GithubPL.pl

/* <Username: jg495  Name: GuanJun>
The program not produce multiple answers.
1. <Number of elements in the list binding Q after executing s1(Q,100) is 1747> 
<The main idea for s1:
	step 1: using Sieve of Eratosthenes method to get prime number list which is less than 100
	step 2: geting composite number list according to prime number list	which is less than 100
	step 3: combining the prime number list with prime list[P,C,P+C,P*C] and combining the composite number list with 
			composite number list[C1,C2,C1+C2,C1*C2]. In this step,firstly I am going to get rid of the pairs which its sum
			more than 100. Secondly, after adding the first filter, I am using mergesort 
			to order the list by it's product,afterthen,I will move out the list which it's product is single(compare the product of 
			the list and it's next list, if they are equal, then save all the list which have the same product with the first list.
			Otherwise, delete that list).
	step 4: mergesort the list by it's sum			
 >
I have not used built-ins.
2. <Number of elements in the list binding Q after executing s2(Q,100) is 145 >
<step 1: using Sieve of Eratosthenes method to get prime number list which is less than 55(53+2,the minimum prime number which is greater than 50 +2)
	step 2: geting composite number list according to prime number list	which is less than 55
	step 3: combining the prime number list with prime list[P,C,P+C,P*C] and combining the composite number list with 
			composite number list[C1,C2,C1+C2,C1*C2]. In this step,firstly I will get rid of some pairs if its value of sum more than 55
		and is an even number. Secondly, after adding the first filter,I am going to delete the list which their sum-2 is prime number or their sum divede 3 is a prime number. 
	step 4: mergesort the list by it's product	>
I have not used built-ins.
3. <Number of elements in the list binding Q after executing s3(Q,100) is 86>
<select the list which their product is unit>
I have not used built-ins.
4.
<select the list which their Sum is unit>
I have not used built-ins.
5. <Number of elements in the list binding Q after executing s4(Q,500) is 1>
s4(Q,500) uses <1,109,560> inferences. */

%--------------------------------------------------------------------------------------------	
s4(Q,Max):-	
		s3(S3Result,Max),
		findTheFinalAnswer(S3Result,S4Result),
		app([S4Result],[],Q).

findTheFinalAnswer([[_,_,S,_],[_,_,S,_]|T],L):-
		deleteDuplicateSum(S,T,Tnew),
		findTheFinalAnswer(Tnew,L).
findTheFinalAnswer([[A1,B1,S1,P1],[_,_,S2,_]|_],[A1,B1,S1,P1]):-
		S1=\=S2.		
				
deleteDuplicateSum(S,[[_,_,S,_]|T],L):-
		deleteDuplicateSum(S,T,L).
deleteDuplicateSum(N,[[A,B,S,P]|T],[[A,B,S,P]|T]):-
		N=\=S.


%--------------------------------------------------------------------------------------------	
s3(Q,Max):-
		s2(S2Result,Max),
		getSingleProduct(S2Result,[],Result),
		merge_sort2(Result,Q).

getSingleProduct([[_,_,_,P],[_,_,_,P]|T],L,R):-	
		deleteDuplicateNum(P,T,Tnew),
		getSingleProduct(Tnew,L,R).
					
getSingleProduct([[A1,B1,C1,D1],[A2,B2,C2,D2]|T],L,R):-	
		D1=\=D2,
		app([[A1,B1,C1,D1]],L,Lnew),
		getSingleProduct([[A2,B2,C2,D2]|T],Lnew,R).									
getSingleProduct([],Ls,Ls).								
getSingleProduct([H|T],L,R):- 
		T = [],
		app([H],L,R).

deleteDuplicateNum(P,[[_,_,_,P]|T],R):-
		deleteDuplicateNum(P,T,R).
deleteDuplicateNum(N,[[A,B,C,D]|T],[[A,B,C,D]|T]):-
		N=\=D.
deleteDuplicateNum(_,[],[]).	


