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.

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

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

$1={\dfrac {5\times 5\times 5-5}{5!}}$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$36={\dfrac {5^{5}-5}{5!}}+5+5$

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

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

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

$40=55-5-5-5$

$41=\phi ^{5}-{\dfrac {1}{\phi ^{5}}}+5\times 5+5$

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

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

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

$44=\left(5-{\dfrac {5}{5}}\right)\left(\phi ^{5}-{\dfrac {1}{\phi ^{5}}}\right)$

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

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