######################################################################
## RingRing.txt Save this file as  RingRing.txt  to use it,          #
# stay in the                                                        #
## same directory, get into Maple (by typing: maple <Enter> )        #
## and then type:  read RingRing.txt <Enter>                         #
## Then follow the instructions given there                          #
##                                                                   #
## Written by Doron Zeilberger, Rutgers University ,                 #
##  DoronZeil at gmail dot com                                       # 
######################################################################


print(`First Written: June 2021: tested for Maple 2018 `):
print(`Version  : June 2021:  `):
print():
print(`This is RingRing.txt, A Maple package to solve and generate Ring Ring puzzles as appearing (starting June 2021) in the New York Times magazine`):
print(`created by Prasanna Seshadri`):
print(`accompanying Shalsoh B. Ekhad's book  `):
print(`Fifty Ring Ring puzzles `):
print(`  `):

print():
print(`The most current version is available on WWW at:`):
print(` http://sites.math.rutgers.edu/~zeilberg/tokhniot/RingRing.txt .`):
print(`Please report all bugs to: DoronZeil at gmail dot com .`):
print():
print(`For general help, and a list of the MAIN functions,`):
print(` type "ezra();". For specific help type "ezra(procedure_name);" `):
print(`For a list of the supporting functions type: ezra1();`):
print():
 



ezraDB:=proc()
if args=NULL then

print(`The prec-computed datab-base procedures are`):
print(`  DB66, DB77, DB88, NYT`):

else
ezra(args):

fi:

end:

ezra1:=proc()
if args=NULL then

print(`The SUPPORTING procedures are`):
print(` CP, FindNewRecs, IniReg, Kids, MakePuz1, PeriE, PeriV, Rec, RecCov`):

else
ezra(args):

fi:

end:

ezra:=proc()
if args=NULL then

print(` RingRing.txt: A Maple package for  solving and creating RingRing puzzles`):

print(`The MAIN procedures are`):
print(` MakePuzBook, MakePuzSol, PlotPuz, PlotSol, RingRing `):

elif nargs=1 and args[1]=CP then
print(`CP(S): given a set of lattice points, S, finds the point [i,j] such that i is as small as possible and for that i, j is as small as possible . Try:`):
print(`CP({[0,0],[1,0],[2,0],[2,2],[-2,-5]});`):


elif nargs=1 and args[1]=CP then
print(`CP(S): given a set of lattice points, S, finds the point [i,j] such that i is as small as possible and for that i, j is as small as possible . Try:`):
print(`CP({[0,0],[1,0],[2,0],[2,2],[-2,-5]});`):


elif nargs=1 and args[1]=DB66 then
print(`DB66(): a list of 50 pre-computed  Ring Ring 6 by 6 puzzles and their solutions. Try:`):
print(`P:=DB66()[1]:  S:=RingRing(6,6,P[1]): evalb(nops(S)=1);  evalb(P[2]=S[1]); PlotSol(6,6,op(P)); `):


elif nargs=1 and args[1]=DB77 then
print(`DB77(): a list of 50 pre-computed  Ring Ring 7 by 7 puzzles and their solutions. Try:`):
print(`P:=DB77()[1]:  S:=RingRing(7,7,P[1]): evalb(nops(S)=1);  evalb(P[2]=S[1]); PlotSol(7,7,op(P)); `):


elif nargs=1 and args[1]=DB88 then
print(`DB88(): a list of 50 pre-computed  Ring Ring 8 by 8 puzzles and their solutions. Try:`):
print(`P:=DB88()[1]:  S:=RingRing(8,8,P[1]): evalb(nops(S)=1);  evalb(P[2]=S[1]); PlotSol(8,8,op(P)); `):



elif nargs=1 and args[1]=NYT then
print(`NYT(): a list of Ring Ring puzzles (so far 8) on  a 10 by 10 grid, that appeared  in the New York Times created by  Prasanna Seshadri`):
print(`P:=NYT()[1]:  S:=RingRing(10,10,P): evalb(nops(S)=1);  PlotSol(10,10,P,S[1]); `):


elif nargs=1 and args[1]=FindNewRecs then
print(`FindNewRecs(a,b,S,FE): given positive integers a and b, a set of already occupied lattice points S`):
print(`and a set of already-used edges, FE`):
print(`outputs the set all legal new rectangles followed by the new occupied set, and the new set of used edges`):
print(`FindNewRecs(4,4,{},{});`):


elif nargs=1 and args[1]=IniReg then
print(`IniReg(a,b,S): inputs positive integers a and b and a subset of lattice points S, outputs the set of all`):
print(`sets of rectangles that solve the Ring Ring puzzle where the region to be covered is`):
print(`the rectangle [0,a-1]x[0,b-1] minus the set S. For example, for the puzzle of New Times magazine (created by Prassana  Seshadri) Sunday,  June 20, 2021, type:`):
print(`IniReg(8,8,{[7,0],[3,1],[4,1],[0,2],[7,4],[5,5],[1,6],[2,6]});`):

elif nargs=1 and args[1]=Kids then
print(`KidS(St): Inputs a state St=[NeverUsed,UsedOnce,ForbiddenEdges] outputs the set of pairs [R,St1] obtained by  inserting legally a rectangle, R, that is a subset of NeverUsed union UsedOnce`):
print(`followed by the new state where the vertices of R intersect NeverUsed get moved to UsedOnce, the vertices of $ intersect UsedOnce are removed, and the new edges of Rec are joined to`):
print(`ForbiddenEdges. Try:`):
print(`Kids([IniReg(8,8,{[7,0],[3,1],[4,1],[0,2],[7,4],[5,5],[1,6],[2,6]}),{},{}]);`):

elif nargs=1 and args[1]=MakePuzBook then
print(`MakePuzBook(a,b,K,M): Creates a list of M RingRing puzzles along with their solution for an a by b rectangle by calling MakePuzSol(a,b,K) M times. Try:`):
print(`MakePuzBook(6,6,100,10);`):

elif nargs=1 and args[1]=MakePuzSol then
print(`MakePuzSol(a,b,K): Makes a random puzzle and its unique solution for an a by b rectangle but trying out MakePusSol1(a,b) K times. If it does not find anything, it returns FAIL.Try:`):
print(`MakePuzSol(6,6,100);`):

elif nargs=1 and args[1]=MakePuzSol1 then
print(`MakePuzSol1(a,b): Makes a random puzzle and its solution for an a by b rectangle if there is no unique solution or or takes more than 2 seconds, it returns FAIL. try:`):
print(`MakePuzSol1(6,6);`):

elif nargs=1 and args[1]=PeriE then
print(`PeriE(pt,a,b): The set of edges in perimeter of an a by b rectangle  whose bottom  left vertex is the lattice point pt. In other words [0,a-1]x[0,b-1]+pt`):
print(`Try:`):
print(`PeriE([2,3],3,2):`):

elif nargs=1 and args[1]=PeriV then
print(`PeriV(pt,a,b): The set of vertices in perimeter of an a by b rectangle  whose bottom  left vertex is the lattice point pt. In other words [0,a-1]x[0,b-1]+pt`):
print(`Try: `):
print(`PeriV([2,3],[3,2]):`):

elif nargs=1 and args[1]=PlotPuz then
print(`PlotPuz(a,b,S): plots the puzzle (a,b,S)`):
print(``):
print(`Try`):


print(`PlotPuz(8,8,{[7,0],[3,1],[4,1],[0,2],[7,4],[5,5],[1,6],[2,6]});`):
print(`PlotPuz(8,8,{[5,0],[6,0],[7,0],[2,2],[5,3],[0,4],[2,4],[3,5],[3,7],[7,7]});`):
print(``):
print(`For the puzzle of mid June, 2021 type:`):
print(`PlotPuz(8,8,{[7,0],[7,1],[0,2],[1,2],[4,2],[7,2],[0,3],[4,6],[3,7],[4,7]});`):
print(``):
print(`For the puzzle of June 27, 2021 type:`):
print(`PlotPuz(10,10,{[5,1],[6,1],[5,2],[4,3],[5,3],[4,4],[1,5],[8,5],[3,6],[4,6],[6,6],[2,7],[3,7],[9,9]});`):

elif nargs=1 and args[1]=PlotSol then
print(`PlotSol(a,b,S,SOL): plots the solution SOL of the Ring Ring puzzle (a,b,S). You need to enter S as well. `):
print(``):
print(`Try: `):

print(`S:= {[7,0],[3,1],[4,1],[0,2],[7,4],[5,5],[1,6],[2,6]}; PlotSol(8,8,S,RingRing(8,8,S));`):
print(``):
print(`S:={[5,0],[6,0],[7,0],[2,2],[5,3],[0,4],[2,4],[3,5],[3,7],[7,7]} ; PlotSol(8,8,S,RingRing(8,8,S));`):
print(`For the puzzle of mid June, 2021 type:`):
print(`S:={[7,0],[7,1],[0,2],[1,2],[4,2],[7,2],[0,3],[4,6],[3,7],[4,7]} ; PlotSol(8,8,S,RingRing(8,8,S));`):
print(``):
print(`For the puzzle of June 27, 2021 type:`):
print(`S:={[5,1],[6,1],[5,2],[4,3],[5,3],[4,4],[1,5],[8,5],[3,6],[4,6],[6,6],[2,7],[3,7],[9,9]}; PlotSol(10,10,S,RingRing(10,10,S));`):



elif nargs=1 and args[1]=Rec then
print(`Rec(a,b): The rectangle [0,a-1]x[0,b-1]. Try:`):
print(`Rec(8,8);`):

elif nargs=1 and args[1]=RecCov then
print(`RecCov(St): The set of coverings of St[1] that obey the rules and avoid Forbidden Edges`):

elif nargs=1 and args[1]=RingRing then

print(`RingRing(a,b,S): inputs a Ring Ring puzzle in the rectangle [0,a-1]x[0,b-1] with the lattice points of S deleted, outputs the set of solutions to a Ring Ring puzzle. Try:`):

