The webmaster of this site has received a few interesting submissions over the year which are worth sharing, but won't always qualify on the other pages due to their exquisiteness level. Therefore, to qualify for inclusion on this page, a submission must tickle the webmaster's fancy.

$\dfrac{\sigma(\Gamma(5) + 7 - 3)}{2} = 28$ | A perfect solution. A perfect
number is defined as a number which is equal to the sum of its
proper divisors. If all divisors are included, the sum
will be twice the number. This level 6.4 First Four Primes
solution incorporates the definition of a perfect number for the
second perfect number, 28. Submitted by Chris Imm of
Prairie Village, Kansas, in December, 2001. |

$10^{(3/.3)^{\Gamma(3)}} = \text{googol}$ | Googol and still counting. This
level 5.4 Letters of JCCC solution was the first of several
googol solutions submitted by Brian Hollenbeck of Emporia,
Kansas, in March, 2002. |

$\dfrac{\arccos(.5)}{(2 \times 3)^{\circ}} + 7 = 17$ | Non-trivial trigonometry. This
level 4.6 First Four Primes solution uses an inverse trig
function to produce a 60-degree angle. Submitted by Mikel King
of Paola, Kansas, in May 2002. |

$\tan \left(\left( \Gamma\left( \dfrac{3}{7-5} \right) \right)^2 \right) = 1$ | Gamma at a half-integer. Since the gamma
function and the factorial function are related by a
horizontal translation (graphically speaking), the
gamma function forms a very convenient alternative to the
factorial for Integermania. But the factorial is defined
only for non-negative integers, whereas the gamma function is
defined on the entire complex plane, except for the non-positive
integers. This level 5.4 First Four Primes solution uses
the value of the gamma function at a half-integer.
Submitted by Andrew Bennett of Manhattan, Kansas, in May, 2003. |

$\pi(d(((p(p(0!)))!)!)) = 10$ | Creation out of nothing.
With enough unary functions, all of the integers could probably be
created from the singleton set {0}. Number
theory appears to provide enough variety to do it using at most
level 6 functions (although the unary surcharge eventually
creates higher level solutions). This level 7.4 solution
from the set {0} suggests how it could be done. In
this problem, $p(n)$ is the $n$th prime, $d(n)$ is the number of
divisors of $n$, and $\pi(n)$ is the number of primes at most
$n$. Submitted by the webmaster, June, 2003. |

$- \log \sqrt{0! \% \cdots \%} = n$ | Creation out of nothing, completed.
With $n$ percent signs, this expression equals $n$. Thus all the integers
have been created, albeit at level $4.8+0.2n$.
Submitted by the webmaster, May, 2012. |

$(\sqrt{4} + \sqrt{-4})^{4+4} = 4096$ | Power of a complex number. This
level 3.6 Four Fours solution uses the square root of a negative number to produce a positive integer. Submitted by Bruce Manoly of Crescent City, California, in May 2024. |