2050.21 – Four Fours

There are many possibilities. Here is one:

$$\begin{aligned}1 &= \dfrac{(4 + 4)}{(4 + 4)} \\[1em]2 &= \dfrac{(4 * 4)}{(4 + 4)} \\[1em]3 &= \dfrac{(4 + 4 + 4)}{4} \\[1em]4 &= \sqrt{4 + 4 + 4 + 4} \\[1em]5 &= \dfrac{(4 * 4 + 4)}{4} \\[1em]6 &= \dfrac{(4 + 4)}{4} + 4 \\[1em]7 &= 4 + \sqrt 4 + \dfrac{4}{4} \\[1em]8 &= 4 + \left(4 * \dfrac{4}{4}\right) \\[1em]9 &= 4 + 4 + \dfrac{4}{4} \\[1em]10 &= \sqrt 4 + 4 * \dfrac{4}{\sqrt{4}} \\[1em]11 &= 4 + 4 + 4 - \left\lfloor\sqrt{\sqrt{4}}\right\rfloor \\[1em]12 &= 4 * 4 - \sqrt{4} \sqrt{4} \\[1em]13 &= 4 + 4 + 4 + \left\lfloor\sqrt{\sqrt{4}}\right\rfloor \\[1em]14 &= 4 * 4 - \dfrac{4}{\sqrt{4}} \\[1em]15 &= 4 * 4 - \dfrac{4}{4} \\[1em]16 &= 4 * 4 * \left(\dfrac{4}{4}\right) \\[1em]17 &= 4 * 4 + \dfrac{4}{4} \\[1em]18 &= 4 * 4 + \dfrac{4}{\sqrt{4}} \\[1em]19 &= 4 * 4 + \sqrt{4} + \left\lfloor\sqrt{\sqrt{4}}\right\rfloor \\[1em]20 &= 4 * \left(4 + \dfrac{4}{4}\right)\end{aligned}$$

Note: $\left\lfloor x \right\rfloor$ is the greatest integer function. That is $\left\lfloor x \right\rfloor$ is the greatest integer less than or equal to $x$. For example, and most notoriously: $\left\lfloor\sqrt{\sqrt{4}}\right\rfloor = \left\lfloor\sqrt{2}\right\rfloor= \left\lfloor 1.4 \right\rfloor = 1$, a handy way to make a $1$ out of just one $4$.