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 36 137 158 over 200  
"First Five Naturals"
{1, 2, 3, 4, 5}		 75 565 873 over 1000    
"First Four Composites"
{4, 6, 8, 9}		 30 162 540 838 over 1000  
"First Four Naturals"
{1, 2, 3, 4}		 28 88 381 392 over 500  
"First Four Primes"
{2, 3, 5, 7}		 25 123 430 430 over 1000  
"Quattro Ones"
{1, 1, 1, 1}		 4 13 15 over 40    
"Four Fours"
{4, 4, 4, 4}		 9 22 132 276 over 1000  
"Four Nines"
{9, 9, 9, 9}		 3 12 123 over 250    
"JCCC Letters"
{3, 3, 3, 10}		 14 24 67 253 468 over 500
"JCCC Zips"
{6, 6, 2, 1, 0}		 26 136 over 500      

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 nL having the property that for every positive integer less than or equal to nL there exists a solution to the Integermania problem using set A with exquisiteness level less than L+1.  Notationally, we can write:
Exq(A,L) = nL,   or   Exq(A) = {n1,n2,...,nL,...}.

Why are Some Sets Highly Exquisite?

Some Facts: