From charlesreid1

See also: Five Fives

Four Fours

The goal of this puzzle is to combine 4 4's with any other mathematical symbol, excepting numbers, to produce every whole number from 1 to 20.

You can extend this to 5 5's, and 6 6's, and so on.

A good strategy is to compile a long list of all the numbers you get when you combine one 4, two 4's, three 4's, and so on. This helps you chain together expressions.

Four Fours/Table of 4s - a table of various combinations of 4s

Starting with 4s:


1 = \dfrac{4+4}{4+4}


2 = \dfrac{4 \times 4}{4 + 4}


3 = \dfrac{4 + 4 + 4}{4}


4 = \sqrt{4} \times \dfrac{4 + 4}{4}


5 = \dfrac{4 \times 4 + 4}{4}


6 = 4 \times \dfrac{ \ln{\left( 4+4 \right)} }{ \ln{4} }


7 = 4 + \sqrt{4} + \dfrac{4}{4}


8 = 4 + 4 \left( \dfrac{4}{4} \right)


8 = \sqrt{4} + \sqrt{4} + \sqrt{4} + \sqrt{4}


9 = 4 + 4 + \dfrac{4}{4}


10 = 4 + 4 + 4 - \sqrt{4}


11 = (4 \times 4) - (4 + \dfrac{4}{4})


11 = \dfrac{44}{\sqrt{4} \sqrt{4}}


12 = 4 + 4 + \sqrt{4} + \sqrt{4}


13 = \dfrac{44}{4} + \sqrt{4}


14 = 4 \times \sqrt{4} \times \sqrt{4} - \sqrt{4}


15 = 4 \times 4 - \dfrac{4}{4}


16 = \sqrt{4} \sqrt{4} \sqrt{4} \sqrt{4}


16 = 4 + 4 + 4 + 4


17 = 4 \times 4 + \dfrac{4}{4}


18 = 4 \times 4 + \dfrac{4}{\sqrt{4}}


18 = 4^{\sqrt{4}} + \dfrac{4}{\sqrt{4}}


19 = 4 \times 4 + 4 - i^{4}


20 = 4 \times 4 + \sqrt{ 4 \times 4 }


20 = \sqrt{4} \sqrt{4} + 4^{\sqrt{4}}


21 = 4 \times 4 + 4 + i^{4}


22 = \dfrac{ 
\ln{ 
\left( 
\left(\sqrt{4}\right)^{44} 
\right) 
}
}{ \ln{(4)} }


23 = 4! - i^{4}


24 = 4! \times i^{4}


25 = 4! + i^{4}


26 = 4! + \dfrac{4+4}{4}


27 = 4! + \dfrac{ \ln{(4+4)} }{ \ln{\sqrt{4}} }


28 = 4 (\sqrt{4} + i^{4} + 4)


29 = 4! + 4 + \dfrac{4}{4}


30 = (4 + i^4)(4 + \sqrt{4})


31 = 4 ( 4 + 4 ) - i^4


32 = \dfrac{ 4 \times 4 \times 4 }{ \sqrt{4} }


33 = 4 ( 4 + 4 ) + i^4


34 = 4(4+4) + \sqrt{4}


35 = (4+\sqrt{4})^{\sqrt{4}} - i^4


36 = \left( 4 + \dfrac{4}{\sqrt{4}} \right)^{\sqrt{4}}


36 = 4 \left( \sqrt{4} + i^{4} \right)^{\sqrt{4}}


36 = 4 \left( 4 \sqrt{4} + i^4 \right)


36 = 4! + 4 + 4 + 4


37 = (4+\sqrt{4})^{\sqrt{4}}


38 = \left( 4 + \sqrt{4} \right)^{\sqrt{4}} + \sqrt{4}


39 = 4! + 4 \times 4 - i^4


40 = 4 (4+4+\sqrt{4})


40 = (4+4)(4+i^4)


41 = 4! + 4 \times 4 + i^4


42 = (4!)(\sqrt{4}) - (4+\sqrt{4})


43 = (4!)(\sqrt{4}) - (4+i^4)


44 = (4!)(\sqrt{4}) - (\sqrt{4} + \sqrt{4})


44 = \sqrt{4} \left( 4! - \dfrac{4}{\sqrt{4}} \right)


45 = (4! - \sqrt{4}) + (4! - i^4)


