Maths on a Mug 4

This is one of my favourite puzzles. It is a simple, life long game: Using exactly four 4s make as many numbers as you can. You can use the operators \(+,−,×,\div,!,\hat{},.,\sqrt{},\Gamma\) and concatenation. You must use all four 4s. My advice, start at 0 and just work your way up, I save doing these puzzles in my head for times when I have to physically, but not mentally, present.

Let me start you off with a couple:

\[\begin{eqnarray*} 0 &=& 44 - 44, \\ 1 &=& \frac{44}{44},\\ 2 &=& \frac{4}{4} + \frac{4}{4}. \end{eqnarray*}\]

The value on the mug is a solution for the number 169.

You can quite comfortabely get the first 100 numbers, and then things start to ramp up in difficulty. If you are really keen, the answer to the first 40,000 numbers is available here. Credit to David A. Wheeler, who compiled the “The Definitive Four Fours Answer Key”.

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