Four Fours

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.

Numbers Puzzle/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} $

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

$ 6 = 4 = \dfrac{4+4}{4} $

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

Five Fives

Extending this idea, we can take a crack at the game of Five Fives.

$ 5 = \dfrac{ \sqrt{5}^{\sqrt{5}} \sqrt{5} }{ 5 \times 5 } $

$ 6 = 5 + \dfrac{5 \times 5}{5 \times 5} $

$ 7 = 5 + \dfrac{5}{5} + \dfrac{5}{5} $

$ 8 = 5 + \dfrac{5+5+5}{5} $

$ 9 = \sqrt{5} \sqrt{5} + 5 - \dfrac{5}{5} $

$ 10 = \dfrac{5 \times 5 + 5 \times 5}{5} $

$ 11 = \dfrac{5 \times 5 + 5}{5} + 5 $

$ 12 = 5 + 5 + \dfrac{5+5}{5} $

$ 13 = 5 + 5 + 5 - \dfrac{ \ln{5} }{ \ln{\sqrt{5}} } $

$ 14 = 5 + 5 + 5 - \dfrac{5}{5} $

$ 15 = \left( \dfrac{5+5}{5} \right) \times 5 + 5 $

$ 16 = 5 + 5 + 5 + \dfrac{5}{5} $

$ 17 = 5 + 5 + 5 + \dfrac{ \ln{5} }{ \ln{\sqrt{5}} } $

$ 18 = 5 \times 5 - 5 - \dfrac{\ln{5}}{\ln{\sqrt{5}}} $

$ 19 = 5 \times 5 - 5 - \dfrac{5}{5} $

$ 20 = \dfrac{5}{5} \left( 5 \times 5 - 5 \right) $