print(`RingRing(8,8,{[7,0],[3,1],[4,1],[0,2],[7,4],[5,5],[1,6],[2,6]});`):
print(`RingRing(8,8,{[5,0],[6,0],[7,0],[2,2],[5,3],[0,4],[2,4],[3,5],[3,7],[7,7]});`):
print(``):
print(`For the puzzle of mid June, 2021 type:`):
print(`RingRing(8,8,{[7,0],[7,1],[0,2],[1,2],[4,2],[7,2],[0,3],[4,6],[3,7],[4,7]});`):

print(``):
print(`For the puzzle of June 27, 2021 type:`):
print(`RingRing(10,10,{[5,1],[6,1],[5,2],[4,3],[5,3],[4,4],[1,5],[8,5],[3,6],[4,6],[6,6],[2,7],[3,7],[9,9]});`):

print(``):
print(`For the puzzle of July 4, 2021 type:`):
print(`P:={[1,1],[6,1],[7,1],[8,1],[1,2],[8,2],[1,3],[1,4],[2,4],[3,5],[4,5],[5,5],[3,6],[6,6],[6,7],[6,8],[7,8],[8,8]};`):
print(`SOL:=RingRing(10,10,P);`):
print(`nops(SOL);`):
print(`PlotSol(10,10,P,SOL[1]);`);


print(``):
print(`For the puzzle of July 11, 2021 type:`):
print(`P:={[3,0],[8,1],[0,2],[8,2],[5,3],[7,4],[0,5],[3,5],[6,5],[7,5],[2,6],[3,6],[6,6],[2,7],[4,8],[9,9]};`):
print(`SOL:=RingRing(10,10,P);`):
print(`nops(SOL);`):
print(`PlotSol(10,10,P,SOL[1]);`);


print(``):
print(`For the puzzle of July 18, 2021 type:`):
print(`P:={[2,0],[3,0],[6,0],[9,2],[1,3],[3,4],[6,4],[0,6],[3,6],[6,6],[1,8],[4,8],[7,8],[8,8]};`):
print(`SOL:=RingRing(10,10,P);`):
print(`nops(SOL);`):
print(`PlotSol(10,10,P,SOL[1]);`);



print(``):
print(`For the puzzle of July 25, 2021 type:`):
print(`P:={[0,0],[1,0],[2,0],[4,2],[5,2],[4,4], [4,5],[6,5],[7,5],[1,6],[2,6],[7,6],[0,8], [7,8],[8,8],[0,9],[3,9],[4,9]};`):
print(`SOL:=RingRing(10,10,P);`):
print(`nops(SOL);`):
print(`PlotSol(10,10,P,SOL[1]);`);


print(``):
print(`For the puzzle of Aug. 1, 2021 type:`):
print(`P:={[0,0],[1,0],[4,0],[8,1],[1,2],[5,2],[6,2],[8,2],[5,3],[6,3],[2,4],[3,4],[6,4],[7,4], [2,5],[3,5],[0,6],[3,6],[4,6],[3,8], [5,8],[5,9]};`):
print(`SOL:=RingRing(10,10,P);`):
print(`nops(SOL);`):
print(`PlotSol(10,10,P,SOL[1]);`);


print(``):
print(`For the puzzle of Aug. 8, 2021 type:`):
print(`P:={[6,0],[1,1],[2,1],[6,1],[8,1],[8,2],[2,3],[4,3],[5,3],[8,3],[4,5],[5,5],[0,6],[4,7],[5,7],[7,7],[4,9],[5,9]};`):
print(`SOL:=RingRing(10,10,P);`):
print(`nops(SOL);`):
print(`PlotSol(10,10,P,SOL[1]);`);


print(``):
print(`For the puzzle of Aug. 15, 2021 type:`):
print(`P:={[4,2],[4,3],[5,4],[5,5],[0,6],[2,6],[3,7],[6,7],[3,9],[6,9]};`):
print(`SOL:=RingRing(10,10,P);`):
print(`nops(SOL);`):
print(`PlotSol(10,10,P,SOL[1]);`);


print(``):
print(`For the puzzle of Aug. 29, 2021 type:`):
print(`P:={[3,0],[6,0],[1,3],[3,3],[4,3],[5,3],[6,3],[3,4],[6,4],[9,4],[3,6],[6,6],[4,7],[5,7],[1,8],[4,9],[5,9],[9,9]};`):
print(`SOL:=RingRing(10,10,P);`):
print(`nops(SOL);`):
print(`PlotSol(10,10,P,SOL[1]);`);


print(``):
print(`For the puzzle of Sept. 5, 2021 type:`):
print(`P:={[0,0],[0,1],[6,1],[9,2],[3,3],[4,3],[3,4],[4,4],[9,6],[0,7],[8,7],[9,7],[0,8],[1,8],[6,8],[0,9],[1,9],[2,9],[5,9],[6,9]};`):
print(`SOL:=RingRing(10,10,P);`):
print(`nops(SOL);`):
print(`PlotSol(10,10,P,SOL[1]);`);


else
 print(`There is no such thing as`, args):

fi:


end:

with(plots):


