Can You Be As Clever As Dirac For A Little Bit

I’ve been reading Graham Farmelo’s The Strangest Man: The Hidden Life of Paul Dirac, which is a quite good biography about a really interestingly odd man and important physicist. Among the things mentioned is that at one point Dirac was invited in to one of those number-challenge puzzles that even today sometimes make the rounds of the Internet. This one is to construct whole numbers using exactly four 2’s and the normal, non-exotic operations — addition, subtraction, exponentials, roots, the sort of thing you can learn without having to study calculus. For example:

1 = \left(2 \div 2\right) \cdot \left(2 \div 2\right)
2 = 2 \cdot 2^{\left(2 - 2\right)}
3 = 2 + \left(\frac{2}{2}\right)^2
4 = 2 + 2 + 2 - 2

Now these aren’t unique; for example, you could also form 2 by writing 2 \div 2 + 2 \div 2, or as 2^{\left(2 + 2\right)\div 2} . But the game is to form as many whole numbers as you can, and to find the highest number you can.

Dirac went to work and, complained his friends, broke the game because he found a formula that can any positive whole number, using exactly four 2’s.

I couldn’t think of it, and had to look to the endnotes to find what it was, but you might be smarter than me, and might have fun playing around with it before giving up and looking in the endnotes yourself. The important things are, it has to produce any positive integer, it has to use exactly four 2’s (although they may be used stupidly, as in the examples I gave above), and it has to use only common arithmetic operators (an ambiguous term, I admit, but, if you can find it on a non-scientific calculator or in a high school algebra textbook outside the chapter warming you up to calculus you’re probably fine). Good luck.