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) $42 = 5 + 5 + \left( \dfrac{ \ln{5} }{ \ln{\sqrt{5}} } \right)^5$ $43 = 55 - \dfrac{5!}{5+5}$ $44 = \left( 5 - \dfrac{5}{5} \right) \left( \phi^5 - \dfrac{1}{\phi^5} \right)$ $45 = 55 - \sqrt{5} \left( \sqrt{5} + \sqrt{5} \right)$ $46 = 55 - \left( 5 + 5 \right) + \left( \ln{e} \right)^5$