Four Fours

See also: Five Fives

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:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$16=4+4+4+4$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$36=4!+4+4+4$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$56=4!+4!+4+4$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$93=4\cdot 4!-4+i$

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

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

$96=4((4!-4)+4)$

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

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

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

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