#start pre-computed puzzles and solutions
#DB66(): A pre-computed list of 50 5 by 6 Ring Ring puzzles
DB66:=proc():
[[{[0, 0], [1, 4], [2, 2], [2, 4], [4, 1], [4, 2]}, {[[0, 1], [4, 5]], [[1, 0],
[5, 4]], [[4, 4], [2, 2]]}], [{[0, 0], [0, 1], [0, 2], [0, 5], [1, 5], [3, 4],
[4, 1], [4, 4]}, {[[0, 3], [2, 2]], [[1, 0], [2, 3]], [[2, 3], [4, 3]], [[3, 0]
, [3, 3]]}], [{[0, 0], [0, 1], [1, 4], [2, 1], [3, 5], [4, 5], [5, 0], [5, 5]},
{[[0, 2], [3, 4]], [[1, 0], [4, 4]], [[3, 1], [3, 4]]}], [{[0, 0], [0, 4], [0,
5], [2, 4], [3, 1], [3, 4], [5, 0], [5, 1], [5, 4], [5, 5]}, {[[0, 1], [3, 3]],
[[1, 0], [4, 6]], [[3, 2], [3, 2]]}], [{[0, 0], [0, 1], [0, 2], [0, 3], [2, 5],
[3, 2], [3, 5], [4, 5], [5, 4], [5, 5]}, {[[0, 4], [2, 2]], [[1, 0], [5, 4]], [
[2, 1], [3, 4]]}], [{[0, 0], [0, 1], [0, 5], [1, 5], [2, 2], [3, 0], [3, 4], [4
, 0], [4, 4], [5, 0]}, {[[0, 2], [2, 3]], [[1, 0], [2, 2]], [[2, 3], [4, 3]], [
[3, 1], [3, 2]]}], [{[0, 2], [2, 0], [2, 1], [3, 0], [4, 0], [4, 3], [5, 0], [5
, 1], [5, 2], [5, 3]}, {[[0, 0], [2, 2]], [[0, 3], [4, 3]], [[1, 2], [2, 3]], [
[3, 1], [2, 2]], [[4, 4], [2, 2]]}], [{[0, 0], [0, 5], [1, 3], [1, 5], [3, 0],
[4, 0], [4, 4], [5, 0]}, {[[0, 1], [4, 4]], [[1, 0], [2, 3]], [[2, 3], [4, 3]],
[[4, 1], [2, 2]]}], [{[0, 0], [1, 0], [1, 4], [4, 0], [5, 0], [5, 1], [5, 4], [
5, 5]}, {[[0, 1], [5, 5]], [[1, 2], [5, 2]], [[2, 0], [2, 5]]}], [{[0, 0], [0,
4], [0, 5], [2, 4], [3, 1], [5, 0], [5, 1], [5, 5]}, {[[0, 1], [3, 3]], [[1, 0]
, [4, 6]], [[3, 2], [3, 3]]}], [{[0, 0], [0, 1], [0, 2], [0, 3], [1, 0], [3, 2]
, [3, 4], [4, 4]}, {[[0, 4], [2, 2]], [[1, 1], [4, 3]], [[2, 0], [4, 6]]}], [{[
0, 4], [0, 5], [1, 5], [2, 5], [3, 1], [4, 4], [5, 0], [5, 1]}, {[[0, 0], [5, 4
]], [[1, 1], [2, 4]], [[3, 2], [3, 4]]}], [{[0, 2], [0, 3], [0, 4], [0, 5], [1,
2], [2, 4], [3, 1], [3, 4], [4, 4], [5, 0], [5, 1], [5, 2]}, {[[0, 0], [2, 2]],
[[1, 3], [5, 3]], [[2, 0], [3, 3]]}], [{[0, 0], [0, 3], [0, 4], [0, 5], [1, 0],
[4, 0], [5, 0], [5, 1], [5, 2], [5, 3], [5, 4], [5, 5]}, {[[0, 1], [5, 2]], [[1
, 3], [4, 3]], [[2, 0], [2, 5]]}], [{[0, 5], [1, 1], [1, 2], [1, 3], [1, 5], [2
, 5], [4, 3], [4, 4], [5, 0], [5, 1]}, {[[0, 0], [3, 5]], [[3, 0], [2, 2]], [[3
, 2], [3, 4]]}], [{[0, 2], [1, 2], [1, 4], [2, 0], [2, 1], [2, 2], [5, 0], [5,
1]}, {[[0, 0], [2, 2]], [[0, 3], [3, 3]], [[3, 0], [2, 2]], [[3, 2], [3, 2]], [
[3, 4], [3, 2]]}], [{[0, 2], [0, 5], [1, 2], [1, 5], [2, 0], [2, 1], [2, 2], [2
, 3], [5, 2], [5, 3], [5, 4], [5, 5]}, {[[0, 0], [2, 2]], [[0, 3], [2, 2]], [[2
, 4], [3, 2]], [[3, 0], [3, 2]], [[3, 2], [2, 2]]}], [{[0, 0], [0, 1], [1, 0],
[2, 0], [2, 3], [3, 0], [4, 3], [4, 4]}, {[[0, 2], [6, 4]], [[1, 1], [3, 4]], [
[4, 0], [2, 2]]}], [{[0, 0], [0, 4], [0, 5], [1, 0], [1, 5], [5, 0], [5, 4], [5
, 5]}, {[[0, 1], [6, 3]], [[1, 2], [3, 3]], [[2, 0], [3, 6]]}], [{[0, 4], [0, 5
], [1, 1], [4, 5], [5, 0], [5, 5]}, {[[0, 0], [5, 4]], [[1, 2], [3, 4]], [[2, 1
], [4, 4]]}], [{[0, 0], [3, 0], [3, 4], [4, 2], [4, 4], [5, 2]}, {[[0, 1], [4,
2]], [[0, 3], [6, 3]], [[1, 0], [2, 5]], [[4, 0], [2, 2]]}], [{[0, 0], [0, 1],
[0, 2], [1, 0], [1, 4], [2, 0], [3, 0], [3, 3], [4, 0], [5, 0], [5, 4], [5, 5]}
, {[[0, 3], [3, 3]], [[1, 1], [3, 2]], [[3, 4], [2, 2]], [[4, 1], [2, 3]]}], [{
[0, 2], [0, 3], [0, 4], [0, 5], [2, 0], [3, 0], [4, 0], [4, 2], [4, 3], [4, 5],
[5, 0], [5, 5]}, {[[0, 0], [2, 2]], [[1, 2], [3, 4]], [[2, 1], [4, 4]]}], [{[0,
0], [0, 3], [0, 4], [0, 5], [1, 0], [2, 0], [2, 1], [2, 2], [2, 4], [4, 1], [4,
3], [5, 3]}, {[[0, 1], [2, 2]], [[1, 3], [3, 3]], [[3, 0], [3, 3]], [[4, 4], [2
, 2]]}], [{[0, 2], [0, 5], [1, 2], [1, 5], [2, 0], [2, 1], [2, 2], [2, 5], [3,
0], [3, 3], [4, 0], [4, 3], [5, 0], [5, 3], [5, 4], [5, 5]}, {[[0, 0], [2, 2]],
[[0, 3], [3, 2]], [[3, 1], [3, 2]], [[3, 4], [2, 2]]}], [{[0, 0], [0, 5], [1, 0
], [2, 0], [3, 4], [4, 4], [5, 0], [5, 1]}, {[[0, 1], [3, 4]], [[1, 2], [5, 4]]
, [[3, 0], [2, 4]]}], [{[0, 2], [0, 3], [0, 4], [0, 5], [2, 0], [2, 4], [3, 0],
[3, 4], [5, 2], [5, 3], [5, 4], [5, 5]}, {[[0, 0], [2, 2]], [[1, 2], [4, 4]], [
[2, 1], [2, 3]], [[4, 0], [2, 2]]}], [{[1, 3], [1, 4], [2, 0], [3, 0], [4, 0],
[4, 2], [4, 3], [4, 5], [5, 0], [5, 5]}, {[[0, 0], [2, 2]], [[0, 2], [4, 4]], [
[2, 1], [4, 4]]}], [{[0, 0], [0, 1], [1, 4], [2, 4], [3, 0], [4, 0], [5, 0], [5
, 5]}, {[[0, 2], [5, 4]], [[1, 0], [2, 4]], [[3, 1], [3, 4]]}], [{[0, 0], [0, 5
], [1, 5], [2, 5], [3, 0], [3, 5], [4, 2], [5, 2]}, {[[0, 1], [4, 4]], [[1, 0],
[2, 4]], [[4, 0], [2, 2]], [[4, 3], [2, 3]]}], [{[0, 0], [0, 1], [0, 2], [0, 3]
, [2, 1], [3, 1], [3, 4], [4, 4], [5, 0], [5, 1]}, {[[0, 4], [2, 2]], [[1, 0],
[4, 4]], [[2, 2], [4, 4]]}], [{[0, 0], [0, 1], [0, 4], [0, 5], [1, 5], [4, 0],
[4, 4], [5, 0]}, {[[0, 2], [5, 2]], [[1, 0], [3, 5]], [[2, 1], [4, 5]]}], [{[0,
3], [0, 4], [0, 5], [1, 5], [3, 0], [5, 3], [5, 4], [5, 5]}, {[[0, 0], [3, 3]],
[[1, 1], [3, 4]], [[2, 3], [3, 3]], [[4, 0], [2, 3]]}], [{[0, 0], [0, 1], [0, 4
], [0, 5], [1, 0], [1, 5], [5, 0], [5, 5]}, {[[0, 2], [4, 2]], [[1, 1], [5, 4]]
, [[2, 0], [3, 6]]}], [{[0, 0], [0, 5], [1, 5], [2, 5], [3, 4], [3, 5], [5, 0],
[5, 3]}, {[[0, 1], [3, 4]], [[1, 0], [4, 4]], [[3, 1], [3, 2]], [[4, 4], [2, 2]
]}], [{[0, 4], [0, 5], [4, 0], [5, 0], [5, 1], [5, 5]}, {[[0, 0], [4, 4]], [[1,
1], [4, 5]], [[2, 2], [4, 3]]}], [{[1, 1], [2, 1], [3, 4], [4, 0], [4, 4], [5,
0]}, {[[0, 0], [4, 3]], [[0, 3], [2, 3]], [[2, 3], [4, 3]], [[4, 1], [2, 2]]}],
[{[0, 2], [0, 3], [3, 0], [4, 0], [4, 2], [4, 3], [4, 4], [5, 0]}, {[[0, 0], [3
, 2]], [[0, 4], [3, 2]], [[1, 2], [2, 2]], [[3, 1], [3, 5]]}], [{[0, 0], [0, 4]
, [0, 5], [1, 5], [4, 0], [4, 4], [5, 0], [5, 1]}, {[[0, 1], [5, 3]], [[1, 0],
[3, 5]], [[2, 2], [4, 4]]}], [{[0, 5], [1, 5], [2, 0], [2, 5], [3, 5], [4, 3],
[4, 5], [5, 5]}, {[[0, 0], [2, 3]], [[0, 3], [3, 2]], [[2, 1], [3, 2]], [[3, 0]
, [3, 5]]}], [{[0, 0], [0, 5], [1, 0], [1, 5], [2, 0], [2, 3], [2, 4], [2, 5],
[3, 5], [4, 1], [4, 5], [5, 3], [5, 4], [5, 5]}, {[[0, 1], [3, 2]], [[0, 3], [2
, 2]], [[3, 0], [3, 3]], [[3, 3], [2, 2]]}], [{[0, 0], [0, 5], [1, 0], [1, 5],
[2, 0], [2, 1], [2, 2], [2, 5], [3, 4], [3, 5], [5, 0], [5, 1]}, {[[0, 1], [2,
2]], [[0, 3], [3, 2]], [[3, 0], [2, 2]], [[3, 2], [3, 2]], [[4, 4], [2, 2]]}],
[{[0, 4], [0, 5], [1, 1], [4, 4], [5, 0], [5, 1]}, {[[0, 0], [5, 4]], [[1, 2],
[5, 4]], [[2, 1], [2, 4]]}], [{[0, 0], [0, 1], [0, 2], [1, 2], [2, 4], [2, 5],
[3, 0], [4, 0], [5, 0], [5, 1], [5, 4], [5, 5]}, {[[0, 3], [2, 3]], [[1, 0], [2
, 2]], [[2, 2], [4, 2]], [[3, 1], [2, 5]]}], [{[0, 2], [0, 3], [0, 4], [0, 5],
[3, 1], [4, 5], [5, 0], [5, 1], [5, 2], [5, 3], [5, 4], [5, 5]}, {[[0, 0], [2,
2]], [[1, 2], [3, 4]], [[2, 0], [3, 5]]}], [{[0, 4], [0, 5], [1, 1], [1, 5], [2
, 1], [2, 5], [3, 5], [4, 5], [5, 4], [5, 5]}, {[[0, 0], [6, 4]], [[1, 2], [2,
3]], [[3, 1], [2, 4]]}], [{[0, 0], [0, 5], [1, 0], [1, 5], [4, 0], [4, 5], [5,
0], [5, 1], [5, 4], [5, 5]}, {[[0, 1], [5, 4]], [[1, 2], [5, 2]], [[2, 0], [2,
6]]}], [{[0, 0], [4, 0], [5, 0], [5, 1], [5, 4], [5, 5]}, {[[0, 1], [5, 5]], [[
1, 0], [3, 5]], [[2, 2], [4, 2]]}], [{[0, 3], [0, 4], [0, 5], [1, 5], [2, 0], [
2, 5], [3, 0], [3, 3], [4, 0], [5, 0]}, {[[0, 0], [2, 3]], [[1, 3], [2, 2]], [[
2, 1], [2, 2]], [[3, 4], [3, 2]], [[4, 1], [2, 3]]}], [{[0, 2], [0, 3], [2, 0],
[2, 5], [3, 0], [3, 5], [4, 0], [4, 2], [4, 3], [4, 5], [5, 0], [5, 5]}, {[[0,
0], [2, 2]], [[0, 4], [2, 2]], [[1, 2], [3, 2]], [[2, 1], [4, 4]]}]]:
end:


