Some answers to the four 4s problem.

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