# Four Fours/Table of 4s

### From charlesreid1

Back to Four Fours

## Contents

## One 4

Believe it or not, the rules allow you to do quite a bit with a single 4. The rules say you may combine 4s with any mathematical symbol except numbers. Thus, in addition to 4 alone, we also have:

The following fractions are also useful:

but these can't appear with 1 in the denominator.

Could possibly add constants (harmonic number), possibly special functions.

Once you allow variables like x into the mix, it's lights out.

## One 4 with Variables

## Two 4s

Note: we aren't using fourth roots or one-quarter powers very much, e.g., . Adding this would greatly expand the possibilities.

(For those keeping score at home, that's 1 decillion 333 nonillion 735 octillion 776 septillion 850 sextillion 284 quintillion 124 quadrillion 449 trillion 81 billion 472 million 843 thousand 776)

## Three 4s

These lists blow up pretty fast... as you can see, focusing on using a smaller number of 4s can force you to be creative. This makes it possible to combine 4 4's beyond the integers from 1 to 20, and keep on going.

One useful template for representing powers of 2 is:

where is any number expressible with two 4's.

Once we can add three 4's, we can start to write expressions like

where P and Q are any expressions involving 1 or 2 fours. If we choose carefully, P and Q will have the same base and different exponents, so we can start to combine integer powers to obtain new integers. This trick will get us out of at least a few jams when constructing the integers from 1 to 100 using only four fours.

(That is, 74 is 4A in hexidecimal, or base 16 = 4 * 4)