 Definition  Levels  Sets  Calculator 
Creating mathematical expressions for Integermania is easy, but creating beautiful expressions is more difficult. First, we have to agree on what constitutes an beautiful expression. Although beauty (in math, we often say "elegance") is in the eye of the beholder, on this site we have attempted to systematize the "exquisiteness" of a solution to the number problems. (We have chosen to avoid defining the term "elegance", instead leaving its connotations of beauty to the subjective realm of mathematics appreciation it already occupies.)
Let $a$, $b$, $c$, and $d$ represent required integers. The exquisiteness of a solution is the highest level operation used in its construction, as given by the following table of levels of operations, plus a possible surcharge. (The background color coding, beginning with the royal purple, moving through the rainbow to yellow and then fading out, is also used in the solutions.) The surcharges are:
We can also define the exquisiteness of an entire set of numbers, according
to the ability in which the smallest positive integers can be created using
values from that set. The exquisiteness of a set of numbers $A$ at level $L$ is
the largest positive integer $n_L$ having the property that for every
positive integer less than or equal to $n_L$ there exists a solution to the
Integermania problem using set $A$ with exquisiteness level
less than $L+1$. Notationally, we can write:
$\operatorname{Exq}(A,L) = n_L$, or
$\operatorname{Exq}(A) = \{n_1, n_2, ..., n_L, ...\}$.
Levels of Exquisiteness (modified as of January, 2007)
Level 1: The basic operations and grouping symbols. 

Level 2: Common operations involving place value. 

Level 3: Exponents, radicals, factorials, and per mille. 

Level 4: Basic algebraic functions. (All will be surcharged except as noted.) 

Level 5: Some arithmetic operations. 

Level 6: Some basic sequences and number theory functions. (All will be surcharged.) 

Level 7: Other advanced functions. (All will be surcharged, except as noted. Only those used so far have been listed here. Others are possible.) 

There are many more functions used in mathematics at the level of calculus and beyond. We cannot possibly list them all, but a good source of additional possibilities can be found at Wolfram Research's Mathematical Functions.