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)

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 1=\left({\dfrac {5+5}{5+5}}\right)^{5}}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 2={\dfrac {5\ln {5}}{\ln {\left({\dfrac {5^{5}}{\sqrt {5}}}\right)}}}}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 3={\dfrac {\log _{5}{(5!+5)^{5}}}{5}}}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 4=5-{\dfrac {5^{5}}{5^{5}}}}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 5={\dfrac {{\sqrt {5}}\times {\sqrt {5}}^{5}}{5\times 5}}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 22=(5\times 5)-\log _{5}{\left(5!+5\right)}}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 23={\dfrac {5!}{5}}-{\dfrac {\sqrt {5\times 5}}{5}}}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 24=5\times 5\times \left({\dfrac {5!+5}{5!}}\right)}

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

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 26=5\times 5+{\dfrac {\sqrt {5\times 5}}{5}}}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 27={\dfrac {5!}{5}}+5-{\dfrac {\ln {5}}{\ln {\sqrt {5}}}}}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 28={\dfrac {5!}{5}}+5-{\dfrac {5}{5}}}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 29=5\times 5+5-{\dfrac {5}{5}}}

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

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 31=5\times 5+5+{\dfrac {5}{5}}}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 32=\left({\dfrac {\ln {5}}{\ln {\dfrac {5}{\sqrt {5}}}}}\right)^{\sqrt {5\times 5}}}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 33=\left({\dfrac {\ln {5}}{\ln {\sqrt {5}}}}\right)^{5}+{\dfrac {5}{5}}}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 34=5\times 5+5+{\dfrac {\ln {5}}{\ln {\sqrt {\sqrt {5}}}}}}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 35={\dfrac {55}{5}}+{\dfrac {5!}{5}}}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 36=5\times 5+{\dfrac {55}{5}}}

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

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 38=\left({\dfrac {5!}{5}}-5\right)\left({\dfrac {\ln {5}}{\ln {\sqrt {5}}}}\right)}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 39={\dfrac {5!}{5}}+5+5+5}

$40=55-5-5-5$

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 41=\phi ^{5}-\phi ^{-5}+5\times 5+5}

(where Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \phi ={\dfrac {1+{\sqrt {5}}}{2}}} is the Golden Ratio)

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 42=5+5+\left({\dfrac {\ln {5}}{\ln {\sqrt {5}}}}\right)^{5}}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 43=55-{\dfrac {5!}{5+5}}}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 44=\left(5-{\dfrac {5}{5}}\right)\left(\phi ^{5}-\phi ^{-5}\right)}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 45=55-{\sqrt {5}}\left({\sqrt {5}}+{\sqrt {5}}\right)}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 46=55-\left(5+5\right)+\left(\ln {e}\right)^{5}}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 47=55-\phi ^{5}+5\phi -5}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 48={\dfrac {5!\ln {\left(5\right)}}{5\ln {\left({\dfrac {5}{\sqrt {5}}}\right)}}}}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 49=55-\left(5+{\dfrac {5}{5}}\right)}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 50=\left({\sqrt {5}}+{\sqrt {5}}\right)\left({\sqrt {5}}\right)\left({\sqrt {5}}\right)\left({\sqrt {5}}\right)}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 51=5\left(5+5\right)+{\dfrac {5}{5}}}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 51=55-\left(5-{\dfrac {5}{5}}\right)}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 52=5\left(\phi ^{5}-\phi ^{-5}\right)-\left(\phi ^{5}-5\phi \right)}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 53=55-5+\left(\phi ^{5}-5\phi \right)}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 54=5+\left(5\times 5+{\dfrac {5!}{5}}\right)}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 55=\left({\sqrt {5}}\right)\left({\sqrt {5}}\right)\left({\dfrac {55}{5}}\right)}

