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Theorems and conjectures about Integermania can be submitted to Steve Wilson for posting on this page. Warning: False conjectures could be bumped off this page by true conjectures!
Definition 1: The exquisiteness of a solution to a Integermania problem is the highest level operation used in its construction, as given by the table of levels of exquisiteness, but the table is not repeated here.
Definition 2: 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
$\operatorname{Exq}(A) = \{n_1, n_2, ..., n_L, ...\}$.
See Exquiteness of Sets for conjectures about different Intergermania problems.
Theorem 1: In an Integermania problem with $n$ values and $k$ available binary operations, where each value is used exactly once, and values are combined by using only the available binary operations, there are at most $\dfrac{k^{n-1}(2n-2)!}{(n-1)!}$ positive integers which can be created. See discussion and proof.
Theorem 2: In an Integermania problem where set $A$ has $n$ values, the level 1 exquisiteness of $A$ will be less than or equal to 1 when $n=1$, and less than or equal to 3 when $n=2$. See proof and comments about larger values of $n$.
Theorem 3: There exists an Integermania solution, without rounding, for every integer $n$ and every non-empty set of integers $A$. See proof and comments.