%--------------------------------------------------------------------------------------------	
s2(Q,Max):-
		Maxnew2 is Max/2,
		getMaxPrime(Maxnew2,MaxPrime),		
		BigestSum is MaxPrime+2,
		getPrimeList(BigestSum,LPrime),				%get Prime number list	< 55
		setTheNum(2,BigestSum,LPrime,[],L1),		%get composite number list	<55
		rev(L1,L1new),
		toGetPLS2(BigestSum,LPrime,L1new,PLList),	%get p*p
		toGetLLS2(BigestSum,L1new,LLList),			%get c*c
		app(PLList,LLList,TList),			
		delete2AddPSumS2(TList,LPrime,[],Result),	%move out if (sum-2) is prime	
		deleteDivideBy3(Result,[],Result2),			%move out if (sum mod 3 =0)and(sum/3) is prime
		merge_sort(Result2,Q).				

deleteDivideBy3([[A,B,S,P]|T],L,R):-
		S mod 3 =\=0,
		deleteDivideBy3(T,[[A,B,S,P]|L],R).
		
deleteDivideBy3([[A,B,S,P]|T],L,R):-
		S mod 3 =:=0,
		Test is S/3,		
		isNot(isPrime(Test,2)),
		deleteDivideBy3(T,[[A,B,S,P]|L],R).
deleteDivideBy3([[_,_,S,_]|T],L,R):-
		S mod 3 =:=0,
		Test is S/3,		
		isPrime(Test,2),
		deleteDivideBy3(T,L,R).
deleteDivideBy3([],L,L).
	
delete2AddPSumS2([[A,B,S,P]|T],LPrime,L,R):-
		TestNumber is S-2,
		isNot(isTheMember(TestNumber,LPrime)),
		delete2AddPSumS2(T,LPrime,[[A,B,S,P]|L],R).
delete2AddPSumS2([[_,_,S,_]|T],LPrime,L,R):-
		TestNumber is S-2,
		isTheMember(TestNumber,LPrime),
		delete2AddPSumS2(T,LPrime,L,R).
delete2AddPSumS2([],_,L,L).		

getMaxPrime(N,R):-
		isNot(isPrime(N,2)),
		Nnew is N+1,
		getMaxPrime(Nnew,R).
getMaxPrime(R,R):-
		isPrime(R,2).


toGetPLS2(Max,L1,L2,R):-
		getPLListS2(Max,L1,L2,[],R).

getPLListS2(_,[],_,Ls,Ls).
getPLListS2(Max,[H|T],L,Ls,R):-
		getEachPLS2(Max,H,L,[],PLnew),
		app(PLnew,Ls,Lsnew),
		getPLListS2(Max,T,L,Lsnew,R).	

getEachPLS2(_,_,[],L,L).
getEachPLS2(Max,E,[H|T],L,R):-
		E+H<Max,	
		Sum is E+H,
		Sum mod 2 =\=0,	
		E<H,				
		Product is E*H,
		app([[E,H,Sum,Product]],L,Lnew),
		getEachPLS2(Max,E,T,Lnew,R).	

getEachPLS2(Max,E,[H|T],L,R):-
		E+H<Max,	
		Sum is E+H,
		Sum mod 2 =\=0,	
		E>H,
		Product is E*H,
		app([[H,E,Sum,Product]],L,Lnew),
		getEachPLS2(Max,E,T,Lnew,R).	
getEachPLS2(Max,E,[H|T],L,R):-
		E+H<Max,	
		Sum is E+H,
		Sum mod 2 =:=0,
		getEachPLS2(Max,E,T,L,R).				
getEachPLS2(Max,E,[H|_],L,L):-
		E+H>=Max.
		
toGetLLS2(Max,L,R):- 
		getLLListS2(Max,L,[],R).
	

getLLListS2(_,[],L1,L1).
getLLListS2(Max,[H|T],L1,R):- 
		getEachLS2(Max,H,T,[],Lnew),
		app(Lnew,L1,L1new),   
		getLLListS2(Max,T,L1new,R).		

getEachLS2(_,_,[],L,L).
getEachLS2(Max,E,[H|T],L,R):-	
		E+H<Max,	
		Sum is E+H,	
		Sum mod 2 =\=0,			
		Product is E*H,		
		app([[E,H,Sum,Product]],L,Lnew),
		getEachLS2(Max,E,T,Lnew,R).										
getEachLS2(Max,E,[H|T],L,R):-	
		E+H<Max,	
		Sum is E+H,	
		Sum mod 2 =:=0,	
		getEachLS2(Max,E,T,L,R).													
getEachLS2(Max,E,[H|_],L,L):-
		E+H>=Max.
	