#DB77(): A list of 50 7 by 7 Ring Ring puzzles (and their solutions)
DB77:=proc():
[[{[0, 0], [0, 1], [0, 2], [0, 3], [2, 1], [4, 3], [5, 1], [5, 3], [5, 6], [6,
3], [6, 6]}, {[[0, 4], [5, 3]], [[1, 0], [3, 3]], [[1, 3], [3, 3]], [[4, 0], [3
, 3]], [[5, 4], [2, 2]]}], [{[0, 0], [0, 3], [1, 0], [1, 3], [2, 0], [2, 5], [2
, 6], [4, 5], [5, 0], [5, 5], [6, 0], [6, 1], [6, 2]}, {[[0, 1], [2, 2]], [[0,
4], [2, 3]], [[2, 1], [4, 4]], [[3, 0], [2, 3]], [[3, 3], [4, 4]]}], [{[0, 0],
[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [0, 6], [2, 1], [2, 5], [3, 1], [3, 5],
[5, 0], [5, 4], [6, 0], [6, 4]}, {[[1, 0], [4, 7]], [[2, 2], [2, 3]], [[5, 1],
[2, 3]], [[5, 5], [2, 2]]}], [{[0, 5], [0, 6], [1, 1], [1, 6], [4, 0], [5, 6],
[6, 2], [6, 3], [6, 4], [6, 5], [6, 6]}, {[[0, 0], [4, 5]], [[1, 2], [5, 4]], [
[2, 1], [3, 6]], [[5, 0], [2, 2]]}], [{[0, 0], [0, 3], [0, 6], [1, 0], [1, 3],
[1, 6], [2, 3], [2, 6], [3, 5], [3, 6], [6, 2], [6, 5], [6, 6]}, {[[0, 1], [2,
2]], [[0, 4], [3, 2]], [[2, 0], [2, 3]], [[3, 3], [4, 2]], [[4, 0], [3, 2]], [[
4, 2], [2, 5]]}], [{[0, 0], [0, 5], [0, 6], [1, 0], [1, 6], [2, 6], [3, 6], [5,
3], [6, 6]}, {[[0, 1], [4, 4]], [[1, 2], [6, 4]], [[2, 0], [3, 4]], [[4, 4], [2
, 3]], [[5, 0], [2, 2]]}], [{[0, 5], [0, 6], [1, 1], [1, 2], [2, 1], [3, 1], [6
, 2]}, {[[0, 0], [5, 5]], [[1, 3], [6, 4]], [[2, 2], [4, 4]], [[5, 0], [2, 2]]}
], [{[0, 0], [1, 0], [3, 3], [5, 0], [6, 0], [6, 1], [6, 6]}, {[[0, 1], [6, 6]]
, [[1, 2], [6, 4]], [[2, 0], [3, 5]]}], [{[0, 6], [1, 1], [1, 4], [1, 6], [3, 0
], [4, 0], [5, 0], [6, 0], [6, 4], [6, 5], [6, 6]}, {[[0, 0], [3, 3]], [[0, 3],
[5, 3]], [[2, 4], [4, 3]], [[3, 1], [2, 2]], [[5, 1], [2, 3]]}], [{[0, 2], [0,
3], [0, 4], [2, 6], [5, 0], [5, 5], [6, 0]}, {[[0, 0], [2, 2]], [[0, 5], [2, 2]
], [[1, 2], [5, 3]], [[2, 0], [3, 6]], [[3, 1], [4, 6]]}], [{[0, 0], [0, 1], [1
, 5], [2, 5], [4, 2], [4, 5], [4, 6], [6, 0], [6, 4]}, {[[0, 2], [4, 5]], [[1,
0], [5, 5]], [[2, 1], [5, 3]], [[5, 5], [2, 2]]}], [{[0, 3], [0, 6], [1, 1], [1
, 3], [1, 6], [2, 1], [2, 5], [2, 6], [4, 5], [5, 0], [6, 0], [6, 1], [6, 2], [
6, 5], [6, 6]}, {[[0, 0], [5, 3]], [[0, 4], [2, 2]], [[2, 3], [5, 2]], [[3, 1],
[3, 6]]}], [{[0, 0], [0, 1], [0, 2], [0, 3], [0, 6], [1, 6], [6, 0], [6, 1], [6
, 6]}, {[[0, 4], [2, 2]], [[1, 0], [3, 2]], [[1, 2], [2, 2]], [[2, 4], [4, 3]],
[[3, 2], [4, 4]], [[4, 0], [2, 4]]}], [{[0, 5], [0, 6], [1, 6], [2, 2], [2, 3],
[2, 6], [5, 0], [6, 0], [6, 1]}, {[[0, 0], [5, 5]], [[1, 1], [5, 5]], [[3, 2],
[4, 5]]}], [{[0, 0], [0, 4], [1, 0], [2, 0], [3, 0], [3, 5], [3, 6], [5, 5], [6
, 0], [6, 1], [6, 2]}, {[[0, 1], [4, 3]], [[0, 5], [3, 2]], [[1, 2], [5, 3]], [
[4, 0], [2, 2]], [[4, 3], [3, 4]]}], [{[0, 6], [1, 3], [1, 4], [1, 6], [2, 0],
[5, 6], [6, 0], [6, 1], [6, 6]}, {[[0, 0], [2, 2]], [[0, 2], [7, 4]], [[2, 1],
[3, 6]], [[3, 0], [3, 5]]}], [{[0, 0], [0, 1], [0, 2], [0, 5], [0, 6], [1, 0],
[1, 6], [2, 0], [3, 0], [4, 0], [4, 1], [5, 2], [6, 2]}, {[[0, 3], [7, 2]], [[1
, 1], [3, 5]], [[2, 2], [3, 5]], [[5, 0], [2, 2]], [[5, 5], [2, 2]]}], [{[0, 0]
, [0, 3], [0, 6], [1, 0], [1, 3], [1, 6], [2, 0], [2, 6], [4, 4], [6, 0], [6, 1
]}, {[[0, 1], [3, 2]], [[0, 4], [2, 2]], [[2, 3], [4, 3]], [[3, 0], [3, 2]], [[
3, 2], [4, 5]]}], [{[0, 0], [0, 3], [0, 4], [0, 5], [0, 6], [1, 0], [1, 3], [1,
6], [2, 0], [2, 6], [3, 6], [6, 0], [6, 4], [6, 5], [6, 6]}, {[[0, 1], [2, 2]],
[[1, 4], [3, 2]], [[2, 1], [5, 3]], [[3, 0], [3, 3]], [[4, 4], [2, 3]]}], [{[0,
0], [0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [0, 6], [1, 2], [1, 6], [4, 1], [5,
5], [6, 0], [6, 1]}, {[[1, 0], [2, 2]], [[1, 3], [4, 3]], [[2, 2], [5, 5]], [[3
, 0], [3, 5]]}], [{[0, 2], [1, 5], [2, 5], [3, 0], [3, 1], [3, 2], [4, 5], [4,
6], [5, 1]}, {[[0, 0], [3, 2]], [[0, 3], [4, 4]], [[1, 2], [2, 3]], [[4, 0], [3
, 3]], [[4, 3], [3, 2]], [[5, 5], [2, 2]]}], [{[0, 0], [0, 3], [0, 4], [1, 0],
[2, 6], [3, 6], [4, 1], [4, 6], [5, 6], [6, 0], [6, 1], [6, 5], [6, 6]}, {[[0,
1], [4, 2]], [[0, 5], [2, 2]], [[1, 3], [3, 2]], [[2, 0], [4, 6]], [[4, 2], [3,
3]]}], [{[0, 0], [2, 4], [3, 0], [4, 0], [4, 1], [4, 2], [6, 6]}, {[[0, 1], [4,
6]], [[1, 0], [2, 3]], [[1, 3], [6, 3]], [[4, 4], [2, 3]], [[5, 0], [2, 3]]}],
[{[0, 3], [3, 1], [3, 4], [3, 6], [4, 1], [5, 1], [6, 6]}, {[[0, 0], [2, 3]], [
[0, 4], [3, 3]], [[1, 3], [6, 3]], [[2, 0], [5, 3]], [[4, 4], [2, 3]]}], [{[0,
4], [0, 5], [0, 6], [1, 1], [3, 4], [4, 0], [4, 3], [4, 4], [5, 0], [6, 0], [6,
6]}, {[[0, 0], [4, 4]], [[1, 2], [5, 5]], [[2, 1], [5, 5]]}], [{[0, 0], [0, 1],
[0, 6], [1, 0], [1, 6], [2, 0], [3, 0], [4, 0], [4, 3], [4, 6], [5, 0], [5, 6],
[6, 0], [6, 5], [6, 6]}, {[[0, 2], [6, 4]], [[1, 1], [6, 4]], [[2, 3], [2, 4]]}
], [{[0, 3], [1, 3], [1, 5], [2, 0], [2, 5], [3, 5], [5, 4], [6, 0], [6, 4]}, {
[[0, 0], [2, 3]], [[0, 4], [5, 3]], [[2, 1], [5, 3]], [[3, 0], [3, 3]], [[5, 5]
, [2, 2]]}], [{[0, 0], [0, 4], [1, 0], [1, 2], [1, 4], [2, 0], [2, 5], [2, 6],
[3, 0], [4, 0], [5, 0], [5, 4], [6, 0], [6, 1], [6, 2], [6, 3], [6, 4], [6, 5],
[6, 6]}, {[[0, 1], [6, 3]], [[0, 5], [2, 2]], [[2, 2], [3, 3]], [[3, 5], [3, 2]
]}], [{[0, 0], [0, 4], [0, 5], [0, 6], [1, 0], [1, 2], [1, 6], [2, 6], [3, 6],
[4, 0], [5, 0], [5, 3], [5, 5], [6, 0], [6, 3]}, {[[0, 1], [5, 3]], [[1, 4], [3
, 2]], [[2, 0], [2, 3]], [[4, 4], [3, 3]], [[5, 1], [2, 2]]}], [{[0, 0], [0, 1]
, [0, 5], [0, 6], [1, 6], [5, 6], [6, 0], [6, 1], [6, 2], [6, 3], [6, 4], [6, 5
], [6, 6]}, {[[0, 2], [4, 3]], [[1, 0], [5, 6]], [[2, 1], [3, 6]]}], [{[0, 0],
[1, 0], [1, 5], [2, 0], [2, 3], [2, 5], [3, 0], [4, 0], [5, 6], [6, 2], [6, 6]}
, {[[0, 1], [5, 6]], [[1, 2], [5, 3]], [[3, 3], [4, 3]], [[5, 0], [2, 2]]}], [{
[0, 0], [0, 1], [0, 2], [0, 6], [1, 0], [1, 6], [2, 6], [3, 3], [3, 5], [3, 6],
[4, 0], [4, 6], [5, 0], [5, 6], [6, 0], [6, 5], [6, 6]}, {[[0, 3], [3, 3]], [[1
, 1], [6, 4]], [[2, 0], [2, 3]], [[4, 2], [2, 4]]}], [{[0, 5], [0, 6], [1, 3],
[2, 0], [2, 1], [3, 3], [3, 6], [4, 6], [5, 6], [6, 3], [6, 4], [6, 5], [6, 6]}
, {[[0, 0], [2, 2]], [[0, 2], [3, 3]], [[1, 5], [2, 2]], [[3, 0], [4, 3]], [[3,
4], [3, 2]], [[4, 1], [2, 3]]}], [{[0, 3], [0, 4], [2, 6], [3, 0], [4, 0], [6,
2], [6, 6]}, {[[0, 0], [3, 3]], [[0, 5], [2, 2]], [[1, 1], [4, 4]], [[2, 3], [5
, 3]], [[3, 2], [3, 5]], [[5, 0], [2, 2]]}], [{[0, 2], [0, 5], [0, 6], [1, 6],
[2, 0], [3, 0], [4, 6], [5, 2], [6, 0], [6, 1], [6, 2]}, {[[0, 0], [2, 2]], [[0
, 3], [7, 2]], [[1, 2], [4, 4]], [[2, 1], [2, 6]], [[4, 0], [2, 2]], [[5, 5], [
2, 2]]}], [{[0, 0], [0, 1], [0, 6], [1, 0], [1, 6], [4, 1], [4, 3], [4, 4], [5,
1]}, {[[0, 2], [6, 4]], [[1, 1], [3, 4]], [[2, 0], [5, 7]]}], [{[0, 3], [1, 3],
[1, 5], [2, 0], [2, 1], [2, 2], [3, 2], [4, 2], [4, 6], [5, 2], [5, 6], [6, 0],
[6, 1], [6, 2], [6, 6]}, {[[0, 0], [2, 3]], [[0, 4], [4, 3]], [[2, 3], [3, 3]],
[[3, 0], [3, 2]], [[5, 3], [2, 3]]}], [{[0, 6], [1, 1], [1, 2], [4, 0], [4, 5],
[5, 2], [5, 4], [5, 5], [6, 2]}, {[[0, 0], [4, 6]], [[1, 3], [6, 4]], [[2, 1],
[3, 4]], [[5, 0], [2, 2]]}], [{[0, 3], [0, 4], [2, 6], [3, 6], [4, 0], [4, 5],
[4, 6], [5, 0], [6, 0], [6, 1], [6, 4]}, {[[0, 0], [4, 3]], [[0, 5], [2, 2]], [
[1, 1], [5, 4]], [[2, 3], [2, 3]], [[4, 2], [3, 2]], [[5, 5], [2, 2]]}], [{[0,
0], [0, 6], [3, 3], [5, 1], [5, 5]}, {[[0, 1], [5, 5]], [[1, 0], [6, 7]], [[2,
2], [4, 3]]}], [{[0, 5], [0, 6], [3, 1], [4, 1], [4, 6], [5, 1], [5, 2], [5, 3]
, [5, 4], [5, 6], [6, 6]}, {[[0, 0], [2, 2]], [[0, 2], [5, 3]], [[1, 3], [3, 4]
], [[2, 0], [5, 6]]}], [{[0, 5], [0, 6], [1, 1], [2, 5], [3, 5], [4, 0], [5, 0]
, [6, 0], [6, 6]}, {[[0, 0], [4, 5]], [[1, 2], [5, 5]], [[2, 1], [5, 3]], [[4,
4], [3, 2]]}], [{[0, 3], [2, 6], [3, 1], [4, 1], [5, 1]}, {[[0, 0], [7, 3]], [[
0, 4], [2, 3]], [[1, 1], [2, 3]], [[2, 4], [4, 2]], [[3, 3], [4, 4]]}], [{[0, 0
], [0, 4], [0, 5], [0, 6], [3, 0], [3, 4], [4, 0], [4, 6], [5, 0], [5, 2], [5,
6], [6, 0], [6, 4], [6, 5], [6, 6]}, {[[0, 1], [4, 3]], [[1, 0], [2, 5]], [[1,
5], [3, 2]], [[4, 1], [3, 3]], [[4, 4], [2, 2]]}], [{[0, 0], [0, 3], [0, 4], [0
, 5], [0, 6], [1, 0], [2, 5], [3, 1], [3, 2], [3, 5], [4, 5], [5, 0], [5, 4], [
5, 5], [6, 0]}, {[[0, 1], [2, 2]], [[1, 3], [6, 4]], [[2, 0], [3, 5]], [[5, 1],
[2, 2]]}], [{[0, 0], [0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [0, 6], [1, 0], [2
, 5], [3, 5], [5, 1], [6, 3], [6, 4]}, {[[1, 1], [4, 6]], [[2, 0], [5, 3]], [[2
, 3], [4, 2]], [[5, 5], [2, 2]]}], [{[0, 0], [1, 0], [3, 3], [3, 6], [4, 0], [4
, 4], [5, 0], [5, 4], [6, 0], [6, 1], [6, 2], [6, 3], [6, 4], [6, 5], [6, 6]},
{[[0, 1], [2, 2]], [[0, 3], [3, 4]], [[1, 4], [3, 2]], [[2, 0], [2, 3]], [[4, 1
], [2, 3]], [[4, 5], [2, 2]]}], [{[0, 0], [0, 1], [0, 5], [0, 6], [1, 0], [1, 6
], [2, 0], [2, 6], [3, 0], [3, 6], [4, 0], [4, 1], [4, 4], [4, 5], [4, 6], [5,
6], [6, 2], [6, 3], [6, 6]}, {[[0, 2], [3, 3]], [[1, 1], [3, 5]], [[4, 2], [2,
2]], [[5, 0], [2, 2]], [[5, 4], [2, 2]]}], [{[1, 3], [2, 6], [5, 0], [6, 0], [6
, 1], [6, 5], [6, 6]}, {[[0, 0], [2, 2]], [[0, 2], [7, 3]], [[0, 5], [2, 2]], [
[2, 0], [3, 6]], [[3, 1], [3, 6]]}], [{[0, 0], [0, 3], [0, 4], [1, 0], [2, 0],
[4, 5], [5, 3], [5, 4], [5, 5], [6, 0], [6, 1]}, {[[0, 1], [3, 2]], [[0, 5], [3
, 2]], [[1, 3], [4, 2]], [[3, 0], [3, 2]], [[3, 2], [4, 5]]}]]:

end:




DB88:=proc():
[[{[0, 2], [0, 3], [0, 4], [0, 5], [0, 6], [0, 7], [1, 7], [3, 1], [5, 5], [5,
6], [5, 7], [7, 0], [7, 1], [7, 4]}, {[[0, 0], [2, 2]], [[1, 2], [7, 2]], [[1,
4], [3, 3]], [[2, 0], [3, 8]], [[5, 0], [2, 5]], [[6, 5], [2, 3]]}], [{[0, 0],
[0, 1], [0, 2], [0, 3], [0, 4], [0, 7], [2, 1], [3, 6], [5, 0], [6, 3], [6, 5],
[7, 5]}, {[[0, 5], [3, 2]], [[1, 0], [4, 8]], [[2, 2], [6, 3]], [[3, 1], [3, 5]
], [[5, 6], [3, 2]], [[6, 0], [2, 2]]}], [{[1, 1], [1, 2], [1, 6], [4, 0], [4,
2], [4, 4], [4, 7], [5, 0], [5, 7], [7, 3], [7, 4], [7, 5]}, {[[0, 0], [4, 8]],
[[1, 3], [6, 3]], [[2, 1], [4, 6]], [[6, 0], [2, 3]], [[6, 6], [2, 2]]}], [{[0,
0], [0, 1], [0, 4], [0, 5], [0, 6], [0, 7], [1, 7], [2, 1], [3, 1], [4, 4], [4,
5], [4, 7], [5, 7], [6, 0], [6, 2], [6, 4], [6, 7], [7, 0], [7, 4], [7, 7]}, {[
[0, 2], [4, 2]], [[1, 0], [5, 7]], [[2, 4], [2, 4]], [[4, 1], [4, 3]], [[6, 5],
[2, 2]]}], [{[0, 7], [1, 3], [2, 0], [2, 1], [4, 1], [6, 3], [7, 6], [7, 7]}, {
[[0, 0], [2, 2]], [[0, 2], [3, 5]], [[1, 4], [6, 4]], [[3, 0], [3, 7]], [[4, 2]
, [4, 4]], [[6, 0], [2, 2]]}], [{[0, 0], [0, 7], [1, 3], [1, 4], [1, 5], [1, 7]
, [3, 0], [4, 0], [5, 0], [5, 7], [6, 3], [6, 4]}, {[[0, 1], [6, 6]], [[1, 0],
[2, 3]], [[2, 3], [3, 5]], [[3, 2], [5, 4]], [[6, 0], [2, 2]], [[6, 6], [2, 2]]
}], [{[1, 5], [1, 6], [2, 0], [2, 6], [3, 0], [4, 4], [4, 6], [4, 7], [5, 2], [
6, 6]}, {[[0, 0], [2, 2]], [[0, 2], [4, 2]], [[0, 4], [4, 4]], [[2, 1], [5, 5]]
, [[4, 0], [4, 4]], [[5, 4], [3, 4]]}], [{[0, 3], [0, 4], [0, 7], [1, 3], [2, 0
], [2, 3], [3, 0], [3, 3], [4, 0], [4, 6], [5, 0], [6, 0], [7, 0], [7, 1], [7,
4], [7, 5]}, {[[0, 0], [2, 3]], [[0, 5], [4, 2]], [[1, 4], [5, 4]], [[2, 1], [2
, 2]], [[4, 1], [3, 5]], [[5, 2], [3, 2]], [[6, 6], [2, 2]]}], [{[0, 0], [0, 3]
, [0, 7], [3, 4], [4, 1], [4, 2], [4, 4], [6, 5], [7, 4], [7, 5]}, {[[0, 1], [4
, 2]], [[0, 4], [3, 3]], [[1, 0], [7, 4]], [[1, 5], [3, 3]], [[4, 5], [2, 3]],
[[5, 1], [2, 4]], [[6, 6], [2, 2]]}], [{[0, 2], [0, 3], [0, 4], [3, 0], [5, 2],
[5, 5], [7, 0], [7, 7]}, {[[0, 0], [3, 2]], [[0, 5], [4, 3]], [[1, 2], [2, 2]],
[[1, 4], [7, 3]], [[3, 1], [5, 3]], [[4, 0], [3, 8]]}], [{[1, 3], [1, 4], [1, 5
], [1, 6], [2, 0], [2, 1], [2, 6], [3, 0], [4, 0], [5, 7], [6, 3], [6, 4], [6,
5], [6, 7], [7, 0], [7, 7]}, {[[0, 0], [2, 2]], [[0, 2], [5, 6]], [[2, 3], [4,
3]], [[3, 1], [5, 6]], [[5, 0], [2, 3]]}], [{[0, 0], [0, 4], [0, 5], [0, 6], [0
, 7], [1, 0], [1, 7], [2, 0], [2, 4], [3, 0], [3, 4], [4, 5], [4, 7], [5, 1], [
5, 7], [7, 0], [7, 1], [7, 2], [7, 3], [7, 4]}, {[[0, 1], [4, 3]], [[1, 2], [5,
5]], [[2, 5], [2, 3]], [[4, 0], [3, 5]], [[6, 5], [2, 3]]}], [{[0, 0], [1, 4],
[1, 5], [1, 6], [2, 6], [3, 0], [4, 0], [4, 7], [5, 3], [5, 7], [6, 7], [7, 3],
[7, 4], [7, 5], [7, 6], [7, 7]}, {[[0, 1], [4, 7]], [[1, 0], [2, 4]], [[2, 4],
[4, 2]], [[4, 1], [3, 6]], [[5, 0], [3, 3]]}], [{[0, 0], [0, 1], [0, 7], [1, 0]
, [2, 0], [2, 3], [3, 5], [3, 6], [5, 0], [5, 3], [6, 0], [7, 0], [7, 1], [7, 5
], [7, 6], [7, 7]}, {[[0, 2], [8, 3]], [[0, 5], [3, 2]], [[1, 1], [6, 7]], [[3,
0], [2, 4]], [[4, 5], [2, 2]]}], [{[0, 0], [0, 3], [0, 4], [0, 5], [0, 6], [0,
7], [3, 4], [3, 5], [4, 0], [4, 7], [5, 7], [6, 4], [6, 7], [7, 3], [7, 4], [7,
7]}, {[[0, 1], [3, 2]], [[1, 0], [3, 4]], [[1, 4], [2, 2]], [[1, 6], [3, 2]], [
[4, 1], [3, 3]], [[4, 4], [2, 3]], [[5, 0], [3, 3]], [[6, 5], [2, 2]]}], [{[0,
0], [0, 3], [0, 4], [0, 5], [0, 6], [0, 7], [1, 0], [1, 3], [1, 7], [2, 0], [3,
0], [3, 1], [5, 6], [6, 2], [6, 4], [6, 5], [6, 6], [7, 0], [7, 1], [7, 2]}, {[
[0, 1], [3, 2]], [[1, 4], [4, 3]], [[2, 3], [6, 5]], [[3, 2], [3, 4]], [[4, 0],
[3, 2]]}], [{[0, 0], [0, 6], [0, 7], [1, 0], [1, 2], [1, 3], [1, 4], [4, 3], [4
, 6], [5, 6], [6, 1], [7, 5], [7, 6], [7, 7]}, {[[0, 1], [6, 5]], [[1, 6], [2,
2]], [[2, 0], [6, 5]], [[3, 2], [4, 6]]}], [{[0, 0], [0, 6], [0, 7], [1, 0], [1
, 2], [1, 3], [1, 4], [3, 7], [4, 7], [5, 1], [5, 3], [7, 0], [7, 1], [7, 7]},
{[[0, 1], [5, 5]], [[1, 6], [2, 2]], [[2, 0], [5, 5]], [[3, 2], [5, 5]], [[5, 5
], [2, 3]]}], [{[0, 7], [1, 7], [2, 0], [5, 0], [6, 0], [6, 2], [6, 6], [7, 0]}
, {[[0, 0], [2, 2]], [[0, 2], [6, 5]], [[1, 3], [6, 3]], [[2, 1], [6, 7]], [[3,
0], [2, 5]]}], [{[0, 0], [0, 1], [0, 2], [0, 3], [0, 7], [1, 0], [1, 3], [2, 0]
, [2, 3], [3, 0], [3, 1], [4, 6], [6, 4], [7, 4]}, {[[0, 4], [4, 3]], [[1, 1],
[2, 2]], [[1, 5], [5, 3]], [[3, 2], [5, 2]], [[4, 0], [2, 5]], [[6, 0], [2, 2]]
, [[6, 5], [2, 3]]}], [{[0, 0], [0, 4], [0, 5], [1, 0], [3, 7], [4, 7], [5, 4],
[5, 7], [6, 0], [6, 7], [7, 0], [7, 1], [7, 6], [7, 7]}, {[[0, 1], [2, 3]], [[0
, 6], [3, 2]], [[1, 4], [2, 2]], [[2, 0], [4, 4]], [[3, 1], [4, 6]], [[4, 2], [
4, 4]]}], [{[0, 7], [2, 4], [2, 5], [3, 0], [4, 0], [5, 0], [5, 1], [6, 0], [6,
4], [6, 5], [6, 6], [7, 0]}, {[[0, 0], [3, 2]], [[0, 2], [6, 5]], [[1, 3], [7,
5]], [[3, 1], [2, 5]], [[6, 1], [2, 2]]}], [{[1, 1], [1, 2], [1, 3], [2, 5], [4
, 7], [5, 4], [5, 5], [5, 6], [5, 7], [6, 1], [6, 3], [6, 7], [7, 3], [7, 7]},
{[[0, 0], [4, 8]], [[1, 4], [4, 3]], [[2, 1], [4, 3]], [[4, 0], [4, 3]], [[6, 4
], [2, 3]]}], [{[0, 0], [0, 7], [1, 0], [4, 2], [4, 7], [5, 5], [6, 0], [6, 5],
[7, 0], [7, 1], [7, 2], [7, 3], [7, 4], [7, 5]}, {[[0, 1], [7, 4]], [[0, 5], [5
, 2]], [[1, 2], [3, 6]], [[2, 0], [4, 4]], [[5, 6], [3, 2]]}], [{[0, 6], [0, 7]
, [1, 1], [3, 0], [3, 7], [4, 5], [4, 7], [5, 7], [7, 0], [7, 1]}, {[[0, 0], [3
, 6]], [[1, 2], [7, 3]], [[1, 6], [2, 2]], [[3, 1], [3, 6]], [[4, 0], [3, 4]],
