2050.22 – Four Fours: Up to Forty

In a previous problem, you found mathematical expressions equal to each of the whole numbers from 1 to 20 using exactly four 4's plus any mathematical symbols you wanted to use. Using the same rules, make the whole numbers 21 to 40.

Solution

21=4!−4 −(4÷4)22=4!−4 +4−423=4!−4 +(4÷4)24=(4×4)+4+425=4!+4 −(4÷4)26=4!+4+4427=4!+4+(4÷4)28=4!+4−4+429=4!+4+(4÷4)30=4!+4+4 +4 or 4+4+4.431=4!+44+4!32=4!+4+4+4 or (4÷.4)+4!−433=4−.4.4+4!34=4!+(4×4)+435=4+.4.4+4!36=4!+4+4+437=4!+44+4!38=44−4−439=4!+4+4.440=4!+4!−4−4\begin{aligned}21 &= 4! - \sqrt 4 - (4 \div 4) \\[1em]22 &= 4! - \sqrt 4 + 4 - 4 \\[1em]23 &= 4! - \sqrt 4 + (4 \div 4) \\[1em]24 &= (4 \times 4) + 4 + 4 \\[1em]25 &= 4! + \sqrt 4 - (4 \div 4) \\[1em]26 &= 4! + \dfrac{4 + 4}{4} \\[1em]27 &= 4! + \sqrt 4 + (4 \div 4) \\[1em]28 &= 4! + 4 - 4 + 4 \\[1em]29 &= 4! + 4 + (4 \div 4) \\[1em]30 &= 4! + \sqrt 4 + \sqrt 4 + \sqrt 4 \text{ or } \dfrac{4 + 4 + 4}{.4} \\[1em]31 &= \dfrac{4! + 4}{4} + 4! \\[1em]32 &= 4! + 4 + \sqrt 4 + \sqrt 4 \text{ or } (4 \div .4) + 4! - \sqrt 4 \\[1em]33 &= \dfrac{4 - .4}{.4} + 4! \\[1em]34 &= 4! + (4 \times \sqrt 4) + \sqrt 4 \\[1em]35 &= \dfrac{4 + .4}{.4} + 4! \\[1em]36 &= 4! + 4 + 4 + 4 \\[1em]37 &= \dfrac{4! + \sqrt 4}{\sqrt 4} + 4! \\[1em]38 &= 44 - \sqrt 4 - 4 \\[1em]39 &= 4! + \dfrac{4 + \sqrt 4}{.4} \\[1em]40 &= 4! + 4! - 4 - 4\end{aligned}

Fun Fact: One can find expressions like this for every counting number up to 100 using these functions and tricks How far can youyou get?