Four Fours

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:

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

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

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

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

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

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

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

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

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

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

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

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

$10=\left({\frac {4}{4}}+4\right){\sqrt {4}}$

$11=4^{\sqrt {4}}-(4+i^{4})$

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

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

$12=\left(4-{\frac {4}{4}}\right)\times 4$

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

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

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

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

$16=4+4+4+4$

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

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

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

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

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

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

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 20=4\times \left(4+{\frac {4}{4}}\right)}

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

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

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 23=4!-i^{4}+4-4}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 24=4!\times i^{4}+4-4}

$25=4!+i^{4}+4-4$

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

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

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

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

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

$31=4(4+4)-i^{4}$

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

$33=4(4+4)+i^{4}$

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

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

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

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

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

$36=4!+4+4+4$

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

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

$39=4!+4\times 4-i^{4}$

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

$40=(4+4)(4+i^{4})$

$41=4!+4\times 4+i^{4}$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$56=4!+4!+4+4$

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

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

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

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

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

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 63 = \dfrac{4^4 - 4}{4} }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 64 = (\sqrt{4})^{\sqrt{4}+\sqrt{4}+\sqrt{4}} }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 65 = (\sqrt{4})^{\sqrt{4}+4} + i^4 }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 66 = \dfrac{4^4}{4} + \sqrt{4} }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 67 = 44 + 4! - i^4 }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 68 = \dfrac{4^4}{4} + 4 }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 69 = (4! - i^4)(4-i^4) }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 70 = 44 + 4! + \sqrt{4} }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 71 = \sqrt{4}(4!) + 4! + i^4 }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 72 = 4! \times \dfrac{ \log(4+4) }{ \log{\sqrt{4}} } }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 73 = 4!\left( \sqrt{4} + i^4 \right) + i^4 }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 73 = 4! \times \sqrt{4} + 4! + i^4 }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 73 = 4! \times (4 - i^4) + i^4 }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 74 = \left( 4! + \sqrt{4} \right) + 4! \sqrt{4} }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 74 = 4! \left( 4 - i^4 \right) + \sqrt{4} }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 75 = (4! + i^4)(\sqrt{4} + i^4) }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 76 = 4! \left( \sqrt{4} + i^4 \right) + 4 }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 76 = 4! \left( 4 - i^4 \right) + 4 }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 77 = 4! \cdot 4 - \left( 4 \ln \left( e \cdot e^4 \right) - \ln \left( e \right) \right) }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 78 = (4-i^4)(4!+\sqrt{4}) }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 79 = (4F)_{4 \cdot 4} \cdot i^4 }

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 80 = \sqrt{4}^{4} \left( 4 + i^4 \right) }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 81 = \left( 4 \sqrt{4} + i^4 \right)^{\sqrt{4}} }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 82 = 4 \cdot (4! - 4) + \sqrt{4} }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 83 = \sqrt{4} \cdot 44 - 4 - \ln (e) }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 84 = 4 ( ( 4! - 4) + i^4) }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 85 = 44 \sqrt{4} - 4 + \ln (e) }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 86 = \sqrt{4} (44 - i^4) }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 87 = 44 \sqrt{4} - i^4 }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 88 = 44 \left( \dfrac{4}{\sqrt{4}} \right) }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 89 = 44 \sqrt{4} + i^4 }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 90 = 44 \sqrt{4} + \sqrt{4} }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 91 = 4 \cdot 4! - 4 - i^4 }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 92 = 4 \left( 4! - \dfrac{4}{4} \right) }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 93 = 4 \cdot 4! - 4 + i^{4} }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 94 = 4 \cdot 4! - \dfrac{4}{\sqrt{4}} }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 95 = 4 \cdot 4! - \dfrac{4}{4} }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 96 = 4 ( ( 4! - 4 ) + 4 ) }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 97 = 4 \cdot 4! + \dfrac{\log{(4)}}{\log{(4)}} }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 98 = \sqrt{4} \left( 4 \sqrt{4} - \ln{(e)}\right)^{\sqrt{4}} }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 99 = \left( 4 \cdot 4! \right) + \left( 4 - i^4 \right) }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 100 = 4 \left( 4! + \dfrac{\log{(4)}}{\log{(4)}} \right) }