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:

${\displaystyle 1=i^{4}}$

${\displaystyle 2={\sqrt {4}}}$

${\displaystyle 24=4!}$

The following fractions are also useful:

${\displaystyle {\dfrac {1}{4}}}$

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

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

but these can't appear with 1 in the denominator.

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

${\displaystyle 5={\dfrac {\ln {\left({\dfrac {\ln {\left({\sqrt {\sqrt {\sqrt {\sqrt {\sqrt {x}}}}}}\right)}}{\ln {x}}}\right)}}{\ln {\sqrt {4}}}}}$

## Two 4s

Note: we aren't using fourth roots or one-quarter powers very much, e.g., ${\displaystyle {\sqrt {2}}={\sqrt[{4}]{4}}=4^{\frac {1}{4}}}$. Adding this would greatly expand the possibilities.

${\displaystyle 1={\dfrac {4}{4}}}$

${\displaystyle 1=\log _{4}(4)}$

${\displaystyle 2=4-{\sqrt {4}}}$

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

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

${\displaystyle 3=4-i^{4}}$

${\displaystyle 4={\sqrt {4}}\times {\sqrt {4}}}$

${\displaystyle 4={\sqrt {4}}+{\sqrt {4}}}$

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

${\displaystyle 5=4+i^{4}}$

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

${\displaystyle 6=4+{\sqrt {4}}}$

${\displaystyle 6=(4-i^{4})!}$

${\displaystyle 8=4+4}$

${\displaystyle 8=4{\sqrt {4}}}$

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

${\displaystyle 16=4\times 4}$

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

${\displaystyle 16={\sqrt {4^{4}}}}$

${\displaystyle 20=4!-4}$

${\displaystyle 22=4!-{\sqrt {4}}}$

${\displaystyle 23=4!-i^{4}}$

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

${\displaystyle 25=4!+i^{4}}$

${\displaystyle 26=4!+{\sqrt {4}}}$

${\displaystyle 28=4!+4}$

${\displaystyle 44}$

${\displaystyle 48=4!\times {\sqrt {4}}}$

${\displaystyle 96=4!\times 4}$

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

${\displaystyle 256=4^{4}}$

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

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

${\displaystyle 331,776=(4!)^{4}}$

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

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

${\displaystyle 1,333,735,776,850,284,124,449,081,472,843,776={4!}^{4!}}$

(For those keeping score at home, that's 1 decillion 333 nonillion 735 octillion 776 septillion 850 sextillion 284 quintillion 124 quadrillion 449 trillion 81 billion 472 million 843 thousand 776)

## 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.

One useful template for representing powers of 2 is:

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

where ${\displaystyle n}$ is any number expressible with two 4's.

Once we can add three 4's, we can start to write expressions like

${\displaystyle {\dfrac {\ln {P}}{\ln {Q}}}}$

where P and Q are any expressions involving 1 or 2 fours. If we choose carefully, P and Q will have the same base and different exponents, so we can start to combine integer powers to obtain new integers. This trick will get us out of at least a few jams when constructing the integers from 1 to 100 using only four fours.

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

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

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

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

${\displaystyle 4=4+4-4}$

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

${\displaystyle 7=4+4-i^{4}}$

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

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

${\displaystyle 9=4+4+i^{4}}$

${\displaystyle 10=4+4+{\sqrt {4}}}$

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

${\displaystyle 12=4+4+4}$

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

${\displaystyle 20=4\times 4+4}$

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

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

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

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

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

${\displaystyle 32=4(4+4)}$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

${\displaystyle 80=4(4!-4)}$

${\displaystyle 116=((4+i^{4})!-4}$

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

${\displaystyle 119=((4+i^{4})!-i^{4})}$

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

${\displaystyle 121=((4+i^{4})!+i^{4})}$

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

${\displaystyle 124=((4+i^{4})!+4}$

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

${\displaystyle 252=4^{4}-4}$

${\displaystyle 254=4^{4}-{\sqrt {4}}}$

${\displaystyle 258=4^{4}+{\sqrt {4}}}$

${\displaystyle 260=4^{4}+4}$

${\displaystyle 1,024=4^{4}\times 4}$

${\displaystyle 4,096=(4+4)^{4}}$

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