Conjectures about the exquisiteness of different sets of values in Integermania can be submitted to Steve Wilson for posting on this page. Submissions should either be related to sets on this web site, or include conjectures about two or more levels.

Set: $A$ |
Level 1 | Level 2 | Level 3 | Level 4 | Level 5 | Level 6 |

"Dave's Birthday" {1, 7, 7, 8} |
2 | 36 | 158 | 228 | 270 | over 279 |

"First Four Composites" {4, 6, 8, 9} |
30 | 162 | 540 | 838 | over 1000 | |

"First Four Naturals" {1, 2, 3, 4} |
28 | 88 | 392 | 786 | over 820 | |

"First Four Nonsquares" {2, 3, 5, 6} |
37 | 201 | over 260 | |||

"First Four Odds" {1, 3, 5, 7} |
17 | 88 | 268 | over 370 | ||

"First Four Primes" {2, 3, 5, 7} |
25 | 123 | 430 | 430 | over 1000 | |

"Quattro Ones" {1, 1, 1, 1} |
4 | 13 | 15 | 57 | 136 | over 139 |

"Four Fours" {4, 4, 4, 4} |
9 | 22 | 132 | 276 | over 1100 | |

"Four Nines" {9, 9, 9, 9} |
3 | 12 | 123 | 261 | over 310 | |

"JCCC Letters" {3, 3, 3, 10} |
14 | 24 | 87 | 337 | over 460 | |

"Mile and a Foot" {5, 2, 8, 1} |
51 | 92 | 292 | over 520 | ||

"Ralph's Birthyear" {1, 9, 4, 8} |
0 | 132 | ||||

"First Five Naturals" {1, 2, 3, 4, 5} |
75 | 565 | 873 | over 1000 | ||

"Zip 66210" {6, 6, 2, 1, 0} |
26 | 136 | 871 | 876 | over 1000 |

**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
$\operatorname{Exq}(A) = \{n_1, n_2, ..., n_L, ...\}$.

**Why are Some Sets Highly Exquisite?**

**Some Facts:**

- If set $A$ consists of the 4 values $\{a_1, a_2, a_3, a_4\}$, and $1 \le a_i \le 9$, then the largest possible exquisiteness at level 1 is achieved by the set {1,2,5,8}.
- If set $A$ consists of the 4 values $\{a_1, a_2, a_3, a_4\}$, and $1 \le a_i \le 99$, then the largest possible exquisiteness at level 1 is achieved by the set {2,3,4,22}.