%--------------------------------------------------------------------------------------------		
s1(Q,Max):-
		Maxnew is Max-1,
		getPrimeList(Maxnew,LPrime),			
		Maxnew2 is Max/2,
		getPrimeList(Maxnew2,LPrimeS),		%get Prime number List		
		setTheNum(2,Maxnew,LPrime,[],L1),	%get composite number List		
		toGetPL(Max,LPrimeS,L1,PLList),		%prime list * composit list	
		toGetLL(Max,L1,LLList),				%composit list * composit list	
		app(PLList,LLList,Result),
		merge_sort(Result,Result2),
		getFinalAns(Result2,[],Result3),	%delete unit product
		merge_sort2(Result3,Q).

getFinalAns([[A1,B1,S1,P1],[A2,B2,S2,P2]|T],L,R):-
		P1=:=P2,
		doMore(P1,[[A1,B1,S1,P1],[A2,B2,S2,P2]|T],[],NewList,NewT),
		app(NewList,L,Lnew),
		getFinalAns(NewT,Lnew,R).
getFinalAns([[_,_,_,P1],[A,B,S,P2]|T],L,R):-
		P1=\=P2,
		getFinalAns([[A,B,S,P2]|T],L,R).		
getFinalAns([_|T],R,R):-
		T=[].		
getFinalAns([],R,R).


doMore(N,[[A,B,S,P]|T],Ls,R,NewT):-
		N=:=P,
		doMore(N,T,[[A,B,S,P]|Ls],R,NewT).
doMore(N,[[A,B,S,P]|T],R,R,[[A,B,S,P]|T]):-
		N=\=P.		
doMore(_,[],R,R,[]).


toGetPL(Max,L1,L2,R):-
		getPLList(Max,L1,L2,[],R).

getPLList(_,[],_,Ls,Ls).
getPLList(Max,[H|T],L,Ls,R):-
		getEachPL(Max,H,L,[],PLnew),
		app(PLnew,Ls,Lsnew),
		getPLList(Max,T,L,Lsnew,R).	

getEachPL(_,_,[],L,L).
getEachPL(Max,E,[H|T],L,R):-
		E+H=<Max,		
		E<H,
		Sum is E+H,
		Product is E*H,
		app([[E,H,Sum,Product]],L,Lnew),
		getEachPL(Max,E,T,Lnew,R).	
getEachPL(Max,E,[H|T],L,R):-
		E+H=<Max,		
		E>H,
		Sum is E+H,
		Product is E*H,
		app([[H,E,Sum,Product]],L,Lnew),
		getEachPL(Max,E,T,Lnew,R).						
getEachPL(Max,E,[H|T],L,R):-
		E+H>Max,						
		getEachPL(Max,E,T,L,R).
		
toGetLL(Max,L,R):- 
		rev(L,Lnew),
		getLLList(Max,Lnew,Lnew,[],R).
	

getLLList(_,[],_,L2,L2).
getLLList(Max,[H|T],L1,L2,R):- 
		getEachL(Max,H,L1,[],Lnew),
		app(Lnew,L2,L2new),
		getLLList(Max,T,L1,L2new,R).		

getEachL(_,_,[],L,L).
getEachL(Max,E,[H|T],L,R):-
		E<H,
		E+H=<Max,		
		Sum is E+H,
		Product is E*H,		
		app([[E,H,Sum,Product]],L,Lnew),
		getEachL(Max,E,T,Lnew,R).														
getEachL(Max,E,[H|T],L,R):-
		E>=H,
		E+H=<Max,
		getEachL(Max,E,T,L,R).											
getEachL(Max,E,[H|_],L,L):-
		E+H>Max.
		
setTheNum(Max,Max,_,L,L).				
setTheNum(N,Max,[],L,R):-
		N<Max,
		Nnew is N+1,
		setTheNum(Nnew,Max,[],[N|L],R).	
setTheNum(N,Max,[H|T],L,R):-		
		N<Max,
		N=\=H,
		Nnew is N+1,
		setTheNum(Nnew,Max,[H|T],[N|L],R).
setTheNum(N,Max,[H|T],L,R):-		
		N<Max,
		N=:=H,
		Nnew is N+1,
		setTheNum(Nnew,Max,T,L,R).	

