Coursework.pl

/* <Username: jat52 	Name:James Treasure>

THIS WORK IS ENTIRELY MY OWN.

The program does not produce multiple answers.

I have not used built-ins.

1. <Number of elements in the list binding Q after executing s1(Q,100) is 1747>
<First I make a list of every potential quad before anything at all has been said.
Remove unique products>

2. <Number of elements in the list binding Q after executing s2(Q,100) is 145>
<Generate a list of numbers that S must be. Remove any quad where S is not in
that list. Sort the list using merge sort so that the list is ordered in
ascending values of P>

3. <Number of elements in the list binding Q after executing s3(Q,100) is 86>
<Remove duplicate producuts. Sort the list using merge sort so that the list
is ordered in ascending values of S>

4.
<Remove duplicate sums leaving the final answer.>

5. <Number of elements in the list binding Q after executing s4(Q,500) is 1>
s4(Q,500) uses 7,629,274,328 inferences. */

%------s1------
get_quads(MaxSum, Result) :-
    MaxSum> 1,								
    get_quads(2, MaxSum, 2, Result).		
get_quads(Sum, Sum, Sum, []).            % 'Sum' has reached max value
get_quads(Sum, MaxSum, Sum, Result) :-   
    Sum < MaxSum,
    Sum1 is Sum + 1,
    get_quads(Sum1, MaxSum, 2, Result).	%3,100,1,Result
get_quads(Sum, MaxSum, X, [[X, Y, Sum, P]|Result]) :- 
	Sum =< MaxSum,	
	X < Sum,
	Y is Sum - X, 
	X < Y,
	P is X*Y,
	X1 is X + 1, 
	get_quads(Sum, MaxSum, X1, Result).
get_quads(Sum, MaxSum, X, Result) :-
	X1 is X + 1,
    get_quads(Sum, MaxSum, X1, Result).
 
fourth([_,_,_,F|_], F).
 
get_fourth([], _, 0).
get_fourth([H|T], X, R) :- 
	fourth(H, X),!,
	get_fourth(T, X, RR), 
	R is RR + 1.
get_fourth([_|T], X, R) :- 
	get_fourth(T, X, R).

remove_unique([], [], _).
remove_unique([H|T], [H|R], A) :- 
	fourth(H, X), 
	get_fourth(A, X, C),
	C > 1, 
	remove_unique(T, R, A).
remove_unique([_|T], R, A) :- 
	remove_unique(T, R, A).
remove_unique(L, R) :-  
	remove_unique(L, R, L), !.
 
s1(Q,N):-
	get_quads(N,Numbers), !,
	remove_unique(Numbers,Q).

%------s2------
prime(2).
prime(N) :- 
	sqrt(N,S), 
	check(N,S,3).	

check(_,S,D) :- D>S.	
check(N,S,D) :-
	N =\= D*(N//D), 
	D1 is D+2,		
	check(N,S,D1).

prime_list(B,L) :-
 	 prime_list(2,B,L).	

prime_list(A,B,[]) :-
	 A > B, 		
	 !.

prime_list(A,B,[A|L]) :- 
	prime(A), 		
	!, 
   	next(A,A1),		
   	prime_list(A1,B,L).

prime_list(A,B,L) :- 
   next(A,A1),
   prime_list(A1,B,L).

next(2,3) :- !.
next(A,A1) :- A1 is A + 2.

add_to_list([],[]).
add_to_list([H|T],[Y|List]):-
	Y is H + 2,
	add_to_list(T,List).

remove_even([],[]).
remove_even([H|T], R):- 
    H mod 2 =:= 0,
    remove_even(T, R).
remove_even([H|T1], [H|T2]):-                            
    remove_even(T1, T2).

generateNumbers(0,[]).
generateNumbers(N,[Head|Tail]):-
	N > 0,						%N must be greater than 0. Base case.
    Head is N,					%Sets the head of list to N
    N1 is N-1,					%Decrements N by 1
    generateNumbers(N1,Tail).	%Recurses on generateNumbers

mysubtract([], _, []).
mysubtract([Head|Tail], L2, L3) :-
        member(Head, L2),
        !,
        mysubtract(Tail, L2, L3).
mysubtract([Head|Tail1], L2, [Head|Tail3]) :-
        mysubtract(Tail1, L2, Tail3).

