## Four Fours

The goal of this puzzle is to combine 4 4's with any other mathematical symbol, excepting numbers, to produce every whole number from 1 to 20.

You can extend this to 5 5's, and 6 6's, and so on.

A good strategy is to compile a long list of all the numbers you get when you combine one 4, two 4's, three 4's, and so on. This helps you chain together expressions.

Four Fours/Table of 4s - a table of various combinations of 4s

Starting with 4s:

${\displaystyle 1={\dfrac {4+4}{4+4}}}$

${\displaystyle 2={\dfrac {4\times 4}{4+4}}}$

${\displaystyle 2=4-4+4-{\sqrt {4}}}$

${\displaystyle 3={\dfrac {4+4+4}{4}}}$

${\displaystyle 3={\dfrac {\ln {\left(4+4\right)}}{\ln {\sqrt {4}}}}}$

${\displaystyle 4={\sqrt {4}}\times {\dfrac {4+4}{4}}}$

${\displaystyle 5={\dfrac {4\times 4+4}{4}}}$

${\displaystyle 6=4\times {\dfrac {\ln {\left(4+4\right)}}{\ln {4}}}}$

${\displaystyle 7=4+{\sqrt {4}}+{\dfrac {4}{4}}}$

${\displaystyle 8=4+4\left({\dfrac {4}{4}}\right)}$

${\displaystyle 8={\sqrt {4}}+{\sqrt {4}}+{\sqrt {4}}+{\sqrt {4}}}$

${\displaystyle 9=4+4+{\dfrac {4}{4}}}$

${\displaystyle 10=4+4+4-{\sqrt {4}}}$

${\displaystyle 10=\left({\frac {4}{4}}+4\right){\sqrt {4}}}$

${\displaystyle 11=4^{\sqrt {4}}-(4+i^{4})}$

${\displaystyle 11={\dfrac {44}{{\sqrt {4}}{\sqrt {4}}}}}$

${\displaystyle 12=4+4+{\sqrt {4}}+{\sqrt {4}}}$

${\displaystyle 12=\left(4-{\frac {4}{4}}\right)\times 4}$

${\displaystyle 13={\dfrac {44}{4}}+{\sqrt {4}}}$

${\displaystyle 14=4\times {\sqrt {4}}\times {\sqrt {4}}-{\sqrt {4}}}$

${\displaystyle 15=4\times 4-{\dfrac {4}{4}}}$

${\displaystyle 16={\sqrt {4}}{\sqrt {4}}{\sqrt {4}}{\sqrt {4}}}$

${\displaystyle 16=4+4+4+4}$

${\displaystyle 17=4\times 4+{\dfrac {4}{4}}}$

${\displaystyle 18=4\times 4+{\dfrac {4}{\sqrt {4}}}}$

${\displaystyle 18=4^{\sqrt {4}}+{\dfrac {4}{\sqrt {4}}}}$

${\displaystyle 19=4\times 4+4-i^{4}}$

${\displaystyle 20=4\times 4+{\sqrt {4\times 4}}}$

${\displaystyle 20={\sqrt {4}}{\sqrt {4}}+4^{\sqrt {4}}}$

${\displaystyle 20=4\times \left(4+{\frac {4}{4}}\right)}$

${\displaystyle 21=4\times 4+4+i^{4}}$

${\displaystyle 22={\dfrac {\ln {\left(\left({\sqrt {4}}\right)^{44}\right)}}{\ln {(4)}}}}$

${\displaystyle 23=4!-i^{4}}$

${\displaystyle 24=4!\times i^{4}}$

${\displaystyle 25=4!+i^{4}}$

${\displaystyle 26=4!+{\dfrac {4+4}{4}}}$

${\displaystyle 27=4!+{\dfrac {\ln {(4+4)}}{\ln {\sqrt {4}}}}}$

${\displaystyle 28=4({\sqrt {4}}+i^{4}+4)}$

${\displaystyle 29=4!+4+{\dfrac {4}{4}}}$

${\displaystyle 30=(4+i^{4})(4+{\sqrt {4}})}$

${\displaystyle 31=4(4+4)-i^{4}}$

${\displaystyle 32={\dfrac {4\times 4\times 4}{\sqrt {4}}}}$

${\displaystyle 33=4(4+4)+i^{4}}$

${\displaystyle 34=4(4+4)+{\sqrt {4}}}$

${\displaystyle 35=(4+{\sqrt {4}})^{\sqrt {4}}-i^{4}}$

${\displaystyle 36=\left(4+{\dfrac {4}{\sqrt {4}}}\right)^{\sqrt {4}}}$

${\displaystyle 36=4\left({\sqrt {4}}+i^{4}\right)^{\sqrt {4}}}$

${\displaystyle 36=4\left(4{\sqrt {4}}+i^{4}\right)}$

${\displaystyle 36=4!+4+4+4}$

${\displaystyle 37=(4+{\sqrt {4}})^{\sqrt {4}}}$

${\displaystyle 38=\left(4+{\sqrt {4}}\right)^{\sqrt {4}}+{\sqrt {4}}}$

${\displaystyle 39=4!+4\times 4-i^{4}}$

${\displaystyle 40=4(4+4+{\sqrt {4}})}$

${\displaystyle 40=(4+4)(4+i^{4})}$

${\displaystyle 41=4!+4\times 4+i^{4}}$

${\displaystyle 42=(4!)({\sqrt {4}})-(4+{\sqrt {4}})}$

${\displaystyle 43=(4!)({\sqrt {4}})-(4+i^{4})}$

${\displaystyle 44=(4!)({\sqrt {4}})-({\sqrt {4}}+{\sqrt {4}})}$

${\displaystyle 44={\sqrt {4}}\left(4!-{\dfrac {4}{\sqrt {4}}}\right)}$

${\displaystyle 45=(4!-{\sqrt {4}})+(4!-i^{4})}$

${\displaystyle 45=(4!-{\sqrt {4}})({\sqrt {4}})+i^{4}}$

${\displaystyle 46=4!+4!-{\dfrac {4}{\sqrt {4}}}}$

${\displaystyle 46={\sqrt {4}}(4!)-{\dfrac {4}{\sqrt {4}}}}$

${\displaystyle 46={\sqrt {4}}\left(4!-{\sqrt {4}}\right)+{\sqrt {4}}}$

${\displaystyle 47=4!{\sqrt {4}}-{\dfrac {4}{4}}}$

${\displaystyle 48=(4!)({\sqrt {4}})\left({\dfrac {4}{4}}\right)}$

${\displaystyle 49=({\sqrt {4}})(4!)+{\dfrac {4}{4}}}$

${\displaystyle 50=({\sqrt {4}})(4!)+{\dfrac {4}{\sqrt {4}}}}$

${\displaystyle 51=({\sqrt {4}})(4!)+4-i^{4}}$

${\displaystyle 52=(4!){\sqrt {4}}+{\sqrt {4}}{\sqrt {4}}}$

${\displaystyle 53=(4!)({\sqrt {4}})+4+i^{4}}$

${\displaystyle 54=4!+4!+4+{\sqrt {4}}}$

${\displaystyle 55=(4!+4)\times {\sqrt {4}}-i^{4}}$

${\displaystyle 56=4!\left({\sqrt {4}}+{\dfrac {i^{4}}{4}}\right)}$

${\displaystyle 56=4!+4!+4+4}$

${\displaystyle 57=(4!+4)\times {\sqrt {4}}+i^{4}}$

${\displaystyle 58=(4!+4)\times {\sqrt {4}}+{\sqrt {4}}}$

${\displaystyle 59={\dfrac {(4+i^{4})!-{\sqrt {4}}}{\sqrt {4}}}}$

${\displaystyle 60=(4!+4)\times {\sqrt {4}}+4}$

${\displaystyle 61={\dfrac {(4+i^{4})!+{\sqrt {4}}}{\sqrt {4}}}}$

${\displaystyle 62={\dfrac {(4+i^{4})!+4}{\sqrt {4}}}}$

${\displaystyle 63={\dfrac {4^{4}-4}{4}}}$

${\displaystyle 64=({\sqrt {4}})^{{\sqrt {4}}+{\sqrt {4}}+{\sqrt {4}}}}$

${\displaystyle 65=({\sqrt {4}})^{{\sqrt {4}}+4}+i^{4}}$

${\displaystyle 66={\dfrac {4^{4}}{4}}+{\sqrt {4}}}$

${\displaystyle 67=44+4!-i^{4}}$

${\displaystyle 68={\dfrac {4^{4}}{4}}+4}$

${\displaystyle 69=(4!-i^{4})(4-i^{4})}$

${\displaystyle 70=44+4!+{\sqrt {4}}}$

${\displaystyle 71={\sqrt {4}}(4!)+4!+i^{4}}$

${\displaystyle 72=4!\times {\dfrac {\log(4+4)}{\log {\sqrt {4}}}}}$

${\displaystyle 73=4!\left({\sqrt {4}}+i^{4}\right)+i^{4}}$

${\displaystyle 73=4!\times {\sqrt {4}}+4!+i^{4}}$

${\displaystyle 73=4!\times (4-i^{4})+i^{4}}$

${\displaystyle 74=\left(4!+{\sqrt {4}}\right)+4!{\sqrt {4}}}$

${\displaystyle 74=4!\left(4-i^{4}\right)+{\sqrt {4}}}$

${\displaystyle 75=(4!+i^{4})({\sqrt {4}}+i^{4})}$

${\displaystyle 76=4!\left({\sqrt {4}}+i^{4}\right)+4}$

${\displaystyle 76=4!\left(4-i^{4}\right)+4}$

${\displaystyle 77=4!\cdot 4-\left(4\ln \left(e\cdot e^{4}\right)-\ln \left(e\right)\right)}$

${\displaystyle 78=(4-i^{4})(4!+{\sqrt {4}})}$

${\displaystyle 79=(4F)_{4\cdot 4}\cdot i^{4}}$

(That is, the number 4F in hex, or base 16.)

${\displaystyle 80={\sqrt {4}}^{4}\left(4+i^{4}\right)}$

${\displaystyle 81=\left(4{\sqrt {4}}+i^{4}\right)^{\sqrt {4}}}$

${\displaystyle 82=4\cdot (4!-4)+{\sqrt {4}}}$

${\displaystyle 83={\sqrt {4}}\cdot 44-4-\ln(e)}$

${\displaystyle 84=4((4!-4)+i^{4})}$

${\displaystyle 85=44{\sqrt {4}}-4+\ln(e)}$

${\displaystyle 86={\sqrt {4}}(44-i^{4})}$

${\displaystyle 87=44{\sqrt {4}}-i^{4}}$

${\displaystyle 88=44\left({\dfrac {4}{\sqrt {4}}}\right)}$

${\displaystyle 89=44{\sqrt {4}}+i^{4}}$

${\displaystyle 90=44{\sqrt {4}}+{\sqrt {4}}}$

${\displaystyle 91=4\cdot 4!-4-i^{4}}$

${\displaystyle 92=4\left(4!-{\dfrac {4}{4}}\right)}$

${\displaystyle 93=4\cdot 4!-4+i}$

${\displaystyle 94=4\cdot 4!-{\dfrac {4}{\sqrt {4}}}}$

${\displaystyle 95=4\cdot 4!-{\dfrac {4}{4}}}$

${\displaystyle 96=4((4!-4)+4)}$

${\displaystyle 97=4\cdot 4!+{\dfrac {\log {(4)}}{\log {(4)}}}}$

${\displaystyle 98={\sqrt {4}}\left(4{\sqrt {4}}-\ln {(e)}\right)^{\sqrt {4}}}$

${\displaystyle 99=\left(4\cdot 4!\right)+\left(4-i^{4}\right)}$

${\displaystyle 100=4\left(4!+{\dfrac {\log {(4)}}{\log {(4)}}}\right)}$