45 = (4! - \sqrt{4})( \sqrt{4} ) + i^4


46 = 4! + 4! - \dfrac{4}{\sqrt{4}}


46 = \sqrt{4}(4!) - \dfrac{4}{\sqrt{4}}


46 = \sqrt{4} \left( 4! - \sqrt{4} \right) + \sqrt{4}


47 = 4! \sqrt{4} - \dfrac{4}{4}


48 = (4!)(\sqrt{4}) \left( \dfrac{4}{4} \right)


49 = (\sqrt{4})(4!) + \dfrac{4}{4}


50 = (\sqrt{4})(4!) + \dfrac{4}{\sqrt{4}}


51 = (\sqrt{4})(4!) + 4 - i^{4}


52 = (4!)(\sqrt{4} + \sqrt{4}\sqrt{4}


53 = (4!)(\sqrt{4}) + 4 + i^4


54 = 4! + 4! + 4 + \sqrt{4}


55 = (4!+4) \times \sqrt{4} - i^4


56 = 4! \left( \sqrt{4} + \dfrac{i^4}{4} \right)


56 = 4! + 4! + 4 + 4


57 = (4!+4) \times \sqrt{4} + i^4


58 = (4!+4) \times \sqrt{4} + \sqrt{4}


59 = \dfrac{ (4+i^4)! - \sqrt{4} }{\sqrt{4}}


60 = (4!+4) \times \sqrt{4} + 4


61 = \dfrac{ (4+i^4)! + \sqrt{4}}{\sqrt{4}}


62 = \dfrac{(4+i^4)!+4}{\sqrt{4}}


63 = \dfrac{4^4 - 4}{4}


64 = (\sqrt{4})^{\sqrt{4}+\sqrt{4}+\sqrt{4}}


65 = (\sqrt{4})^{\sqrt{4}+4} + i^4


66 = \dfrac{4^4}{4} + \sqrt{4}


67 = 44 + 4! - i^4


68 = \dfrac{4^4}{4} + 4


69 = (4! - i^4)(4-i^4)


70 = 44 + 4! + \sqrt{4}


71 = \sqrt{4}(4!) + 4! + i^4


72 = 4! \times \dfrac{ \log(4+4) }{ \log{\sqrt{4}} }


73 = 4!\left( \sqrt{4} + i^4 \right) + i^4


73 = 4! \times \sqrt{4} + 4! + i^4


73 = 4! \times (4 - i^4) + i^4


74 = \left( 4! + \sqrt{4} \right) + 4! \sqrt{4}


74 = 4! \left( 4 - i^4 \right) + \sqrt{4}


75 = (4! + i^4)(\sqrt{4} + i^4)


76 = 4! \left( \sqrt{4} + i^4 \right) + 4


76 = 4! \left( 4 - i^4 \right) + 4


77 = 4! \cdot 4 - \left( 4 \ln \left( e \cdot e^4 \right) - \ln \left( e \right) \right)


78 = (4-i^4)(4!+\sqrt{4})


79 = (4F)_{4 \cdot 4} \cdot i^4

(That is, the number 4F in hex, or base 16.)


80 = \sqrt{4}^{4} \left( 4 + i^4 \right)


81 = \left( 4 \sqrt{4} + i^4 \right)^{\sqrt{4}}


82 = 4 \cdot (4! - 4) + \sqrt{4}


83 = \sqrt{4} \cdot 44 - 4 - \ln (e)


84 = 4 ( ( 4! - 4) + i^4)


85 = 44 \sqrt{4} - 4 + \ln (e)


86 = \sqrt{4} (44 - i^4)


87 = 44 \sqrt{4} - i^4


88 = 44 \left( \dfrac{4}{\sqrt{4}} \right)


89 = 44 \sqrt{4} + i^4


90 = 44 \sqrt{4} + \sqrt{4}


91 = 4 \cdot 4! - 4 - i^4


92 = 4 \left( 4! - \dfrac{4}{4} \right)


93 = 4 \cdot 4! - 4 + i


94 = 4 \cdot 4! - \dfrac{4}{\sqrt{4}}


95 = 4 \cdot 4! - \dfrac{4}{4}


96 = 4 ( ( 4! - 4 ) + 4 )


97 = 4 \cdot 4! + \dfrac{\log{(4)}}{\log{(4)}}


98 = \sqrt{4} \left( 4 \sqrt{4} - \ln{(e)}\right)^{\sqrt{4}}


99 = \left( 4 \cdot 4! \right) + \left( 4 - i^4 \right)


100 = 4 \left( 4! + \dfrac{\log{(4)}}{\log{(4)}} \right)

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