remove_greater([],[]).
remove_greater([H|T],[H|List]):-
	H < 55,
	H \= 51,
	remove_greater(T,List).
remove_greater([_|T],List):-
	remove_greater(T,List).

not_member(_,[]).
not_member(Element,[H|Tail]) :-
	Element \= H,
	not_member(Element, Tail),!.

member_third(H,List):-
	third(H,Third),
	not_member(Third,List).

s2_remove([],_,[]).
s2_remove([H|T], TheList, Result) :-
	member_third(H, TheList),
	s2_remove(T, TheList, Result),!.
s2_remove([A|T], TheList, [A|Result]) :-
	s2_remove(T, TheList, Result).

split_list([], [], []).
split_list([A], [A], []).
split_list([A, B|T], [A|Ta], [B|Tb]) :-
	split_list(T, Ta, Tb).

merge_by_product(A, [], A).
merge_by_product([], A, A).
merge_by_product([A|Ta], [B|Tb], [A|S]) :-
	A = [_, _, _, P1],
	B = [_, _, _, P2],
	P1 =< P2,
	merge_by_product(Ta, [B|Tb], S).
merge_by_product([A|Ta], [B|Tb], [B|S]) :-
	A = [_, _, _, P1],
	B = [_, _, _, P2],
	P1 > P2,
	merge_by_product([A|Ta], Tb, S).

sort_by_product([], []).
sort_by_product([A], [A]).
sort_by_product([A, B|T], S) :-
	split_list([A, B|T], L1, L2),
	sort_by_product(L1, S1),
	sort_by_product(L2, S2),
	merge_by_product(S1, S2, S).


s2(Q,End):-
	generateNumbers(End,Numbers),
	remove_greater(Numbers,RemovedNumbers),
	prime_list(End,Primes),
	remove_greater(Primes,RemovedPrimes),
	add_to_list(RemovedPrimes,NewList),
	mysubtract(RemovedNumbers,NewList,SubList),
	remove_even(SubList,SPossible),
	s1(S1,End),
	s2_remove(S1,SPossible,AA),!,
	sort_by_product(AA,Q),!.

%------s3------
remove_duplicte_products([], [], _).
 
remove_duplicte_products([H|T], [H|R], A) :- 
	last(H, X),
	get_fourth(A, X, C),
	C =:= 1, 
	remove_duplicte_products(T, R, A).
 
remove_duplicte_products([_|T], R, A) :- 
	remove_duplicte_products(T, R, A).
 
remove_duplicte_products(L, R) :-  
	remove_duplicte_products(L, R, L), !.

merge_by_sum(A, [], A).
merge_by_sum([], A, A).
merge_by_sum([A|Ta], [B|Tb], [A|S]) :-
	A = [_, _, S1, _],
	B = [_, _, S2, _],
	S1 =< S2,
	merge_by_sum(Ta, [B|Tb], S).

merge_by_sum([A|Ta], [B|Tb], [B|S]) :-
	A = [_, _, S1, _],
	B = [_, _, S2, _],
	S1 > S2,
	merge_by_sum([A|Ta], Tb, S).

sort_by_sum([], []).
sort_by_sum([A], [A]).
sort_by_sum([A, B|T], S) :-
	split_list([A, B|T], L1, L2),
	sort_by_sum(L1, S1),
	sort_by_sum(L2, S2),
	merge_by_sum(S1, S2, S).

s3(Q,N):-
	s2(Result,N),
	remove_duplicte_products(Result,AA),!,
	sort_by_sum(AA,Q),!.

%------s4------
third([_,_,Third|_], Third).
 
get_third([], _, 0).
get_third([H|T], X, R) :- 
	third(H, X),!,
	get_third(T, X, RR), 
	R is RR + 1.
get_third([_|T], X, R) :- 
	get_third(T, X, R).

remove_duplicate_sums([], [], _).
remove_duplicate_sums([H|T], [H|R], A) :- 
	third(H, X),
	get_third(A, X, C),
	C =:= 1, 
	remove_duplicate_sums(T, R, A).
remove_duplicate_sums([_|T], R, A) :- 
	remove_duplicate_sums(T, R, A).
remove_duplicate_sums(L, R) :-  
	remove_duplicate_sums(L, R, L), !.

s4(Q,N):-
	s3(Result,N),!,
	remove_duplicate_sums(Result,Q).


/*
?- consult('jat52.pl').
% jat52.pl compiled 0.01 sec, 76 clauses
true.
?- time(s1(Q,100)).
% 11,107,031 inferences, 1.701 CPU in 1.731 seconds (98% CPU, 6528885 Lips)
Q = [[3,4,7,12],[2,6,8,12],[3,6,9,18],[4,5,9,20],[4,6,10,24],[2,9,11,18],[3,8,11,24],[4,7,11,28],[5,6,11,30],[2,10,12,20],[4,8,12,32],[3,10,13,30],[4,9,13,36],[5,8,13,40],[6,7,13,42],[2,12,14,24],[4,10,14,40],[5,9,14,45],[6,8,14,48],[3,12,15,36],[4,11,15,44],[5,10,15,50],[6,9,15,54],[7,8,15,56],[2,14,16,28],[4,12,16,48],[6,10,16,60],[7,9,16,63],[2,15,17,30],[3,14,17,42],[4,13,17,52],[5,12,17,60],[6,11,17,66],[7,10,17,70],[8,9,17,72],[2,16,18,32],[3,15,18,45],[4,14,18,56],[6,12,18,72],[8,10,18,80],[3,16,19,48],[4,15,19,60],[5,14,19,70],[6,13,19,78],[7,12,19,84],[8,11,19,88],[9,10,19,90],[2,18,20,36],[4,16,20,64],[5,15,20,75],[6,14,20,84],[8,12,20,96],[9,11,20,99],[3,18,21,54],[4,17,21,68],[5,16,21,80],[6,15,21,90],[7,14,21,98],[8,13,21,104],[9,12,21,108],[10,11,21,110],[2,20,22,40],[4,18,22,72],[6,16,22,96],[7,15,22,105],[8,14,22,112],[9,13,22,117],[10,12,22,120],[2,21,23,42],[3,20,23,60],[4,19,23,76],[5,18,23,90],[6,17,23,102],[7,16,23,112],[8,15,23,120],[9,14,23,126],[10,13,23,130],[11,12,23,132],[2,22,24,44],[3,21,24,63],[4,20,24,80],[6,18,24,108],[8,16,24,128],[9,15,24,135],[10,14,24,140],[3,22,25,66],[4,21,25,84],[5,20,25,100],[6,19,25,114],[7,18,25,126],[8,17,25,136],[9,16,25,144],[10,15,25,150],[11,14,25,154],[12,13,25,156],[2,24,26,48],[4,22,26,88],[5,21,26,105],[6,20,26,120],[8,18,26,144],[9,17,26,153],[10,16,26,160],[11,15,26,165],[12,14,26,168],[2,25,27,50],[3,24,27,72],[4,23,27,92],[5,22,27,110],[6,21,27,126],[7,20,27,140],[8,19,27,152],[9,18,27,162],[10,17,27,170],[11,16,27,176],[12,15,27,180],[13,14,27,182],[2,26,28,52],[3,25,28,75],[4,24,28,96],[6,22,28,132],[7,21,28,147],[8,20,28,160],[9,19,28,171],[10,18,28,180],[12,16,28,192],[13,15,28,195],[2,27,29,54],[3,26,29,78],[4,25,29,100],[5,24,29,120],[6,23,29,138],[7,22,29,154],[8,21,29,168],[9,20,29,180],[10,19,29,190],[11,18,29,198],[12,17,29,204],[13,16,29,208],[14,15,29,210],[2,28,30,56],[4,26,30,104],[6,24,30,144],[8,22,30,176],[9,21,30,189],[10,20,30,200],[12,18,30,216],[14,16,30,224],[3,28,31,84],[4,27,31,108],[5,26,31,130],[6,25,31,150],[7,24,31,168],[8,23,31,184],[9,22,31,198],[10,21,31,210],[11,20,31,220],[12,19,31,228],[13,18,31,234],[14,17,31,238],[15,16,31,240],[2,30,32,60],[4,28,32,112],[5,27,32,135],[6,26,32,156],[7,25,32,175],[8,24,32,192],[9,23,32,207],[10,22,32,220],[11,21,32,231],[12,20,32,240],[14,18,32,252],[15,17,32,255],[3,30,33,90],[4,29,33,116],[5,28,33,140],[6,27,33,162],[7,26,33,182],[8,25,33,200],[9,24,33,216],[10,23,33,230],[12,21,33,252],[13,20,33,260],[14,19,33,266],[15,18,33,270],[16,17,33,272],[2,32,34,64],[4,30,34,120],[6,28,34,168],[7,27,34,189],[8,26,34,208],[9,25,34,225],[10,24,34,240],[12,22,34,264],[13,21,34,273],[14,20,34,280],[15,19,34,285],[16,18,34,288],[2,33,35,66],[3,32,35,96],[4,31,35,124],[5,30,35,150],[6,29,35,174],[7,28,35,196],[8,27,35,216],[9,26,35,234],[10,25,35,250],[11,24,35,264],[12,23,35,276],[13,22,35,286],[14,21,35,294],[15,20,35,300],[16,19,35,304],[17,18,35,306],[2,34,36,68],[3,33,36,99],[4,32,36,128],[6,30,36,180],[8,28,36,224],[9,27,36,243],[10,26,36,260],[11,25,36,275],[12,24,36,288],[14,22,36,308],[15,21,36,315],[16,20,36,320],[2,35,37,70],[3,34,37,102],[4,33,37,132],[5,32,37,160],[6,31,37,186],[7,30,37,210],[8,29,37,232],[9,28,37,252],[10,27,37,270],[11,26,37,286],[12,25,37,300],[13,24,37,312],[14,23,37,322],[15,22,37,330],[16,21,37,336],[17,20,37,340],[18,19,37,342],[2,36,38,72],[3,35,38,105],[4,34,38,136],[5,33,38,165],[6,32,38,192],[8,30,38,240],[9,29,38,261],[10,28,38,280],[11,27,38,297],[12,26,38,312],[13,25,38,325],[14,24,38,336],[15,23,38,345],[16,22,38,352],[17,21,38,357],[18,20,38,360],[3,36,39,108],[4,35,39,140],[5,34,39,170],[6,33,39,198],[7,32,39,224],[8,31,39,248],[9,30,39,270],[10,29,39,290],[11,28,39,308],[12,27,39,324],[14,25,39,350],[15,24,39,360],[16,23,39,368],[17,22,39,374],[18,21,39,378],[19,20,39,380],[2,38,40,76],[4,36,40,144],[5,35,40,175],[6,34,40,204],[7,33,40,231],[8,32,40,256],[9,31,40,279],[10,30,40,300],[12,28,40,336],[13,27,40,351],[14,26,40,364],[15,25,40,375],[16,24,40,384],[18,22,40,396],[19,21,40,399],[2,39,41,78],[3,38,41,114],[4,37,41,148],[5,36,41,180],[6,35,41,210],[7,34,41,238],[8,33,41,264],[9,32,41,288],[10,31,41,310],[11,30,41,330],[12,29,41,348],[13,28,41,364],[14,27,41,378],[15,26,41,390],[16,25,41,400],[17,24,41,408],[18,23,41,414],[19,22,41,418],[20,21,41,420],[2,40,42,80],[3,39,42,117],[4,38,42,152],[6,36,42,216],[7,35,42,245],[8,34,42,272],[9,33,42,297],[10,32,42,320],[12,30,42,360],[14,28,42,392],[15,27,42,405],[16,26,42,416],[17,25,42,425],[18,24,42,432],[20,22,42,440],[3,40,43,120],[4,39,43,156],[5,38,43,190],[6,37,43,222],[7,36,43,252],[8,35,43,280],[9,34,43,306],[10,33,43,330],[11,32,43,352],[12,31,43,372],[13,30,43,390],[14,29,43,406],[15,28,43,420],[16,27,43,432],[17,26,43,442],[18,25,43,450],[19,24,43,456],[20,23,43,460],[21,22,43,462],[2,42,44,84],[4,40,44,160],[5,39,44,195],[6,38,44,228],[8,36,44,288],[9,35,44,315],[10,34,44,340],[12,32,44,384],[14,30,44,420],[15,29,44,435],[16,28,44,448],[17,27,44,459],[18,26,44,468],[19,25,44,4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],[24,62,86,1488],[26,60,86,1560],[28,58,86,1624],[30,56,86,1680],[32,54,86,1728],[34,52,86,1768],[36,50,86,1800],[38,48,86,1824],[42,44,86,1848],[2,85,87,170],[3,84,87,252],[5,82,87,410],[6,81,87,486],[7,80,87,560],[9,78,87,702],[10,77,87,770],[11,76,87,836],[12,75,87,900],[13,74,87,962],[15,72,87,1080],[17,70,87,1190],[18,69,87,1242],[19,68,87,1292],[21,66,87,1386],[22,65,87,1430],[23,64,87,1472],[24,63,87,1512],[25,62,87,1550],[27