[[6, 5], [2, 3]]}], [{[1, 3], [2, 3], [2, 5], [2, 6], [2, 7], [3, 7], [4, 1], [
5, 4], [6, 4], [6, 5], [6, 7], [7, 7]}, {[[0, 0], [3, 2]], [[0, 2], [5, 3]], [[
0, 5], [2, 3]], [[3, 0], [5, 7]], [[4, 5], [2, 3]], [[5, 1], [2, 3]]}], [{[0, 0
], [0, 1], [0, 7], [1, 3], [1, 4], [1, 5], [1, 7], [3, 0], [3, 5], [4, 0], [4,
5], [5, 4], [5, 5], [7, 2]}, {[[0, 2], [7, 5]], [[1, 0], [2, 2]], [[2, 3], [6,
5]], [[3, 1], [2, 4]], [[5, 0], [3, 2]]}], [{[0, 6], [0, 7], [1, 1], [1, 4], [3
, 0], [4, 0], [4, 5], [4, 6], [5, 0], [7, 2], [7, 3], [7, 4]}, {[[0, 0], [3, 3]
], [[0, 3], [3, 3]], [[1, 6], [2, 2]], [[3, 1], [3, 7]], [[4, 2], [3, 3]], [[6,
0], [2, 2]], [[6, 5], [2, 3]]}], [{[0, 0], [0, 1], [0, 2], [0, 7], [2, 1], [2,
5], [5, 0], [5, 7], [6, 0], [6, 4], [6, 5], [6, 7], [7, 0], [7, 7]}, {[[0, 3],
[8, 4]], [[1, 0], [4, 3]], [[1, 4], [4, 4]], [[3, 1], [3, 5]], [[6, 1], [2, 2]]
}], [{[0, 0], [0, 1], [1, 0], [1, 5], [1, 6], [2, 6], [4, 2], [4, 7], [5, 7], [
6, 7], [7, 0], [7, 1], [7, 2], [7, 7]}, {[[0, 2], [4, 6]], [[1, 1], [5, 4]], [[
2, 0], [5, 6]], [[4, 3], [4, 4]]}], [{[0, 7], [2, 2], [3, 2], [4, 2], [6, 0], [
6, 2], [6, 6], [7, 0]}, {[[0, 0], [6, 7]], [[1, 1], [7, 7]], [[2, 3], [2, 3]],
[[4, 3], [3, 3]]}], [{[1, 1], [2, 1], [5, 1], [5, 6], [6, 1], [6, 2], [7, 6], [
7, 7]}, {[[0, 0], [8, 6]], [[0, 6], [2, 2]], [[1, 2], [5, 3]], [[2, 3], [5, 5]]
, [[3, 1], [2, 6]]}], [{[0, 0], [0, 1], [0, 4], [0, 5], [0, 6], [0, 7], [2, 1],
[3, 1], [3, 2], [3, 3], [4, 3], [6, 0], [6, 3], [6, 5], [6, 7], [7, 0], [7, 3],
[7, 7]}, {[[0, 2], [3, 2]], [[1, 0], [5, 8]], [[2, 4], [2, 3]], [[4, 1], [4, 2]
], [[4, 4], [4, 3]]}], [{[0, 0], [0, 4], [0, 7], [1, 4], [1, 7], [2, 4], [2, 5]
, [2, 6], [2, 7], [3, 0], [3, 7], [4, 0], [4, 7], [5, 0], [6, 0], [6, 2], [6, 4
], [6, 6], [7, 0], [7, 4]}, {[[0, 1], [8, 3]], [[0, 5], [2, 2]], [[1, 0], [2, 3
]], [[3, 2], [3, 3]], [[3, 5], [2, 2]], [[5, 5], [3, 3]]}], [{[0, 7], [1, 1], [
1, 2], [1, 3], [1, 7], [2, 1], [2, 5], [3, 1], [3, 5], [4, 7], [6, 1], [6, 2],
[6, 3], [6, 4]}, {[[0, 0], [5, 5]], [[0, 5], [2, 2]], [[2, 2], [2, 2]], [[2, 6]
, [2, 2]], [[4, 5], [3, 2]], [[5, 0], [3, 8]]}], [{[0, 0], [0, 4], [1, 0], [1,
6], [5, 2], [7, 0]}, {[[0, 1], [8, 3]], [[0, 5], [5, 3]], [[1, 2], [4, 3]], [[2
, 0], [5, 7]], [[5, 4], [3, 4]]}], [{[0, 4], [0, 5], [0, 6], [0, 7], [2, 2], [5
, 4], [5, 6], [6, 0], [7, 0], [7, 1], [7, 6], [7, 7]}, {[[0, 0], [6, 4]], [[1,
1], [6, 7]], [[2, 4], [3, 3]], [[3, 2], [5, 4]]}], [{[0, 0], [0, 1], [0, 2], [0
, 3], [0, 6], [0, 7], [1, 2], [4, 0], [4, 1], [6, 2], [6, 4], [6, 5], [6, 6], [
7, 2]}, {[[0, 4], [5, 2]], [[1, 0], [3, 2]], [[1, 3], [7, 5]], [[2, 2], [4, 5]]
, [[5, 0], [3, 2]]}], [{[0, 2], [0, 3], [0, 4], [0, 5], [3, 0], [3, 1], [3, 4],
[3, 6], [4, 0], [4, 6], [5, 0], [5, 6], [7, 6], [7, 7]}, {[[0, 0], [3, 2]], [[0
, 6], [2, 2]], [[1, 2], [7, 4]], [[2, 3], [5, 5]], [[4, 1], [2, 4]], [[6, 0], [
2, 2]]}], [{[0, 4], [1, 6], [2, 6], [3, 0], [4, 3], [4, 4], [5, 6], [6, 3], [7,
0], [7, 5], [7, 6], [7, 7]}, {[[0, 0], [3, 4]], [[0, 5], [4, 3]], [[1, 1], [3,
4]], [[4, 0], [3, 3]], [[4, 5], [3, 3]], [[5, 1], [3, 4]]}], [{[0, 0], [0, 1],
[0, 7], [1, 0], [1, 7], [2, 0], [3, 6], [4, 6], [5, 1], [6, 7], [7, 0], [7, 1],
[7, 4], [7, 7]}, {[[0, 2], [3, 3]], [[0, 5], [2, 2]], [[1, 1], [4, 3]], [[2, 5]
, [4, 3]], [[3, 0], [4, 5]], [[5, 2], [3, 2]], [[6, 5], [2, 2]]}], [{[0, 3], [0
, 6], [0, 7], [1, 1], [2, 6], [3, 0], [3, 1], [3, 5], [3, 6], [4, 0], [4, 6], [
5, 0], [5, 6], [6, 0], [6, 2], [7, 0], [7, 6], [7, 7]}, {[[0, 0], [3, 3]], [[0,
4], [3, 2]], [[1, 3], [6, 5]], [[3, 2], [3, 3]], [[4, 1], [4, 5]]}], [{[0, 0],
[0, 7], [1, 0], [1, 2], [1, 5], [1, 7], [3, 2], [3, 5], [6, 3], [6, 4]}, {[[0,
1], [5, 6]], [[1, 3], [5, 2]], [[2, 0], [6, 8]], [[5, 1], [2, 2]], [[5, 5], [2,
2]]}], [{[0, 0], [0, 1], [0, 2], [0, 3], [0, 7], [1, 7], [2, 7], [3, 7], [4, 0]
, [5, 0], [6, 0], [6, 2], [6, 3], [6, 5], [7, 0], [7, 5], [7, 6], [7, 7]}, {[[0
, 4], [2, 3]], [[1, 0], [3, 4]], [[2, 1], [6, 4]], [[2, 5], [2, 2]], [[4, 2], [
2, 4]], [[4, 6], [3, 2]]}], [{[0, 2], [0, 7], [1, 5], [1, 7], [2, 0], [2, 7], [
3, 0], [3, 7], [4, 0], [4, 7], [6, 1], [7, 7]}, {[[0, 0], [2, 2]], [[0, 3], [8,
4]], [[1, 2], [3, 3]], [[2, 1], [3, 5]], [[5, 0], [3, 3]], [[5, 4], [2, 4]]}],
[{[0, 7], [1, 3], [1, 4], [1, 5], [1, 7], [2, 0], [2, 1], [2, 7], [3, 0], [3, 3
], [3, 7], [4, 7], [5, 6], [5, 7], [7, 0], [7, 1], [7, 2], [7, 3], [7, 4], [7,
5]}, {[[0, 0], [2, 2]], [[0, 2], [3, 5]], [[3, 1], [3, 2]], [[3, 4], [2, 3]], [
[4, 0], [3, 4]], [[5, 4], [2, 2]], [[6, 6], [2, 2]]}], [{[0, 0], [0, 3], [0, 7]
, [1, 0], [1, 3], [1, 7], [2, 3], [2, 4], [2, 5], [3, 1], [3, 3], [5, 4], [6, 4
], [6, 6], [7, 0], [7, 4]}, {[[0, 1], [2, 2]], [[0, 4], [2, 3]], [[2, 0], [5, 3
]], [[2, 6], [3, 2]], [[3, 4], [2, 2]], [[4, 1], [4, 3]], [[5, 5], [3, 3]]}], [
{[0, 0], [0, 4], [0, 5], [0, 6], [0, 7], [4, 1], [5, 3], [6, 3], [6, 7], [7, 7]
}, {[[0, 1], [4, 3]], [[1, 0], [7, 5]], [[1, 5], [5, 3]], [[2, 2], [3, 5]], [[5
, 1], [2, 2]], [[6, 5], [2, 2]]}], [{[1, 3], [1, 6], [2, 0], [2, 6], [3, 0], [3
, 6], [4, 0], [4, 1], [5, 4], [5, 5], [7, 0], [7, 1], [7, 2], [7, 7]}, {[[0, 0]
, [2, 2]], [[0, 2], [7, 6]], [[1, 4], [3, 2]], [[2, 1], [2, 3]], [[4, 3], [4, 4
]], [[5, 0], [2, 2]]}], [{[0, 0], [0, 4], [0, 5], [1, 0], [1, 2], [3, 2], [3, 6
], [5, 0], [5, 4], [5, 7], [6, 0], [6, 4], [6, 7], [7, 0], [7, 4], [7, 7]}, {[[
0, 1], [6, 3]], [[0, 6], [2, 2]], [[1, 4], [3, 2]], [[2, 0], [3, 8]], [[5, 5],
[3, 2]], [[6, 1], [2, 3]]}]]:
end:



