From charlesreid1

Back to Four Fours

One 4

Believe it or not, the rules allow you to do quite a bit with a single 4. The rules say you may combine 4s with any mathematical symbol except numbers. Thus, in addition to 4 alone, we also have:


1 = i^{4}


2 = \sqrt{4}


24 = 4!

The following fractions are also useful:


\dfrac{1}{4}


\dfrac{1}{2} = \dfrac{1}{\sqrt{4}}


\dfrac{1}{24} = \dfrac{1}{4!}

Could possibly add constants (harmonic number), possibly special functions.

Once you allow variables like x into the mix, it's lights out.

One 4 with Variables


5 = 
\dfrac{ 

\ln{
\left(
\dfrac{

\ln{ \left( \sqrt{ \sqrt{ \sqrt{ \sqrt{ \sqrt{ x } } } } } \right) }

}{
\ln{x}
}
\right)
}

}{

\ln{\sqrt{4}}

}

Two 4s


1 = \dfrac{4}{4}


1 = \log_{4}(4)


2 = 4 - \sqrt{4}


2 = \dfrac{4}{\sqrt{4}}


3 = \sqrt{4} + i^{4}


3 = 4 - i^{4}


4 = \sqrt{4} \times \sqrt{4}


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


4 = \dfrac{4}{i^4}


5 = 4 + i^{4}


5 = \sqrt{ 4! + i^4 }


6 = 4 + \sqrt{4}


6 = (4 - i^4)!


8 = 4+4


8 = 4 \sqrt{4}


12 = \dfrac{4!}{\sqrt{4}}


16 = 4 \times 4


16 = 4^{\sqrt{4}} = (\sqrt{4})^4


16 = \sqrt{4^4}


20 = 4! - 4


22 = 4! - \sqrt{4}


23 = 4! - i^4


24 = \sqrt{ (4!)^{\sqrt{4}} }


25 = 4! + i^4


26 = 4! + \sqrt{4}


28 = 4! + 4


44


48 = 4! \times \sqrt{4}


96 = 4! \times 4


120 = (4 + i^4)!


256 = 4^4


576 = (4!)^{\sqrt{4}}


720 = (4 + \sqrt{4})!


331,776 = (4!)^4


16,777,216 = \sqrt{4}^{4!}


281,474,976,710,656 = 4^{4!}

Three 4s

These lists blow up pretty fast... as you can see, focusing on using a smaller number of 4s can force you to be creative. This makes it possible to combine 4 4's beyond the integers from 1 to 20, and keep on going...


2^n = (\sqrt{4})^n

where n is any number expressible with two 4's.


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


2 = \sqrt{4} \times \left( \dfrac{4}{4} \right)


3 = \dfrac{ \ln{(4+4)} }{ \ln{\sqrt{4}} }


4 = 4 \times \left( \dfrac{4}{4} \right)


4 = 4 + 4 - 4


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


7 = 4 + 4 - i^{4}


7 = 4 + \sqrt{4} + i^{4}


8 = 4 \times \left( \dfrac{4}{\sqrt{4}} \right)


9 = 4 + 4 + i^{4}



10 = 4 + 4 + \sqrt{4}


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


12 = 4+4+4


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


20 = 4 \times 4 + 4


26 = 4! + 4 - \sqrt{4}


28 = 4! + \sqrt{4} + \sqrt{4}


30 = 4! + 4 + \sqrt{4}


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


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


32 = 4(4+4)


32 = 4^{\sqrt{4}} \sqrt{4}


36 = (4+\sqrt{4})^{\sqrt{4}}


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


47 = (4!)(\sqrt{4}) - i^4


49 = (4!)(\sqrt{4}) + i^4


50 = 4! + 4! + \sqrt{4}


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


64 = \dfrac{4^4}{4}


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


74 = (4A)_{4 \cdot 4}

(That is, 74 is 4A in hexidecimal, or base 16 = 4 * 4)


75 = (4B)_{4 \cdot 4}


76 = (4C)_{4 \cdot 4}


77 = (4D)_{4 \cdot 4}


78 = (4E)_{4 \cdot 4}


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


80 = 4(4! - 4)


116 = ( (4 + i^4)! - 4


118 = ( (4 + i^4)! - \sqrt{4}


119 = ( (4 + i^4)! - i^4)


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


121 = ( (4 + i^4)! + i^4)


122 = ( (4 + i^4)! + \sqrt{4})


124 = ( (4 + i^4)! + 4


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


252 = 4^4 - 4


254 = 4^4 - \sqrt{4}


258 = 4^4 + \sqrt{4}


260 = 4^4 + 4


1,024 = 4^4 \times 4


4,096 = (4+4)^{4}


65,536 = (4 \times 4)^{4}