,60,87,1620],[30,57,87,1710],[31,56,87,1736],[32,55,87,1760],[33,54,87,1782],[35,52,87,1820],[36,51,87,1836],[37,50,87,1850],[39,48,87,1872],[42,45,87,1890],[2,86,88,172],[3,85,88,255],[4,84,88,336],[6,82,88,492],[7,81,88,567],[8,80,88,640],[10,78,88,780],[12,76,88,912],[13,75,88,975],[14,74,88,1036],[16,72,88,1152],[18,70,88,1260],[19,69,88,1311],[20,68,88,1360],[22,66,88,1452],[24,64,88,1536],[25,63,88,1575],[26,62,88,1612],[28,60,88,1680],[30,58,88,1740],[32,56,88,1792],[34,54,88,1836],[36,52,88,1872],[40,48,88,1920],[42,46,88,1932],[2,87,89,174],[3,86,89,258],[4,85,89,340],[5,84,89,420],[7,82,89,574],[8,81,89,648],[9,80,89,720],[11,78,89,858],[12,77,89,924],[13,76,89,988],[14,75,89,1050],[15,74,89,1110],[17,72,89,1224],[19,70,89,1330],[20,69,89,1380],[21,68,89,1428],[23,66,89,1518],[24,65,89,1560],[25,64,89,1600],[26,63,89,1638],[27,62,89,1674],[29,60,89,1740],[31,58,89,1798],[32,57,89,1824],[33,56,89,1848],[35,54,89,1890],[37,52,89,1924],[38,51,89,1938],[39,50,89,1950],[40,49,89,1960],[44,45,89,1980],[2,88,90,176],[3,87,90,261],[4,86,90,344],[5,85,90,425],[6,84,90,504],[8,82,90,656],[10,80,90,800],[12,78,90,936],[14,76,90,1064],[15,75,90,1125],[16,74,90,1184],[18,72,90,1296],[20,70,90,1400],[21,69,90,1449],[22,68,90,1496],[24,66,90,1584],[26,64,90,1664],[28,62,90,1736],[30,60,90,1800],[32,58,90,1856],[34,56,90,1904],[36,54,90,1944],[42,48,90,2016],[3,88,91,264],[4,87,91,348],[5,86,91,430],[6,85,91,510],[7,84,91,588],[9,82,91,738],[10,81,91,810],[11,80,91,880],[13,78,91,1014],[14,77,91,1078],[15,76,91,1140],[16,75,91,1200],[17,74,91,1258],[19,72,91,1368],[21,70,91,1470],[22,69,91,1518],[23,68,91,1564],[25,66,91,1650],[27,64,91,1728],[28,63,91,1764],[29,62,91,1798],[31,60,91,1860],[33,58,91,1914],[34,57,91,1938],[35,56,91,1960],[36,55,91,1980],[40,51,91,2040],[45,46,91,2070],[2,90,92,180],[4,88,92,352],[5,87,92,435],[6,86,92,516],[7,85,92,595],[8,84,92,672],[10,82,92,820],[11,81,92,891],[12,80,92,960],[14,78,92,1092],[15,77,92,1155],[16,76,92,1216],[17,75,92,1275],[18,74,92,1332],[20,72,92,1440],[22,70,92,1540],[24,68,92,1632],[26,66,92,1716],[27,65,92,1755],[28,64,92,1792],[30,62,92,1860],[32,60,92,1920],[34,58,92,1972],[36,56,92,2016],[38,54,92,2052],[40,52,92,2080],[42,50,92,2100],[44,48,92,2112],[2,91,93,182],[3,90,93,270],[5,88,93,440],[6,87,93,522],[7,86,93,602],[8,85,93,680],[9,84,93,756],[11,82,93,902],[12,81,93,972],[13,80,93,1040],[15,78,93,1170],[16,77,93,1232],[17,76,93,1292],[18,75,93,1350],[19,74,93,1406],[21,72,93,1512],[23,70,93,1610],[24,69,93,1656],[25,68,93,1700],[27,66,93,1782],[28,65,93,1820],[29,64,93,1856],[30,63,93,1890],[33,60,93,1980],[35,58,93,2030],[36,57,93,2052],[42,51,93,2142],[45,48,93,2160],[2,92,94,184],[3,91,94,273],[4,90,94,360],[6,88,94,528],[7,87,94,609],[8,86,94,688],[9,85,94,765],[10,84,94,840],[12,82,94,984],[13,81,94,1053],[14,80,94,1120