Five Fives

See also: Four Fours

Five Fives

Extending the idea of Four Fours, where the intention is to write each integer as a combination of 4 4's (any mathematical symbol is allowed except for digits that are not 4), we can create the game of Five Fives.

Write each integer as a combination of 5 5's - no other numerical digits allowed.

(It is useful to build a table of different combinations of 5 to help out: Five Fives/Table of 5s)

$1=\left({\dfrac {5+5}{5+5}}\right)^{5}$

$2={\dfrac {5\ln {5}}{\ln {\left({\dfrac {5^{5}}{\sqrt {5}}}\right)}}}$

$3={\dfrac {\log _{5}{(5!+5)^{5}}}{5}}$

$4=5-{\dfrac {5^{5}}{5^{5}}}$

$5={\dfrac {{\sqrt {5}}\times {\sqrt {5}}^{5}}{5\times 5}}$

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

$6=5+{\dfrac {5\times 5}{5\times 5}}$

$7=5+{\dfrac {5}{5}}+{\dfrac {5}{5}}$

$8=5+{\dfrac {5+5+5}{5}}$

$9={\sqrt {5}}{\sqrt {5}}+5-{\dfrac {5}{5}}$

$10={\dfrac {5\times 5+5\times 5}{5}}$

$11={\dfrac {5\times 5+5}{5}}+5$

$12=5+5+{\dfrac {5+5}{5}}$

$13=5+5+5-{\dfrac {\ln {5}}{\ln {\sqrt {5}}}}$

$14=5+5+5-{\dfrac {5}{5}}$

$15=\left({\dfrac {5+5}{5}}\right)\times 5+5$

$16=5+5+5+{\dfrac {5}{5}}$

$17=5+5+5+{\dfrac {\ln {5}}{\ln {\sqrt {5}}}}$

$18=5\times 5-5-{\dfrac {\ln {5}}{\ln {\sqrt {5}}}}$

$19=5\times 5-5-{\dfrac {5}{5}}$

$20={\dfrac {5}{5}}\left(5\times 5-5\right)$

$21={\dfrac {\sqrt {5^{5}}}{\sqrt {5}}}-{\dfrac {\ln {5}}{\ln {\left({\sqrt {\sqrt {5}}}\right)}}}$

$22=(5\times 5)-\log _{5}{\left(5!+5\right)}$

$23={\dfrac {5!}{5}}-{\dfrac {\sqrt {5\times 5}}{5}}$

$24=5\times 5\times \left({\dfrac {5!+5}{5!}}\right)$

$25={\dfrac {5^{5}}{5}}-5\times 5!$

$26=5\times 5+{\dfrac {\sqrt {5\times 5}}{5}}$

$27={\dfrac {5!}{5}}+5-{\dfrac {\ln {5}}{\ln {\sqrt {5}}}}$

$28={\dfrac {5!}{5}}+5-{\dfrac {5}{5}}$

$29=5\times 5+5-{\dfrac {5}{5}}$

$30=5\left({\dfrac {\ln {5}}{\ln {\sqrt {5}}}}+{\dfrac {\ln {5}}{\ln {\sqrt {\sqrt {5}}}}}\right)$

$31=5\times 5+5+{\dfrac {5}{5}}$

$32=\left({\dfrac {\ln {5}}{\ln {\dfrac {5}{\sqrt {5}}}}}\right)^{\sqrt {5\times 5}}$

$33=\left({\dfrac {\ln {5}}{\ln {\sqrt {5}}}}\right)^{5}+{\dfrac {5}{5}}$

$34=5\times 5+5+{\dfrac {\ln {5}}{\ln {\sqrt {\sqrt {5}}}}}$

$35={\dfrac {55}{5}}+{\dfrac {5!}{5}}$

$36=5\times 5+{\dfrac {55}{5}}$

$37={\dfrac {5!}{5+5}}+5\times 5$

$38=\left({\dfrac {5!}{5}}-5\right)\left({\dfrac {\ln {5}}{\ln {\sqrt {5}}}}\right)$

$39={\dfrac {5!}{5}}+5+5+5$

$40=55-5-5-5$

$41=\phi ^{5}-\phi ^{-5}+5\times 5+5$

(where $\phi ={\dfrac {1+{\sqrt {5}}}{2}}$ is the Golden Ratio)

$42=5+5+\left({\dfrac {\ln {5}}{\ln {\sqrt {5}}}}\right)^{5}$

$43=55-{\dfrac {5!}{5+5}}$

$44=\left(5-{\dfrac {5}{5}}\right)\left(\phi ^{5}-\phi ^{-5}\right)$

$45=55-{\sqrt {5}}\left({\sqrt {5}}+{\sqrt {5}}\right)$

$46=55-\left(5+5\right)+\left(\ln {e}\right)^{5}$

$47=55-\phi ^{5}+5\phi -5$

$48={\dfrac {5!\ln {\left(5\right)}}{5\ln {\left({\dfrac {5}{\sqrt {5}}}\right)}}}$

$49=55-\left(5+{\dfrac {5}{5}}\right)$

$50=\left({\sqrt {5}}+{\sqrt {5}}\right)\left({\sqrt {5}}\right)\left({\sqrt {5}}\right)\left({\sqrt {5}}\right)$

$51=5\left(5+5\right)+{\dfrac {5}{5}}$

$51=55-\left(5-{\dfrac {5}{5}}\right)$

$52=5\left(\phi ^{5}-\phi ^{-5}\right)-\left(\phi ^{5}-5\phi \right)$

$53=55-5+\left(\phi ^{5}-5\phi \right)$

$54=5+\left(5\times 5+{\dfrac {5!}{5}}\right)$

$55=\left({\sqrt {5}}\right)\left({\sqrt {5}}\right)\left({\dfrac {55}{5}}\right)$

$56=\left({\dfrac {5!}{5}}\right)+\left({\dfrac {\ln {\left(5\right)}}{\ln {\left({\sqrt {5}}\right)}}}\right)^{5}$

$57=\left({\dfrac {\ln {\left(5\right)}}{\ln {\left({\sqrt {5}}\right)}}}\right)^{5}+5\times 5$

$58=55+\log _{5}{(5!+5)}$

$59={\dfrac {5!}{\left({\dfrac {\ln {\left(5\right)}}{\ln {\left({\sqrt {5}}\right)}}}\right)}}-{\dfrac {5}{5}}$

$60=5!-{\dfrac {5\times 5!}{5+5}}$

$61={\dfrac {5!}{\left({\dfrac {\ln {\left(5\right)}}{\ln {\left({\sqrt {5}}\right)}}}\right)}}+{\dfrac {5}{5}}$

$61=55+5+{\dfrac {5}{5}}$

$62=5+55+{\dfrac {\ln {\left(5\right)}}{\ln {\left({\sqrt {5}}\right)}}}$

$63={\dfrac {5!}{\left({\dfrac {\ln {\left(5\right)}}{\ln {\left({\sqrt {5}}\right)}}}\right)}}+\left(\phi ^{5}-5\phi \right)$

$64=\left({\dfrac {\ln {5}}{\ln {\sqrt {5}}}}\right)^{5+{\frac {5}{5}}}$

$65=5\left(\phi ^{5}-\phi ^{-5}\right)+5+5$

$66=\left(\phi ^{5}-\phi ^{-5}\right)\left(5+{\frac {5}{5}}\right)$

$67=\left(\phi ^{5}-\phi ^{-5}\right)+5\left(\phi ^{5}-\phi ^{-5}\right)+\ln {e}$

$68=\left(\phi ^{5}-\phi ^{-5}\right)\left(5+\ln {e}\right)+{\dfrac {\ln {\left(5\right)}}{\ln {\left({\sqrt {5}}\right)}}}$

$69=5!-55+{\dfrac {\ln {\left(5\right)}}{\ln {\left({\sqrt {\sqrt {5}}}\right)}}}$

$70=5\times 5\times 5-55$

$71=5!-{\frac {5!}{5}}-\left(5\times 5\right)$

$72={\dfrac {5\times 5!+5!}{5+5}}$

$73=5!!+\phi ^{5}-5\phi +55$

$74=55-5+{\frac {5!}{5}}$

$75=5\times 5\times \log _{5}{\left(5!+5\right)}$

$76={\dfrac {\ln {\left(5\right)}}{\ln {\left({\sqrt {5}}\right)}}}\left(5!!+{\frac {5!}{5}}-\ln {e}\right)$

$77=\left(\phi ^{5}-\phi ^{-5}\right)\left(5+{\dfrac {\ln {\left(5\right)}}{\ln {\left({\sqrt {5}}\right)}}}\right)$

$78=\left(\phi ^{5}-5\phi \right)\left(\phi ^{5}-\phi ^{-5}+5!!\right)$

$79=5\left(\phi ^{5}-\phi ^{-5}\right)-{\frac {5!}{5}}$

$80=\left(\phi ^{5}-5\phi +5\right)\left(5+5\right)$

$81=\left(\phi ^{5}-5\phi \right)^{\left(5-{\frac {5}{5}}\right)}$

$82=5\left(5+5\right)+{\dfrac {\ln {\left(5\right)}}{\ln {\left({\sqrt {\sqrt {\sqrt {\sqrt {\sqrt {5}}}}}}\right)}}}$

$83=5!-5-\left({\dfrac {\ln {5}}{\ln {\left({\sqrt {5}}\right)}}}\right)$

$84=5!-5\times 5-\phi ^{5}+\phi ^{-5}$

$84={\dfrac {\phi \left(\phi ^{5+5}\right)-55}{\phi }}-5$

$85=5!-5\times 5+5-5!!$

$86=5!-{\dfrac {5!}{5}}+5-5!!$

$87=55+\left({\dfrac {\ln {\left(5\right)}}{\ln {\left({\sqrt {5}}\right)}}}\right)^{5}$

$88=\left(\phi ^{5}-\phi ^{-5}\right)\left(\phi ^{5}-5\phi +5\right)$

$89=5!-{\dfrac {5!}{5}}-5+\sin {\left(5\pi \right)}$

$90=\left(5+5\right)\left(5+5+\cos {\left(5\pi \right)}\right)$

$91=5!-{\dfrac {5!}{5}}-5+\sin {\left(5\pi \right)}$

$92=5!-5\times 5-\phi ^{5}+5\phi$

$93=5!-{\dfrac {5!}{5}}-\phi ^{5}+5\phi$

$94={\dfrac {\phi \left(\phi ^{5+5}\right)-55}{\phi }}+5$

$95=5!+5-5-5\times 5$

$96=5!-5\times 5+\sin {\left(5\pi \right)}-\cos {\left(5\pi \right)}$

$97=5!-5\times 5+{\dfrac {\ln {\left(5\right)}}{\ln {\left({\sqrt {5}}\right)}}}$

$98=\phi ^{5}-5\phi +5!-5\times 5$

$99=\left(\phi ^{5}-\phi ^{-5}\right)\left(5+{\dfrac {\ln {\left(5\right)}}{\ln {\left({\sqrt {\sqrt {5}}}\right)}}}\right)$

$100={\dfrac {5\times 5!}{5+{\frac {5}{5}}}}$