getPrimeList(Max,R):-setList(Max,[],R,Max).
setList(1,L,R,Max):-setPrimeNumber(2,L,R,Max).
setList(N,L,R,Max):- 
		N>1,
		Nnew is N-1,
		setList(Nnew,[N|L],R,Max).

setPrimeNumber(Prime,L,R,Max):-
		Prime=<sqrt(Max),
		isPrime(Prime,2),		
		deleteNum(Prime,L,Lnew),
		PrimeNew is Prime+1,		
		setPrimeNumber(PrimeNew,Lnew,R,Max).		
setPrimeNumber(Prime,L,R,Max):-
		Prime=<sqrt(Max),
		isNot(isPrime(Prime,2)),
		PrimeNew is Prime+1,
		setPrimeNumber(PrimeNew,L,R,Max).
setPrimeNumber(Prime,L,L,Max):-
		Prime>sqrt(Max).
				
deleteNum(Prime,L,R):- deleteisNotPrime(Prime,L,[],R).
deleteisNotPrime(Prime,[H|T],L,R):-
		H =\= Prime,
		H mod Prime =:= 0,
		deleteisNotPrime(Prime,T,L,R).
			
deleteisNotPrime(H,[H|T],L,R):-
		deleteisNotPrime(H,T,[H|L],R).		
deleteisNotPrime(Prime,[H|T],L,R):-
		H mod Prime =\= 0,
		deleteisNotPrime(Prime,T,[H|L],R).
deleteisNotPrime(_,[],L,R):-
		rev(L,R).
					
rev(L,R):-  theRev(L,[],R).
theRev([],A,A).
theRev([H|T],A,R):-  theRev(T,[H|A],R). 

%test the number if is prime number		
isPrime(X,X).
isPrime(X,Y):-
		Y<X,
		0 =\= X mod Y,
		Ynew is Y+1,
		isPrime(X,Ynew).	

isNot(Function):-
	Function,
	!,
	fail.
isNot(_).

app([],L,L).
app([E|T],L,[E|M]) :- app(T,L,M).

merge_sort2([],[]).     
merge_sort2([X],[X]).   
merge_sort2(List,Sorted):-
    List=[_,_|_],divide(List,L1,L2),     
	merge_sort2(L1,Sorted1),merge_sort2(L2,Sorted2),  
	merge2(Sorted1,Sorted2,Sorted).                
merge2([],L,L).
merge2(L,[],L):-L\=[].
merge2([[A1,B1,S1,P1]|T1],[[A2,B2,S2,P2]|T2],[[A1,B1,S1,P1]|T]):-S1=<S2,merge2(T1,[[A2,B2,S2,P2]|T2],T).
merge2([[A1,B1,S1,P1]|T1],[[A2,B2,S2,P2]|T2],[[A2,B2,S2,P2]|T]):-S1>S2,merge2([[A1,B1,S1,P1]|T1],T2,T).

merge_sort([],[]).   
merge_sort([X],[X]).  
merge_sort(List,Sorted):-
    List=[_,_|_],divide(List,L1,L2),    
	merge_sort(L1,Sorted1),merge_sort(L2,Sorted2),  
	merge(Sorted1,Sorted2,Sorted).                  
merge([],L,L).
merge(L,[],L):-L\=[].
merge([[A1,B1,S1,P1]|T1],[[A2,B2,S2,P2]|T2],[[A1,B1,S1,P1]|T]):-P1=<P2,merge(T1,[[A2,B2,S2,P2]|T2],T).
merge([[A1,B1,S1,P1]|T1],[[A2,B2,S2,P2]|T2],[[A2,B2,S2,P2]|T]):-P1>P2,merge([[A1,B1,S1,P1]|T1],T2,T).


divide(L,L1,L2):-halve(L,L1,L2).
halve(L,A,B):-hv(L,L,A,B).
   
hv([],R,[],R).   
hv([_],R,[],R).  
hv([_,_|T],[X|L],[X|L1],R):-hv(T,L,L1,R).

isTheMember(E,[E|_]):-!.
isTheMember(E,[_|R]) :- isTheMember(E,R).

