######################################################################
# most current version:                                              #
#http://sites.math.rutgers.edu/~zeilberg/tokhniot/NotAlone.txt       #
## NotAlone.txt Save this file as  NotAlone.txt  to use it,          #
# stay in the                                                        #
## same directory, get into Maple (by typing: maple <Enter> )        #
## and then type:  read  `NotAlone.txt` <Enter>                      #
##you also need the file M88.txt available from                      #
#http://sites.math.rutgers.edu/~zeilberg/tokhniot/M88.txt            #
## Then follow the instructions given there                          #
##                                                                   #
## Written by Doron Zeilberger, Rutgers University ,                 #
##  DoronZeil at gmail dot com                                       # 
######################################################################


print(`First Written:  Aug. 14, 2024: tested for Maple 2020 `):
print(`Version of Sept. 25, 2024, thanks to Christoph Koutschan`):

with(combinat):
with(plots):



read `M88.txt`:

print():
print(`This is  NotAlone.txt, A Maple package`):
print(`to create and solve Not Alone puzzles of the style designed by Prasanna Seshadri `):
print(` that appear in the  New York Times Sunday  magazine (summer 2024).`):

print():
print(`The most current version is available from:`):
print(` http://sites.math.rutgers.edu/~zeilberg/tokhniot/NotAlone.txt .`):
print(`Please report all bugs to: DoronZeil at gmail dot com .`):
print():
print(`-----------------------------`):

print(`For general help, and a list of the MAIN functions,`):
print(` type "ezra();". For specific help type "ezra(procedure_name);" `):
print():
 
print(`-----------------------------`):

print(`For a list of the supporting procedures type ezra1()`):
print(`For specific help type "ezra(procedure_name);" `):
print(`For a list of the supporting functions type: ezra1();`):
print():

print(`-----------------------------`):

 
print(`-----------------------------`):

ezra1:=proc()
if args=NULL then

print(`The SUPPORTING procedures are:  Ank, FC, FC1, FillIn1, FFin, IsBad, Gnk, Gnmk, IsBad, IsBadR, IsGood, IsSubM, MakeBetter, MakeHarder, RanExt, RandPuz, RanSol, RPuzP, Shane`):
print(`  `):

else
ezra(args):

fi:

end:

ezra:=proc()
if args=NULL then

print(`The main procedures are: MakeHardPuz, PlotPuz, Ptor, RPuz, SolveN`):
print(` `):


elif nargs=1 and args[1]=Ank then
print(`Ank(n,k): all 0-1 vectors of length n with  k 1's. Try:`):
print(`Ank(10,4);`):


elif nargs=1 and args[1]=FC then
print(`FC(M,X,k1,k2): Applies FC1(M,X,k1,k2) as much as possible. Try:`):
print(`FC([[1,0,X],[0,1,X]],3,3);`):

elif nargs=1 and args[1]=FC1 then
print(`FC1(M,X,k1,k2): Given  a matrix in {0,1,X} finds whether there is place with 10X and then it returns the same with the X becoming a 0`):
print(`or X01, both horizintally and vertically. (k1,k2) are the Not Alone parameters. Try:`):
print(`FC1([[1,0,X],[0,1,X]],X,3,3);`):

elif nargs=1 and args[1]=FFin then
print(`FFin(M,X,k1,k2) : Given a partial solution of a Not Alone puzzle with still some X finds all the way`):
print(`to fill-it in and still stay legal`):

elif nargs=1 and args[1]=FillIn1 then
print(`FillIn1(M,X,k1,k2): fills in X's where the number of 0's or 1's in a row in exactly k1 and ditto for columns with k2. Try:`):
print(`FillIn1([[X,X,X,1,X,X],[X$6],[X,X,X,1,0,X],[1,0,X,X,X,X],[X$6],[X,0,X,X,X,X]],X,3,3);`):

elif nargs=1 and args[1]=Gnk then
print(`Gnk(n,k): all 0-1 vectors of length n with  k 1's and no one alone, i.e. [0,1,0], and [1,0,1] are forbidden. Try:`):
print(`Gnk(6,3);`):