],[16,78,94,1248],[18,76,94,1368],[19,75,94,1425],[20,74,94,1480],[22,72,94,1584],[24,70,94,1680],[25,69,94,1725],[26,68,94,1768],[28,66,94,1848],[30,64,94,1920],[32,62,94,1984],[34,60,94,2040],[39,55,94,2145],[40,54,94,2160],[42,52,94,2184],[44,50,94,2200],[45,49,94,2205],[2,93,95,186],[3,92,95,276],[4,91,95,364],[5,90,95,450],[7,88,95,616],[8,87,95,696],[9,86,95,774],[10,85,95,850],[11,84,95,924],[13,82,95,1066],[14,81,95,1134],[15,80,95,1200],[17,78,95,1326],[18,77,95,1386],[20,75,95,1500],[21,74,95,1554],[23,72,95,1656],[25,70,95,1750],[26,69,95,1794],[27,68,95,1836],[29,66,95,1914],[30,65,95,1950],[31,64,95,1984],[32,63,95,2016],[33,62,95,2046],[35,60,95,2100],[39,56,95,2184],[40,55,95,2200],[44,51,95,2244],[2,94,96,188],[3,93,96,279],[4,92,96,368],[5,91,96,455],[6,90,96,540],[8,88,96,704],[9,87,96,783],[10,86,96,860],[11,85,96,935],[12,84,96,1008],[14,82,96,1148],[15,81,96,1215],[16,80,96,1280],[18,78,96,1404],[20,76,96,1520],[21,75,96,1575],[22,74,96,1628],[24,72,96,1728],[26,70,96,1820],[28,68,96,1904],[30,66,96,1980],[34,62,96,2108],[36,60,96,2160],[40,56,96,2240],[42,54,96,2268],[2,95,97,190],[3,94,97,282],[4,93,97,372],[5,92,97,460],[6,91,97,546],[7,90,97,630],[9,88,97,792],[10,87,97,870],[11,86,97,946],[12,85,97,1020],[13,84,97,1092],[15,82,97,1230],[16,81,97,1296],[17,80,97,1360],[19,78,97,1482],[20,77,97,1540],[21,76,97,1596],[22,75,97,1650],[23,74,97,1702],[25,72,97,1800],[27,70,97,1890],[28,69,97,1932],[29,68,97,1972],[31,66,97,2046],[32,65,97,2080],[33,64,97,2112],[34,63,97,2142],[40,57,97,2280],[45,52,97,2340],[48,49,97,2352],[2,96,98,192],[3,95,98,285],[4,94,98,376],[5,93,98,465],[6,92,98,552],[7,91,98,637],[8,90,98,720],[10,88,98,880],[11,87,98,957],[12,86,98,1032],[13,85,98,1105],[14,84,98,1176],[16,82,98,1312],[17,81,98,1377],[18,80,98,1440],[20,78,98,1560],[21,77,98,1617],[22,76,98,1672],[23,75,98,1725],[24,74,98,1776],[26,72,98,1872],[28,70,98,1960],[30,68,98,2040],[32,66,98,2112],[33,65,98,2145],[34,64,98,2176],[35,63,98,2205],[38,60,98,2280],[42,56,98,2352],[48,50,98,2400],[3,96,99,288],[4,95,99,380],[5,94,99,470],[6,93,99,558],[7,92,99,644],[8,91,99,728],[9,90,99,810],[11,88,99,968],[12,87,99,1044],[13,86,99,1118],[14,85,99,1190],[15,84,99,1260],[17,82,99,1394],[18,81,99,1458],[19,80,99,1520],[21,78,99,1638],[23,76,99,1748],[24,75,99,1800],[25,74,99,1850],[27,72,99,1944],[29,70,99,2030],[30,69,99,2070],[31,68,99,2108],[35,64,99,2240],[36,63,99,2268],[39,60,99,2340],[2,98,100,196],[4,96,100,384],[5,95,100,475],[6,94,100,564],[7,93,100,651],[8,92,100,736],[9,91,100,819],[10,90,100,900],[12,88,100,1056],[13,87,100,1131],[14,86,100,1204],[15,85,100,1275],[16,84,100,1344],[18,82,100,1476],[19,81,100,1539],[20,80,100,1600],[22,78,100,1716],[24,76,100,1824],[26,74,100,1924],[28,72,100,2016],[30,70,100,2100],[32,68,100,2176],[34,66,100,2244],[40,60,100,2400]].