/*
?- consult(<jg495>).
% jg495 compiled 0.01 sec, 1 clauses
true.
?- time(s1(Q,100)).
% 161,966 inferences, 0.076 CPU in 0.083 seconds (91% CPU, 2124116 Lips)
Q = [[3, 4, 7, 12], [2, 6, 8, 12], [4, 5, 9, 20], [3, 6, 9, 18], [4, 6, 10, 24], [5, 6, 11, 30], [4, 7, 11, 28], [3, 8, 11, 24], [2, 9, 11, 18], [4, 8, 12, 32], [2, 10, 12, 20], [6, 7, 13, 42], [5, 8, 13, 40], [4, 9, 13, 36], [3, 10, 13, 30], [6, 8, 14, 48], [5, 9, 14, 45], [4, 10, 14, 40], [2, 12, 14, 24], [7, 8, 15, 56], [6, 9, 15, 54], [5, 10, 15, 50], [4, 11, 15, 44], [3, 12, 15, 36], [7, 9, 16, 63], [6, 10, 16, 60], [4, 12, 16, 48], [2, 14, 16, 28], [8, 9, 17, 72], [7, 10, 17, 70], [6, 11, 17, 66], [5, 12, 17, 60], [4, 13, 17, 52], [3, 14, 17, 42], [2, 15, 17, 30], [8, 10, 18, 80], [6, 12, 18, 72], [4, 14, 18, 56], [3, 15, 18, 45], [2, 16, 18, 32], [9, 10, 19, 90], [8, 11, 19, 88], [7, 12, 19, 84], [6, 13, 19, 78], [5, 14, 19, 70], [4, 15, 19, 60], [3, 16, 19, 48], [9, 11, 20, 99], [8, 12, 20, 96], [6, 14, 20, 84], [5, 15, 20, 75], [4, 16, 20, 64], [2, 18, 20, 36], [10, 11, 21, 110], [9, 12, 21, 108], [8, 13, 21, 104], [7, 14, 21, 98], [6, 15, 21, 90], [5, 16, 21, 80], [4, 17, 21, 68], [3, 18, 21, 54], [10, 12, 22, 120], [9, 13, 22, 117], [8, 14, 22, 112], [7, 15, 22, 105], [6, 16, 22, 96], [4, 18, 22, 72], [2, 20, 22, 40], [11, 12, 23, 132], [10, 13, 23, 130], [9, 14, 23, 126], [8, 15, 23, 120], [7, 16, 23, 112], [6, 17, 23, 102], [5, 18, 23, 90], [4, 19, 23, 76], [3, 20, 23, 60], [2, 21, 23, 42], [10, 14, 24, 140], [9, 15, 24, 135], [8, 16, 24, 128], [6, 18, 24, 108], [4, 20, 24, 80], [3, 21, 24, 63], [2, 22, 24, 44], [12, 13, 25, 156], [11, 14, 25, 154], [10, 15, 25, 150], [9, 16, 25, 144], [8, 17, 25, 136], [7, 18, 25, 126], [6, 19, 25, 114], [5, 20, 25, 100], [4, 21, 25, 84], [3, 22, 25, 66], [12, 14, 26, 168], [11, 15, 26, 165], [10, 16, 26, 160], [9, 17, 26, 153], [8, 18, 26, 144], [6, 20, 26, 120], [5, 21, 26, 105], [4, 22, 26, 88], [2, 24, 26, 48], [13, 14, 27, 182], [12, 15, 27, 180], [11, 16, 27, 176], [10, 17, 27, 170], [9, 18, 27, 162], [8, 19, 27, 152], [7, 20, 27, 140], [6, 21, 27, 126], [5, 22, 27, 110], [4, 23, 27, 92], [3, 24, 27, 72], [2, 25, 27, 50], [13, 15, 28, 195], [12, 16, 28, 192], [10, 18, 28, 180], [9, 19, 28, 171], [8, 20, 28, 160], [7, 21, 28, 147], [6, 22, 28, 132], [4, 24, 28, 96], [3, 25, 28, 75], [2, 26, 28, 52], [14, 15, 29, 210], [13, 16, 29, 208], [12, 17, 29, 204], [11, 18, 29, 198], [10, 19, 29, 190], [9, 20, 29, 180], [8, 21, 29, 168], [7, 22, 29, 154], [6, 23, 29, 138], [5, 24, 29, 120], [4, 25, 29, 100], [3, 26, 29, 78], [2, 27, 29, 54], [14, 16, 30, 224], [12, 18, 30, 216], [10, 20, 30, 200], [9, 21, 30, 189], [8, 