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 surcharge of $+ 0.2$ for each unary operation used.

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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 May, 2025)

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: $\operatorname{antilog} a$, $\exp a$ (equivalent to $10^a$ and $e^a$, but $10$ and $e$ are constants, 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: $\operatorname{arsinh} a$, $\operatorname{arcosh} a$, $\operatorname{artanh} a$, $\operatorname{arcoth} a$, $\operatorname{arsech} a$, $\operatorname{arcsch} a$
Level 5: Some operations related to factorials.
  • Binomial coefficients: ${}_a C_b$
  • Permutations: ${}_a P_b$
  • 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. (will be surcharged)
  • Triple factorial:   $n!!! = n \times (n-3) \times (n-6) \times ... \times c$,   where $c$ is 1, 2, or 3. (will be surcharged)
  • Gamma: $\Gamma(a)$. When $a$ is a positive integer, then $\Gamma(a) = (a - 1)!$ (will be surcharged)
Level 6: Some other basic math functions.
  • Sequence of prime numbers: $p_a$ (the $a$th prime number, will be surcharged)
  • Fibonacci sequence: $f_a$ (the $a$th term of the Fibonacci sequence, will be surcharged)
  • Floor function: $\lfloor a \rfloor$. This is equivalent to integer division, and to the greatest integer function. (will be surcharged)
  • Ceiling function: $\lceil a \rceil$ (will be surcharged)
  • Division modulo $b$:   $a \mod b$ (the remainder after dividing $a$ by $b$)
  • Greatest common divisors and least common multiples: $\gcd(a,b)$, $\operatorname{lcm}(a,b)$
  • Summations: $\sum_a^b c$ (but an index variable is a variable, and therefore prohibited)
  • Determinants: $\begin{vmatrix} a & b \\ c & d \end{vmatrix}$
Level 7: Other advanced functions. (All will be surcharged, except as noted. Others are possible.)
  • Change of base: $a_b =$ the number obtained by using the numeral $a$ in the base $b$ numeration system (Will not be surcharged.)
  • Derivative: $a'$, the derivative of $a$. When $a$ is a number, then $a'=0$.
  • Euler's totient function: $\varphi(a)$
  • 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: $L_a = L_{a-1} + L_{a-2}$, $L_1 = 1$, $L_2 = 3$
  • Multifactorials other than double or triple: $n!!!!$, $n!!!!!$, etc.
  • Number of divisors of a number: $d(a)$
  • Prime Counting Function: $\pi(a)$, the number of primes at most $a$
  • Sum of divisors of a number: $\sigma(a)$
  • 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.