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\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} } $