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:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$$ 16 = 4 + 4 + 4 + 4 $$

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

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

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

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

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

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

$20 = 4 \times \left( 4 + \frac{4}{4} \right)$

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

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

$23 = 4! - i^{4} + 4 - 4$

$24 = 4! \times i^{4} + 4 - 4$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$$ 67 = 44 + 4! - i^4 $$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$93 = 4 \cdot 4! - 4 + i^{4}$

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

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

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

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

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

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

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

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Four Fours

From charlesreid1

Four Fours

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