22, 30, 176], [6, 24, 30, 144], [4, 26, 30, 104], [2, 28, 30, 56], [15, 16, 31, 240], [14, 17, 31, 238], [13, 18, 31, 234], [12, 19, 31, 228], [11, 20, 31, 220], [10, 21, 31, 210], [9, 22, 31, 198], [8, 23, 31, 184], [7, 24, 31, 168], [6, 25, 31, 150], [5, 26, 31, 130], [4, 27, 31, 108], [3, 28, 31, 84], [15, 17, 32, 255], [14, 18, 32, 252], [12, 20, 32, 240], [11, 21, 32, 231], [10, 22, 32, 220], [9, 23, 32, 207], [8, 24, 32, 192], [7, 25, 32, 175], [6, 26, 32, 156], [5, 27, 32, 135], [4, 28, 32, 112], [2, 30, 32, 60], [16, 17, 33, 272], [15, 18, 33, 270], [14, 19, 33, 266], [13, 20, 33, 260], [12, 21, 33, 252], [10, 23, 33, 230], [9, 24, 33, 216], [8, 25, 33, 200], [7, 26, 33, 182], [6, 27, 33, 162], [5, 28, 33, 140], [4, 29, 33, 116], [3, 30, 33, 90], [16, 18, 34, 288], [15, 19, 34, 285], [14, 20, 34, 280], [13, 21, 34, 273], [12, 22, 34, 264], [10, 24, 34, 240], [9, 25, 34, 225], [8, 26, 34, 208], [7, 27, 34, 189], [6, 28, 34, 168], [4, 30, 34, 120], [2, 32, 34, 64], [17, 18, 35, 306], [16, 19, 35, 304], [15, 20, 35, 300], [14, 21, 35, 294], [13, 22, 35, 286], [12, 23, 35, 276], [11, 24, 35, 264], [10, 25, 35, 250], [9, 26, 35, 234], [8, 27, 35, 216], [7, 28, 35, 196], [6, 29, 35, 174], [5, 30, 35, 150], [4, 31, 35, 124], [3, 32, 35, 96], [2, 33, 35, 66], [16, 20, 36, 320], [15, 21, 36, 315], [14, 22, 36, 308], [12, 24, 36, 288], [11, 25, 36, 275], [10, 26, 36, 260], [9, 27, 36, 243], [8, 28, 36, 224], [6, 30, 36, 180], [4, 32, 36, 128], [3, 33, 36, 99], [2, 34, 36, 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90, 98, 720], [7, 91, 98, 637], [6, 92, 98, 552], [5, 93, 98, 465], [4, 94, 98, 376], [3, 95, 98, 285], [2, 96, 98, 192], [39, 60, 99, 2340], [36, 63, 99, 2268], [35, 64, 99, 2240], [31, 68, 99, 2108], [30, 69, 99, 2070], [29, 70, 99, 2030], [27, 72, 99, 1944], [25, 74, 99, 1850], [24, 75, 99, 1800], [23, 76, 99, 1748], [21, 78, 99, 1638], [19, 80, 99, 1520], [18, 81, 99, 1458], [17, 82, 99, 1394], [15, 84, 99, 1260], [14, 85, 99, 1190], [13, 86, 99, 1118], [12, 87, 99, 1044], [11, 88, 99, 968], [9, 90, 99, 810], [8, 91, 99, 728], [7, 92, 99, 644], [6, 93, 99, 558], [5, 94, 99, 470], [4, 95, 99, 380], [3, 96, 99, 288], [40, 60, 100, 2400], [34, 66, 100, 2244], [32, 68, 100, 2176], [30, 70, 100, 2100], [28, 72, 100, 2016], [26, 74, 100, 1924], [24, 76, 100, 1824], [22, 78, 100, 1716], [20, 80, 100, 1600], [19, 81, 100, 1539], [18, 82, 100, 1476], [16, 84, 100, 1344], [15, 85, 100, 1275], [14, 86, 100, 1204], [13, 87, 100, 1131], [12, 88, 100, 1056], [10, 90, 100, 900], [9, 91, 100, 819], [8, 92, 100, 736], [7, 93, 100, 651], [6, 94, 100, 564], [5, 95, 100, 475], [4, 96, 100, 384], [2, 98, 100, 196]] 

