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} $

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

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

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

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

Five Fives

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