NYT:=proc():
[{[5,1],[6,1],[5,2],[4,3],[5,3],[4,4],[1,5],[8,5],[3,6],[4,6],[6,6],[2,7],[3,7],[9,9]},
{[1,1],[6,1],[7,1],[8,1],[1,2],[8,2],[1,3],[1,4],[2,4],[3,5],[4,5],[5,5],[3,6],[6,6],[6,7],[6,8],[7,8],[8,8]},
{[3,0],[8,1],[0,2],[8,2],[5,3],[7,4],[0,5],[3,5],[6,5],[7,5],[2,6],[3,6],[6,6],[2,7],[4,8],[9,9]},
{[2,0],[3,0],[6,0],[9,2],[1,3],[3,4],[6,4],[0,6],[3,6],[6,6],[1,8],[4,8],[7,8],[8,8]},
{[0,0],[1,0],[2,0],[4,2],[5,2],[4,4], [4,5],[6,5],[7,5],[1,6],[2,6],[7,6],[0,8], [7,8],[8,8],[0,9],[3,9],[4,9]},
{[0,0],[1,0],[4,0],[8,1],[1,2],[5,2],[6,2],[8,2],[5,3],[6,3],[2,4],[3,4],[6,4],[7,4], [2,5],[3,5],[0,6],[3,6],[4,6],[3,8], [5,8],[5,9]},
{[6,0],[1,1],[2,1],[6,1],[8,1],[8,2],[2,3],[4,3],[5,3],[8,3],[4,5],[5,5],[0,6],[4,7],[5,7],[7,7],[4,9],[5,9]},
{[4,2],[4,3],[5,4],[5,5],[0,6],[2,6],[3,7],[6,7],[3,9],[6,9]},
{[3,0],[6,0],[1,3],[3,3],[4,3],[5,3],[6,3],[3,4],[6,4],[9,4],[3,6],[6,6],[4,7],[5,7],[1,8],[4,9],[5,9],[9,9]},
{[0,0],[0,1],[6,1],[9,2],[3,3],[4,3],[3,4],[4,4],[9,6],[0,7],[8,7],[9,7],[0,8],[1,8],[6,8],[0,9],[1,9],[2,9],[5,9],[6,9]}
]:
end:


#End pre-computed puzzles and solutions


#Rec(a,b): The rectangle [0,a-1]x[0,b-1]
Rec:=proc(a,b) local i,j:
{seq(seq([i,j],i=0..a-1),j=0..b-1)}:
end:


