Four Fours/Table of 4s
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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
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)
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)