## Integermania!

#### Exquisiteness of Sets

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}.