Integermania!

Exquisiteness

-- 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:

• $+ 0.2$ for each unary operation used
• $+ 2.0 + 2\text{error}$   for the existence of rounding in a solution, where $\text{error}$ is the difference between the rounded and unrounded values of the expression (note that rounding will also incur unary operation surcharges)

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. Addition: $a+b$ Subtraction: $a-b$ Opposite:   $-b$ (will be surcharged) Multiplication: $a \times b$ Division: $\dfrac{a}{b}$ (Parentheses) Level 2: Common operations involving place value. The decimal point: $.a$ (2.3 will not be surcharged, but .3 will) The bar for a repeating decimal: $.\overline{a}$ (will be surcharged) The percent sign: $a\%$ (will be surcharged, equivalent to $0.01a$) Juxtaposition: $ab$ (equivalent to $10a+b$) Level 3: Exponents, radicals, factorials, and per mille. Exponents: $a^b$ Radicals: $\sqrt{a}$, $\sqrt[b]{a}$ (if the index is assumed, it will be surcharged) Factorials: $a!$ (will be surcharged) Per mille: $a ‰$ (will be surchaged, equivalent to $0.001a$) Level 4: Basic algebraic functions. (All will be surcharged except as noted.) Logarithms: $\log_b a$ (will not be surcharged) Specific Logarithms: $\log a$, $\ln a$ (logs base 10 and $e$, respectively) Exponentials: $\exp a$ (equivalent to $e^a$, but $e$ is a constant, and therefore prohibited) Trigonometrics: $\sin a$, $\cos a$, $\tan a$, $\cot a$, $\sec a$, $\csc a$ Inverse trigonometrics: $\arcsin a$, $\arccos a$, $\arctan a$, $\operatorname{arccot} a$, $\operatorname{arcsec} a$, $\operatorname{arccsc} a$ Hyperbolics: $\sinh a$, $\cosh a$, $\tanh a$, $\coth a$, $\operatorname{sech} a$, $\operatorname{csch} a$ Inverse hyperbolics: $\operatorname{arcsinh} a$, $\operatorname{arccosh} a$, $\operatorname{arctanh} a$, $\operatorname{arccoth} a$, $\operatorname{arcsech} a$, $\operatorname{arccsch} a$ Level 5: Some arithmetic operations. Binomial coefficients: ${}_a C_b$ Permutations: ${}_a P_b$ Gamma: $\Gamma(a)$. When $a$ is a positive integer, then $\Gamma(a) = (a - 1)!$ (will be surcharged) Greatest common divisors and least common multiples: $\gcd(a,b)$, $\operatorname{lcm}(a,b)$ Division modulo $b$:   $a \mod b$ (the remainder after dividing $a$ by $b$) Summations: $\sum_a^b$ (but an index variable is a variable, and therefore prohibited) Determinants: $\begin{vmatrix} a & b \\ c & d \end{vmatrix}$ Change of base: $a_b$ (the number obtained by using the numeral $a$ in the base $b$ numeration system) Level 6: Some basic sequences and number theory functions. (All will be surcharged.) Sequence of prime numbers: $p_a$ (the $a$th prime number) Fibonacci sequence: $f_a$ (the $a$th term of the Fibonacci sequence) Prime Counting Function: $\pi(a)$, the number of primes at most $a$ Number of divisors of a number: $d(a)$ Sum of divisors of a number: $\sigma(a)$ Euler's totient function: $\varphi(a)$ Derivative: $a'$, the derivative of $a$. When $a$ is a number, then $a'=0$. Level 7: Other advanced functions. (All will be surcharged, except as noted. Only those used so far have been listed here. Others are possible.) Double factorial:   $n!! = n \times (n-2) \times (n-4) \times ... \times c$,   where $c$ is 1 if $n$ is odd, and $c$ is 2 if $n$ is even. Grad: $n \text{ grad}$ (1 grad is a hundredth of a right angle, hence $1 \text{ grad} = \dfrac{\pi}{200}$). Kronecker delta: $\delta_a^b$ (equal to 1 if $a = b$, 0 if $a \ne b$) (Will not be surcharged.) Lucas sequence: $L_a = L_{a-1} + L_{a-2}$, $L_1 = 1$, $L_2 = 3$ Triangular numbers: $T_a = {}_{a+1} C_2$ Zeta function: $\zeta(a) = \sum\limits_{k=1}^{\infty} \dfrac{1}{k^a}$

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.