#CP(S): given a set of lattice points, S, finds the point [i,j] such that i is as small as possible and for that i, j is as small as possible . try:
#CP({[0,0],[1,0],[2,0],[2,2],[-2,-5]});
CP:=proc(S) local j0,i0,s,Sb:
i0:=min(seq(s[1],s in S)):
Sb:={}:
for s in S do

if s[1]=i0 then
 Sb:=Sb union {s}:
fi:

od:
j0:=min(seq(s[2], s in Sb)):

[i0,j0]:

end:

#PeriV(pt,a,b): The set of vertices in perimeter of an a by b rectangle  whose bottom  left vertex is the lattice point pt. In other words [0,a-1]x[0,b-1]+pt
#Try:
#PeriV([2,3],[3,2]):
PeriV:=proc(pt,a,b) local i,j:
{seq(pt+[i,0],i=0..a-1),seq(pt+[a-1,j],j=0..b-1),seq(pt+[i,b-1],i=0..a-1),seq(pt+[0,j],j=0..b-1)}
end:


#PeriE(pt,a,b): The set of edges in perimeter of an a by b rectangle  whose bottom  left vertex is the lattice point pt. In other words [0,a-1]x[0,b-1]+pt
#Try:
#PeriE([2,3],3,2):
PeriE:=proc(pt,a,b) local i,j:
{seq({pt+[i,0],pt+[i+1,0]},i=0..a-2),seq({pt+[a-1,j],pt+[a-1,j+1]}, j=0..b-2),seq({pt+[i,b-1],pt+[i-1,b-1]},i=1..a-1),seq({pt+[0,j],pt+[0,j-1]},j=1..b-1)}
end:

#IniReg(a,b,S): inputs positive integers a and b and a subset of lattice points S, outputs the set of all
#sets of rectangles that solve the Ring Ring puzzle where the region to be covered is
#the rectanle [0,a-1]x[0,b-1] minus the set S. For example, for the puzzle of New Times magazine (created by Prassana  Seshadri) Sunday,  June 20, 2021, type:
#IniReg(8,8,{[7,0],[3,1],[4,1],[0,2],[7,4],[5,5],[1,6],[2,6]});
IniReg:=proc(a,b,S):
Rec(a,b) minus S:
end:


#KidS(St): Inputs a state St=[NeverUsed,UsedOnce,ForbiddenEdges] outputs the set of pairs [R,St1] obtained by  inserting legally a rectangle, R, that is a subset of NeverUsed union UsedOnce
#followed by the new state where the vertices of R intersect NeverUsed get moved to UsedOnce, the vertices of $ intersect UsedOnce are removed, and the new edges of Rec are joined to
#ForbiddenEdges. Try:
#Kids([IniReg(8,8,{[7,0],[3,1],[4,1],[0,2],[7,4],[5,5],[1,6],[2,6]}),{},{}]);
Kids:=proc(St) local NeverUsed,UsedOnce,FE, NeverUsed1,UsedOnce1,FE1,gu,pt,Useable,mX,mY,pt1,a1,b1,NewV,NewE:


NeverUsed:=St[1]:

if NeverUsed={} then
   RETURN({{}}):
fi:

UsedOnce:=St[2]:
FE:=St[3]:


pt:=CP(NeverUsed):



Useable:=St[1] union St[2]:



mX:=max(seq(pt1[1]-pt[1],pt1 in Useable))+1:
mY:=max(seq(pt1[2]-pt[2],pt1 in Useable))+1:


gu:={}:

for a1 from 2 to mX do
 for b1 from 2 to mY do
   NewV:=PeriV(pt,a1,b1):
   NewE:=PeriE(pt,a1,b1):
   if NewV minus (NeverUsed union UsedOnce)={} and NewE intersect FE={} then
    NeverUsed1:=NeverUsed minus NewV:
    UsedOnce1:=UsedOnce minus (NewV intersect UsedOnce) union (NewV intersect NeverUsed):
     FE1:=FE union NewE:
    gu:=gu union { [[pt,[a1,b1]], [NeverUsed1,UsedOnce1,FE1]]}:
   fi:
 od:
od:
gu:

end:


#RecCov(St): The set of coverings of St[1] that obey the rules and avoid Forbidden Edges
RecCov:=proc(St) local mu,mu1,gu,gu1,gu11:
option remember:

if St[1]={} then
 RETURN({{}}):
fi:


mu:=Kids(St):



gu:={}:

for mu1 in mu do

 gu1:=RecCov(mu1[2]):

 gu:=gu union {   seq({mu1[1]} union gu11,gu11 in gu1)}:

od:

gu:


end:


#Pinot(R): The four cornvers of the rectangle R=[pt,[a,b]]
Pinot:=proc(R) local pt,a,b:
pt:=R[1]: a:=R[2][1]: b:=R[2][2]:
{pt,pt+[a-1,0],pt+[0,b-1],pt+[a-1,b-1]}:
end:

#IsGoodPinot(SOL): are there no overlapping cornners?
IsGoodPinot:=proc(SOL) local i:
evalb(nops({seq(op(Pinot(SOL[i])),i=1..nops(SOL))})=4*nops(SOL)):
end:

#RingRing(a,b,S): inputs a Ring Ring puzzle in the rectangle [0,a-1]x[0,b-1] with the lattice points of S deleted, outputs the set of solutions to a Ring Ring puzzle. Try:
#RingRing(8,8,{[7,0],[3,1],[4,1],[0,2],[7,4],[5,5],[1,6],[2,6]});
RingRing:=proc(a,b,S) local SOL,i,gu:
SOL:=RecCov([IniReg(a,b,S),{},{}]):

gu:={}:

for i from 1 to nops(SOL) do
 if IsGoodPinot(SOL[i]) then
  gu:=gu union {SOL[i]}:
 fi:
od:
gu:
end:



PlotPuz:=proc(a,b,S) local i,j,s,T,p,pt:

T:=IniReg(a,b,S):

p:=plot([[0,0],[a,0],[a,b],[0,b],[0,0]],thickness=3,axes=none):

for i from 1 to b-1 do
p:=p,plot([[0,i],[a,i]]):
od:
for j from 1 to a-1 do
p:=p,plot([[j,0],[j,b]]):
od:
for s in S do
p:=p,plot([s,s+[1,0],s+[1,1],s+[0,1],s], filled=true, color=black):
od:

display(p):
end:

PlotSol:=proc(a,b,P,SOL) local i,j,s,T,p,pt,re,a1,b1:

T:=IniReg(a,b,P):

p:=plot([[0,0],[a,0],[a,b],[0,b],[0,0]],thickness=3,axes=none):

for i from 1 to b-1 do
p:=p,plot([[0,i],[a,i]]):
od:
for j from 1 to a-1 do
p:=p,plot([[j,0],[j,b]]):
od:
for s in P do
p:=p,plot([s,s+[1,0],s+[1,1],s+[0,1],s], filled=true, color=black):
od:

for re in SOL do

 pt:=re[1]+[1/2,1/2]:
 
 a1:=re[2][1]-1:
 b1:=re[2][2]-1:
 p:=p,plot([pt,pt+[a1,0],pt+[a1,b1],pt+[0,b1],pt],thickness=3,color=blue):
od:

 
display(p):
end:




#FindNewRecs(a,b,S,FE): given positive integers a and b, a set of already occupied lattice points S
#and a set of already-used edges, FE
#outputs the set all legal new rectangles followed by the new occupied set, and the new set of used edges
#FindNewRecs(4,4,{},{});
FindNewRecs:=proc(a,b,S,FE) local a1,b1,Av,gu,pt:

Av:=Rec(a,b) minus S:

gu:={}:

for pt in Av do
 for a1 from 2 to a-pt[1] do
  for b1 from 2 to b-pt[2] do
   if {pt,pt+[a1-1,0],pt+[0,b1-1],pt+[a1-1,b1-1]} intersect S={} and PeriE(pt,a1,b1) intersect FE={} then

   gu:=gu union {[[pt,[a1,b1]], S union PeriV(pt,a1,b1), FE union PeriE(pt,a1,b1) ]  } :
  
  fi:
 od:
od:
od:

gu:

end:


#MakePuzSol1(a,b): Makes a random puzzle and its solution for an a by b rectangle if there is no unique solution or or takes more than 2 seconds, it returns FAIL. try:
#MakePuzSol1(6,6);

MakePuzSol1:=proc(a,b) local gu,S,lu,rec,FE,P,SOL,t0:
t0:=time():
gu:={}:
S:={}:
FE:={}:

lu:=FindNewRecs(a,b,S,FE):

while nops(S)<a*b-1  and lu<>{} and time()-t0 < 0.5 do

lu:=FindNewRecs(a,b,S,FE):

#print(`nops(lu) is`, nops(lu)):

if lu<>{} then
rec:=lu[rand(1..nops(lu))()]:
gu:=gu union {rec[1]}:
S:=rec[2]:
FE:=rec[3]:
#print(`Right now it loos like`):
#print(PlotSol(a,b,Rec(a,b) minus S, gu)):
else

P:=Rec(a,b) minus S:


SOL:=RingRing(a,b,P):


if not member(gu,SOL) then
 print(`Something bad happened`):
fi:

if nops(SOL)<>1 then
 RETURN(FAIL):
else
 RETURN(Rec(a,b) minus S,gu):
fi:

fi:
od:


FAIL:



end:


#MakePuzSol(a,b,K): Makes a puzzle and its solution for an a by b rectangle by trying random puzzles K times
MakePuzSol:=proc(a,b,K) local gu,i:

for i from 1 to K do

gu:=MakePuzSol1(a,b):

if gu<>FAIL then
 RETURN(gu):
fi:
od:

FAIL:

end:

#MakePuzBook(a,b,K,M): Creates a list of M RingRing puzzles along with their solution for an a by b rectangle by calling MakePuzSol(a,b,K) M times. Try:
#MakePuzBook(6,6,100,10);
MakePuzBook:=proc(a,b,K,M) local gu,i,lu:
gu:=[]:

while nops(gu)<M do
lu:=[MakePuzSol(a,b,K)]:

if lu<>FAIL and not member(lu,{op(gu)}) then
  gu:=[op(gu),lu]:
fi:
od:
gu:
end: