How many of the natural numbers do you think you can produce just by combining four 4s with standard mathematical operations?

The answer might blow your mind a little bit. It’s *all of them!*

In particular, you can get any natural number by using the symbols ‘4’, ‘√’, ‘log’, and ‘/’. And in particular, you can do it with just four 4s!

Try it for yourself before moving on!

Alright, so here’s how to do it: Take the first four, and take n square roots of it. The result of this is 4^(.5^{n}). Then take the log base 4 of the result, giving (1/2)^{n}. And *then* take the log base √4/4 (otherwise known as 1/2) of *that* result, and you get n!

(Wait, aren’t we cheating? There’s an n on the right hand side! No, the n is just for our reading comprehension; it just denotes the number of square root symbols we will put in to get out the desired natural number.)

It turns out that this particular strategy is unique to 4. If you tried to do it with four fives, you would not get out the right answer, since the base of the outside logarithm would no longer be 1/2, but √5/5. But a slight modification *can* make this work with *five* fives, as long as you also allow the ‘+’ symbol to be used!

See if you can solve this “five fives” puzzle for yourself before reading on!

…

Alright, here it is:

Nice and simple! We can produce 1/2 as the base of our outside logarithm with three 5s. And of course, this now generalizes to *any* number besides five! In other words, just five copies of any natural number k can be combined with square root symbols, logarithms, and plus signs to generate *all* natural numbers!

I assume this is known, but haven’t ever seen it written anywhere else. I don’t know if there’s any use to this, but it’s a fun puzzle to think about! Can we do better than five copies of k to generate all the natural numbers, using just standard mathematical symbols?