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}}

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

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

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

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}}}

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

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

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 7=5+{\dfrac {5}{5}}+{\dfrac {5}{5}}}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 8=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 9={\sqrt {5}}{\sqrt {5}}+5-{\dfrac {5}{5}}}

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

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

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

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

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

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

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

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

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

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 19=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 20={\dfrac {5}{5}}\left(5\times 5-5\right)}

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

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

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

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

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 25={\dfrac {5^{5}}{5}}-5\times 5!}

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

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

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

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

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

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}}}

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

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

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

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

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

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

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

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

$40=55-5-5-5$

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

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

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

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

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

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

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

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

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

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

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

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)}

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

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)}

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

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

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

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

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

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

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

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

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}}}}

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

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

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 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 (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 69 = 5! - 55 + \dfrac{ \ln{\left(5\right)} }{ \ln{\left(\sqrt{\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 70 = 5 \times 5 \times 5 - 55 }

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 71 = 5! - \frac{5!}{5} - \left( 5 \times 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 72 = \dfrac{ 5 \times 5! + 5! }{ 5 + 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 73 = 5!! + \phi^{5} - 5 \phi + 55 }

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 74 = 55 - 5 + \frac{5!}{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 75 = 5 \times 5 \times \log_{5}{\left(5! + 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 76 = \dfrac{ \ln{\left(5\right)} }{ \ln{\left(\sqrt{ 5 }\right)} } \left( 5!! + \frac{5!}{5} - \ln{e} \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 77 = \left( \phi^5 - \phi^{-5} \right) \left( 5 + \dfrac{ \ln{\left( 5 \right)} }{ \ln{\left(\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 78 = \left( \phi^{5} - 5 \phi \right) \left( \phi^5 - \phi^{-5} + 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 79 = 5 \left( \phi^5 - \phi^{-5} \right) - \frac{5!}{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 80 = \left( \phi^5 - 5 \phi + 5 \right) \left( 5 + 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 81 = \left( \phi^{5} - 5 \phi \right)^{\left(5 - \frac{5}{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 82 = 5 \left( 5 + 5 \right) + \dfrac{ \ln{\left(5\right)} }{ \ln{ \left( \sqrt{ \sqrt{ \sqrt{ \sqrt{ \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 83 = 5! - 5 - \left( \dfrac{ \ln{ 5 } }{ \ln{ \left( \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 84 = 5! - 5 \times 5 - \phi^{5} + \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 84 = \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 85 = 5! - 5 \times 5 + 5 - 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 86 = 5! - \dfrac{5!}{5} + 5 - 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 87 = 55 + \left( \dfrac{ \ln{\left(5\right)} }{ \ln{\left(\sqrt{5}\right)} } \right)^{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 88 = \left( \phi^5 - \phi^{-5} \right) \left( \phi^5 - 5 \phi + 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 89 = 5! - \dfrac{5!}{5} - 5 + \sin{ \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 90 = \left( 5 + 5 \right) \left( 5 + 5 + \cos{\left( 5 \pi \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 91 = 5! - \dfrac{5!}{5} - 5 + \sin{\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 92 = 5! - 5 \times 5 - \phi^5 + 5 \phi }

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

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 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} } }