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)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$32 = \left( \dfrac{ \ln{5} }{ \ln{ \dfrac{5}{\sqrt{5}} } } \right)^{\sqrt{5\times5}}$

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

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

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

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

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

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

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

$$ 40 = 55 - 5 - 5 - 5 $$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Five Fives

From charlesreid1

Five Fives