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

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

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

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 59={\dfrac {5!}{\left({\dfrac {\ln {\left(5\right)}}{\ln {\left({\sqrt {5}}\right)}}}\right)}}-{\dfrac {5}{5}}}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 60=5!-{\dfrac {5\times 5!}{5+5}}}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 61={\dfrac {5!}{\left({\dfrac {\ln {\left(5\right)}}{\ln {\left({\sqrt {5}}\right)}}}\right)}}+{\dfrac {5}{5}}}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 61=55+5+{\dfrac {5}{5}}}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 62=5+55+{\dfrac {\ln {\left(5\right)}}{\ln {\left({\sqrt {5}}\right)}}}}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 63={\dfrac {5!}{\left({\dfrac {\ln {\left(5\right)}}{\ln {\left({\sqrt {5}}\right)}}}\right)}}+\left(\phi ^{5}-5\phi \right)}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 64=\left({\dfrac {\ln {5}}{\ln {\sqrt {5}}}}\right)^{5+{\frac {5}{5}}}}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 65=5\left(\phi ^{5}-\phi ^{-5}\right)+5+5}

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

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 67=\left(\phi ^{5}-\phi ^{-5}\right)+5\left(\phi ^{5}-\phi ^{-5}\right)+\ln {e}}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 68=\left(\phi ^{5}-\phi ^{-5}\right)\left(5+\ln {e}\right)+{\dfrac {\ln {\left(5\right)}}{\ln {\left({\sqrt {5}}\right)}}}}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 69=5!-55+{\dfrac {\ln {\left(5\right)}}{\ln {\left({\sqrt {\sqrt {5}}}\right)}}}}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 70=5\times 5\times 5-55}

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

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 72={\dfrac {5\times 5!+5!}{5+5}}}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 73=5!!+\phi ^{5}-5\phi +55}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 74=55-5+{\frac {5!}{5}}}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 75=5\times 5\times \log _{5}{\left(5!+5\right)}}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 76={\dfrac {\ln {\left(5\right)}}{\ln {\left({\sqrt {5}}\right)}}}\left(5!!+{\frac {5!}{5}}-\ln {e}\right)}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 77=\left(\phi ^{5}-\phi ^{-5}\right)\left(5+{\dfrac {\ln {\left(5\right)}}{\ln {\left({\sqrt {5}}\right)}}}\right)}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 78=\left(\phi ^{5}-5\phi \right)\left(\phi ^{5}-\phi ^{-5}+5!!\right)}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 79=5\left(\phi ^{5}-\phi ^{-5}\right)-{\frac {5!}{5}}}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 80=\left(\phi ^{5}-5\phi +5\right)\left(5+5\right)}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 81=\left(\phi ^{5}-5\phi \right)^{\left(5-{\frac {5}{5}}\right)}}

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

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 83=5!-5-\left({\dfrac {\ln {5}}{\ln {\left({\sqrt {5}}\right)}}}\right)}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 84=5!-5\times 5-\phi ^{5}+\phi ^{-5}}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 84={\dfrac {\phi \left(\phi ^{5+5}\right)-55}{\phi }}-5}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 85=5!-5\times 5+5-5!!}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 86=5!-{\dfrac {5!}{5}}+5-5!!}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 87=55+\left({\dfrac {\ln {\left(5\right)}}{\ln {\left({\sqrt {5}}\right)}}}\right)^{5}}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 88=\left(\phi ^{5}-\phi ^{-5}\right)\left(\phi ^{5}-5\phi +5\right)}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 89=5!-{\dfrac {5!}{5}}-5+\sin {\left(5\pi \right)}}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 90=\left(5+5\right)\left(5+5+\cos {\left(5\pi \right)}\right)}

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

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 92=5!-5\times 5-\phi ^{5}+5\phi }

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

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 94={\dfrac {\phi \left(\phi ^{5+5}\right)-55}{\phi }}+5}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 95 = 5! + 5 - 5 - 5 \times 5 }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 96 = 5! - 5 \times 5 + \sin{\left( 5 \pi \right)} - \cos{\left( 5 \pi \right)} }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 97 = 5! - 5 \times 5 + \dfrac{ \ln{ \left( 5 \right) } }{ \ln{ \left( \sqrt{5} \right)} } }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 98 = \phi^5 - 5 \phi + 5! - 5 \times 5 }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 99 = \left( \phi^5 - \phi^{-5} \right) \left( 5 + \dfrac{ \ln{ \left( 5 \right) } }{ \ln{ \left( \sqrt{ \sqrt{ 5 } } \right) } } \right) }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 100 = \dfrac{ 5 \times 5!}{ 5 + \frac{5}{5} } }