 4} + 4\)
\(10 = {\frac{4}{\sqrt{4}} + 4} + 4\)	\(11 = \frac{({4!} + 4)}{4} + 4\)
\(12 = (4 - \frac{4}{4}) \times 4\)	\(13 = \frac{({4!} \times \sqrt{4} + 4)}{4}\)
\(14 = {{\sqrt{4} + 4} + 4} + 4\)	\(15 = 4 \times 4 - \frac{4}{4}\)
\(16 = \frac{4 ^ 4 / 4}{4}\)	\(17 = \frac{4}{4} + 4 \times 4\)
\(18 = (\frac{\sqrt{4}}{4} + 4) \times 4\)	\(19 = {{4!} - \frac{4}{4}} - 4\)
\(20 = (\frac{4}{4} + 4) \times 4\)

Complicated answers

\(0 = \sqrt{{-(\sqrt{\frac{{4!}}{{4!}}} \times {4!})} + {4!}}\)	\(1 = \sqrt{{\sqrt{{\sqrt{\frac{{4!}}{{4!}}}} ^ {{4!}}}} ^ {{4!}}}\)
\(2 = {-(\frac{{-(\sqrt{{4!} \times {4!}} + {4!})}}{{4!}})}\)	\(3 = {-(\frac{{-({4!} - {-({4!} + {4!})})}}{{4!}})}\)
\(4 = {-(\sqrt{{\sqrt{\frac{{4!}}{{4!}}}} ^ {{4!}}} \times {-4})}\)	\(5 = \sqrt{\sqrt{{\sqrt{\frac{{4!}}{{4!}}}} ^ {{4!}}} + {4!}}\)
\(6 = {-(\frac{{-({-(\frac{{4!}}{{4!}})} \times {4!})}}{{-4}})}\)	\(7 = \sqrt{{-({-(\frac{{4!}}{{4!}})} - {4!})} + {4!}}\)
\(8 = {-({-(\frac{{-({4!} + {4!})}}{{4!}})} \times {-4})}\)	\(9 = {-({-4} - \sqrt{\sqrt{\frac{{4!}}{{4!}}} + {4!}})}\)
\(10 = \sqrt{{-(\sqrt{\frac{{4!}}{{4!}}} + {4!})} \times {-4}}\)	\(11 = {-(\sqrt{\frac{{4!}}{{4!}}} + {-(\frac{{4!}}{\sqrt{4}})})}\)
\(12 = {-(\frac{{-(\sqrt{\frac{{4!}}{{4!}}} \times {4!})}}{\sqrt{4}})}\)	\(13 = {-({-(\frac{{4!}}{{4!}})} + {-(\frac{{4!}}{\sqrt{4}})})}\)
\(14 = {-({-(\frac{\sqrt{{4!} \times {4!}}}{\sqrt{4}})} - \sqrt{4})}\)	\(15 = {-(\frac{{-({-(\frac{{4!}}{{-4}})} + {4!})}}{\sqrt{4}})}\)
\(16 = \sqrt{{{-(\frac{\sqrt{\frac{{4!}}{{4!}}}}{{-4}})}} ^ {{-4}}}\)	\(17 = \sqrt{\frac{{-({-({4!} \times {4!})} - \sqrt{4})}}{\sqrt{4}}}\)
\(18 = {-({-(\frac{\sqrt{{4!} \times {4!}}}{{-4}})} - {4!})}\)	\(19 = {-(\sqrt{\sqrt{\frac{{4!}}{{4!}}} + {4!}} - {4!})}\)
\(20 = {-({-(\sqrt{\frac{{4!}}{{4!}}} \times {4!})} - {-4})}\)