?- time(s2(Q,100)).
% 19,968 inferences, 0.004 CPU in 0.005 seconds (95% CPU, 4437333 Lips)
Q = [[2, 9, 11, 18], [3, 8, 11, 24], [4, 7, 11, 28], [5, 6, 11, 30], [2, 15, 17, 30], [3, 14, 17, 42], [2, 21, 23, 42], [2, 25, 27, 50], [4, 13, 17, 52], [2, 27, 29, 54], [5, 12, 17, 60], [3, 20, 23, 60], [6, 11, 17, 66], [2, 33, 35, 66], [7, 10, 17, 70], [2, 35, 37, 70], [3, 24, 27, 72], [8, 9, 17, 72], [4, 19, 23, 76], [3, 26, 29, 78], [2, 39, 41, 78], [5, 18, 23, 90], [2, 45, 47, 90], [4, 23, 27, 92], [3, 32, 35, 96], [4, 25, 29, 100], [6, 17, 23, 102], [3, 34, 37, 102], [2, 51, 53, 102], [5, 22, 27, 110], [7, 16, 23, 112], [3, 38, 41, 114], [5, 24, 29, 120], [8, 15, 23, 120], [4, 31, 35, 124], [9, 14, 23, 126], [6, 21, 27, 126], [10, 13, 23, 130], [11, 12, 23, 132], [3, 44, 47, 132], [4, 33, 37, 132], [6, 23, 29, 138], [7, 20, 27, 140], [4, 37, 41, 148], [5, 30, 35, 150], [3, 50, 53, 150], [8, 19, 27, 152], [7, 22, 29, 154], [5, 32, 37, 160], [9, 18, 27, 162], [8, 21, 29, 168], [10, 17, 27, 170], [4, 43, 47, 172], [6, 29, 35, 174], [11, 16, 27, 176], [5, 36, 41, 180], [12, 15, 27, 180], [9, 20, 29, 180], [13, 14, 27, 182], [6, 31, 37, 186], [10, 19, 29, 190], [7, 28, 35, 196], [4, 49, 53, 196], [11, 18, 29, 198], [12, 17, 29, 204], [13, 16, 29, 208], [7, 30, 37, 210], [5, 42, 47, 210], [14, 15, 29, 210], [6, 35, 41, 210], [8, 27, 35, 216], [8, 29, 37, 232], [9, 26, 35, 234], [7, 34, 41, 238], [5, 48, 53, 240], [6, 41, 47, 246], [10, 25, 35, 250], [9, 28, 37, 252], [11, 24, 35, 264], [8, 33, 41, 264], [10, 27, 37, 270], [12, 23, 35, 276], [7, 40, 47, 280], [6, 47, 53, 282], [13, 22, 35, 286], [11, 26, 37, 286], [9, 32, 41, 288], [14, 21, 35, 294], [15, 20, 35, 300], [12, 25, 37, 300], [16, 19, 35, 304], [17, 18, 35, 306], [10, 31, 41, 310], [13, 24, 37, 312], [8, 39, 47, 312], [14, 23, 37, 322], [7, 46, 53, 322], [11, 30, 41, 330], [15, 22, 37, 330], [16, 21, 37, 336], [17, 20, 37, 340], [18, 19, 37, 342], [9, 38, 47, 342], [12, 29, 41, 348], [8, 45, 53, 360], [13, 28, 41, 364], [10, 37, 47, 370], [14, 27, 41, 378], [15, 26, 41, 390], [11, 36, 47, 396], [9, 44, 53, 396], [16, 25, 41, 400], [17, 24, 41, 408], [18, 23, 41, 414], [19, 22, 41, 418], [20, 21, 41, 420], [12, 35, 47, 420], [10, 43, 53, 430], [13, 34, 47, 442], [11, 42, 53, 462], [14, 33, 47, 462], [15, 32, 47, 480], [12, 41, 53, 492], [16, 31, 47, 496], [17, 30, 47, 510], [13, 40, 53, 520], [18, 29, 47, 522], [19, 28, 47, 532], [20, 27, 47, 540], [21, 26, 47, 546], [14, 39, 53, 546], [22, 25, 47, 550], [23, 24, 47, 552], [15, 38, 53, 570], [16, 37, 53, 592], [17, 36, 53, 612], [18, 35, 53, 630], [19, 34, 53, 646], [20, 33, 53, 660], [21, 32, 53, 672], [22, 31, 53, 682], [23, 30, 53, 690], [24, 29, 53, 696], [25, 28, 53, 700], [26, 27, 53, 702]] 

?- time(s3(Q,100)).
% 22,249 inferences, 0.005 CPU in 0.005 seconds (93% CPU, 4343811 Lips)
Q = [[4, 7, 11, 28], [3, 8, 11, 24], [2, 9, 11, 18], [4, 13, 17, 52], [10, 13, 23, 130], [7, 16, 23, 112], [4, 19, 23, 76], [13, 14, 27, 182], [11, 16, 27, 176], [10, 17, 27, 170], [9, 18, 27, 162], [8, 19, 27, 152], [7, 20, 27, 140], [5, 22, 27, 110], [4, 23, 27, 92], [2, 25, 27, 50], [13, 16, 29, 208], [12, 17, 29, 204], [11, 18, 29, 198], [10, 19, 29, 190], [8, 21, 29, 168], [7, 22, 29, 154], [6, 23, 29, 138], [4, 25, 29, 100], [2, 27, 29, 54], [17, 18, 35, 306], [16, 19, 35, 304], [14, 21, 35, 294], [12, 23, 35, 276], [10, 25, 35, 250], [9, 26, 35, 234], [8, 27, 35, 216], [6, 29, 35, 174], [4, 31, 35, 124], [3, 32, 35, 96], [17, 20, 37, 340], [16, 21, 37, 336], [10, 27, 37, 270], [9, 28, 37, 252], [8, 29, 37, 232], [6, 31, 37, 186], [5, 32, 37, 160], [19, 22, 41, 418], [18, 23, 41, 414], [17, 24, 41, 408], [16, 25, 41, 400], [15, 26, 41, 390], [14, 27, 41, 378], [13, 28, 41, 364], [12, 29, 41, 348], [10, 31, 41, 310], [9, 32, 41, 288], [7, 34, 41, 238], [4, 37, 41, 148], [3, 38, 41, 114], [23, 24, 47, 552], [22, 25, 47, 550], [20, 27, 47, 540], [19, 28, 47, 532], [18, 29, 47, 522], [17, 30, 47, 510], [16, 31, 47, 496], [15, 32, 47, 480], [13, 34, 47, 442], [10, 37, 47, 370], [7, 40, 47, 280], [6, 41, 47, 246], [4, 43, 47, 172], [26, 27, 53, 702], [25, 28, 53, 700], [24, 29, 53, 696], [23, 30, 53, 690], [22, 31, 53, 682], [21, 32, 53, 672], [20, 33, 53, 660], [19, 34, 53, 646], [18, 35, 53, 630], [17, 36, 53, 612], [16, 37, 53, 592], [15, 38, 53, 570], [13, 40, 53, 520], [12, 41, 53, 492], [10, 43, 53, 430], [8, 45, 53, 360], [6, 47, 53, 282], [5, 48, 53, 240]] 

?- time(s4(Q,100)).
% 22,258 inferences, 0.005 CPU in 0.005 seconds (99% CPU, 4362603 Lips)
Q = [[4, 13, 17, 52]] 

12 ?- time(s4(Q,500)).
% 1,104,872 inferences, 0.296 CPU in 0.307 seconds (96% CPU, 3737221 Lips)
Q = [[4, 13, 17, 52]] 
*/