Four Fours
From charlesreid1
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)} } $
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) $