From charlesreid1

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\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 = \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