?- time(s2(Q,100)).
% 11,163,637 inferences, 1.702 CPU in 1.740 seconds (98% CPU, 6558890 Lips)
Q = [[2,9,11,18],[3,8,11,24],[4,7,11,28],[2,15,17,30],[5,6,11,30],[3,14,17,42],[2,21,23,42],[2,25,27,50],[4,13,17,52],[2,27,29,54],[3,20,23,60],[5,12,17,60],[6,11,17,66],[2,33,35,66],[2,35,37,70],[7,10,17,70],[8,9,17,72],[3,24,27,72],[4,19,23,76],[3,26,29,78],[2,39,41,78],[2,45,47,90],[5,18,23,90],[4,23,27,92],[3,32,35,96],[4,25,29,100],[2,51,53,102],[6,17,23,102],[3,34,37,102],[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],[4,33,37,132],[11,12,23,132],[3,44,47,132],[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],[9,20,29,180],[5,36,41,180],[12,15,27,180],[13,14,27,182],[6,31,37,186],[10,19,29,190],[4,49,53,196],[7,28,35,196],[11,18,29,198],[12,17,29,204],[13,16,29,208],[5,42,47,210],[14,15,29,210],[7,30,37,210],[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],[8,33,41,264],[11,24,35,264],[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],[12,25,37,300],[15,20,35,300],[16,19,35,304],[17,18,35,306],[10,31,41,310],[8,39,47,312],[13,24,37,312],[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],[12,35,47,420],[20,21,41,420],[10,43,53,430],[13,34,47,442],[14,33,47,462],[11,42,53,462],[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],[14,39,53,546],[21,26,47,546],[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)).
% 11,204,219 inferences, 1.723 CPU in 1.754 seconds (98% CPU, 6502267 Lips)
Q = [[2,9,11,18],[4,7,11,28],[3,8,11,24],[4,13,17,52],[4,19,23,76],[10,13,23,130],[7,16,23,112],[7,20,27,140],[8,19,27,152],[5,22,27,110],[11,16,27,176],[9,18,27,162],[2,25,27,50],[13,14,27,182],[4,23,27,92],[10,17,27,170],[13,16,29,208],[8,21,29,168],[11,18,29,198],[4,25,29,100],[2,27,29,54],[10,19,29,190],[7,22,29,154],[6,23,29,138],[12,17,29,204],[17,18,35,306],[3,32,35,96],[12,23,35,276],[14,21,35,294],[8,27,35,216],[6,29,35,174],[4,31,35,124],[9,26,35,234],[10,25,35,250],[16,19,35,304],[9,28,37,252],[5,32,37,160],[6,31,37,186],[8,29,37,232],[16,21,37,336],[10,27,37,270],[17,20,37,340],[14,27,41,378],[7,34,41,238],[12,29,41,348],[3,38,41,114],[18,23,41,414],[16,25,41,400],[13,28,41,364],[4,37,41,148],[10,31,41,310],[15,26,41,390],[9,32,41,288],[19,22,41,418],[17,24,41,408],[15,32,47,480],[22,25,47,550],[4,43,47,172],[16,31,47,496],[19,28,47,532],[6,41,47,246],[23,24,47,552],[18,29,47,522],[17,30,47,510],[7,40,47,280],[20,27,47,540],[10,37,47,370],[13,34,47,442],[21,32,53,672],[13,40,53,520],[25,28,53,700],[17,36,53,612],[6,47,53,282],[23,30,53,690],[15,38,53,570],[19,34,53,646],[10,43,53,430],[12,41,53,492],[22,31,53,682],[5,48,53,240],[26,27,53,702],[8,45,53,360],[18,35,53,630],[24,29,53,696],[16,37,53,592],[20,33,53,660]].

?- time(s4(Q,100)).
% 11,220,296 inferences, 1.676 CPU in 1.709 seconds (98% CPU, 6693793 Lips)
Q = [[4,13,17,52]].
*/