Making any integer with four 2s (2025)

There's a adorable math puzzle that can be engaging to folks on very different levels:

Given exactly four instances of the numeral 2 and several mark natural number, use any mathematical operations to create the mark figure alongside these 2s, using no another digits.

Some examples can be done by basic academy kids:

In center school, kids study concerning exponents, factorials, etc. which expands the range considerably:

Then arrive the tricks; for example, the figure 22 (twenty two) can be seen as a valid use of two 2s, and so on; so we can have:

Getting to 7 is notoriously difficult, but if you authorize equal additional mathematical tools akin the Gamma function, it becomes easy:

The additional math accomplishment group have, the additional numbers they can make. See this thread for several fun concoctions using integrals, repeating fractions and combinatorial operators. One of my favorite examples involves complex numbers:

So the fun doesn't end equal following one graduates from university! In fact, this seems to have been a favorite pastime for mathematicians in the 1920s. Until Paul Dirac ruined it for everyone by finding a broad resolution for all number.

It's all concerning nested quadrate roots:

If the quadrate base is applied n times:

All that's remaining now is several base-2 logarithms:

And another:

This leads to the broad formula:

There's fair one small wrinkle: it uses three instances of the numeral 2, not four. This is uncomplicated to amend, however; since , we can replace any sole numeral alongside that and get exactly four:

One may assertion this is cheating, but it seems to be in row alongside the rules of the puzzle! Note that the being n doesn't really appear anyplace - it's just a helper to figure the figure of repeated quadrate roots. For example, another way to province 7 is:

There are exactly four 2s, and this uses lone reasonable, basic math operations to do the calculation. It's apparent that any figure can be expressed this way; the lone difficulty is correctly example all those quadrate roots!