2050.21 – Four Fours: One to Twenty

You are to find expressions for each whole number from 1 through 20, using in each case exactly four 4’s, plus any mathematical symbols you like: parentheses, plus signs, etc.

Sample: $1 = (4 + 4)/(4+4)$

Solution

There are many possibilities. Here is one:

\begin{aligned} 1 &=& (4 + 4)/(4 + 4) \\ 2 &=& (4 * 4)/(4 + 4) \\ 3 &=& (4 + 4 + 4)/4 \\ 4 &=& \sqrt (4 + 4 + 4 + 4) \\ 5 &=& (4 * 4 + 4)/4 \\ 6 &=& (4 + 4)/4 + 4 \\ 7 &=& 4 + \sqrt 4 + 4/4 \\ 8 &=& 4 + (4 * 4/4) \\ 9 &=& 4 + 4 + 4/4 \\ 10 &=& \sqrt 4 + 4 * 4/\sqrt (4) \\ 11 &=& 4 + 4 + 4 –[\sqrt \sqrt 4] \\ 12 &=& 4 * 4 – \sqrt (4)\sqrt (4) \\ 13 &=& 4 + 4 + 4 + [\sqrt \sqrt 4] \\ 14 &=& 4 * 4 – 4/\sqrt 4 \\ 15 &=& 4 * 4 – 4/4 \\ 16 &=& 4 * 4 * (4/4) \\ 17 &=& 4 * 4 + 4/4 \\ 18 &=& 4 * 4 + 4/\sqrt 4 \\ 19 &=& 4 * 4 + \sqrt 4 + [\sqrt \sqrt 4] \\ 20 &=& 4 * (4 + 4/4) \end{aligned}

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