elif nargs=1 and args[1]=Gnmk then
print(`Gnmk(n,m,k1,k2): all m by n 0-1 matrices where each row avoids [1,0,1],[0,1,0], has k1 1's and no column has [1,0,1],[0,1,0]  and each column has k2 1's. Try`):
print(`Gnmk(6,4,3,2);`):

elif nargs=1 and args[1]=G1nmk then
print(`G1nmk(n,m,k): all m by n 0-1 matrices where each row avoids [1,0,1],[0,1,0], has k 1's and no column has [1,0,1],[0,1,0] `):
print(`G1nmk(6,6,3):`):

elif nargs=1 and args[1]=IsBad then
prinmt(`IsBad(M,k1,k2): Is the matrix M with alphabet {0,1,X} bad in the Not Alone sense and that the number of 0s <=k1 and number of 1's<=k1  in every row`):
print(`and number of 0s and number of 1s<=k2 in every column. Try:`):
print(`M:=Ptor([[X$6],[X,X,1,1,X,X],[X$6],[1,X,1,X,X,X],[X,1,X,X,X,X],[X$6]],X,3,3); IsBad(M,3,3);`):

elif nargs=1 and args[1]=IsBadR then
print(`IsBadR(M,X,k1,k2,R): Given a partial solution M does there exist an  place that`):
print(` changing X to either 0 or 1 will be (R-1)-bad. with R=0 it is the same IsBad(M,k1,k2)`):

elif nargs=1 and args[1]=IsGood then
print(`IsGood(v): Given a 0-1 vector decides whether it is good at the end`):

elif nargs=1 and args[1]=IsSubM then
print(`IsSubM(M1,M2,X): Given  a matrix M1 in {0,1,X} and a matrix M2 of the same dimensions in {0,1} checks that the non-Z entries of M1 are the same as those of M2. Try:`):
print(`IsSubM([[1,X],[X,0]],[[1,0],[0,0]]);`):

elif nargs=1 and args[1]=MakeBetter then
print(`MakeBetter(M,X,k1,k2,d): Given a partial solution,M,  to a (k1,k2) Not Alone puzzle tries to find a better one using IsbadR with d`):
print(`Try:`):
print(`M:=SolveN1([[X$8],[X$8],[X,0,0,X,X,1,0,X],[X,0,1,X,X,0,1,X],[X$8],[X,X,X,1,1,X,X,X],[X$8],[X,X,X,X,X,X,1,X]],X,4,4); MakeBetter(M,X,4,4,2);`):

elif nargs=1 and args[1]=MakeHarder then
print(`MakeHarder(M,k1,k2,X): given a (k1,k2) puzzle in terms of X tries to make it harder`):

elif nargs=1 and args[1]=MakeHardPuz then
print(`MakeHardPuz(n,m,k1,k2,X,c,K,TL). Makes a hard Not Alone puzzle (where all clues are crucial to have a unique solution) on an m by n grid  with time limit TL`):
print(`with k1 1s in every row and k2 1s in every column. Try:`):
print(`MakeHardPuz(6,6,3,3,X,8,1000,5);`):

elif nargs=1 and args[1]=PlotPuz then
print(`PlotPuz(P): prints the puzzle P. Try:`):
print(`PlotPuz([[X,X,0,X,X,0],[X,X,1,X,X,1],[X$6],[X,X,X,1,X,X],[X$6],[X$5,0]]);`):

elif nargs=1 and args[1]=Ptor then
print(`Ptor(M,X,k1,k2): Given a NotAlone puzzle where X is blank 1 is filled circle and 0 is empty circle finds all solutions.  USING BRUTE FORCE, with a data base, M88.txt, created by`):
print(`Christoph Koutschan`):
print(`This file is available from `):
print(``):
print(` http://sites.math.rutgers.edu/~zeilberg/tokhniot/M88.txt `):
print(``):
print(`and should be in the SAME directory as this file, in your computer`):
print(``):
print(`For example, for the Aug. 11, 2024 puzzle by`):
print(`Presanna Seshardi, type:`):
print(`Ptor([[X$6],[X,X,1,1,X,X],[X$6],[1,X,1,X,X,X],[X,1,X,X,X,X],[X$6]],X,3,3);`):
print(`For the Aug. 18, 2024 type`):
print(`Ptor([[X,X,0,X,X,0],[X,X,1,X,X,1],[X$6],[X,X,X,1,X,X],[X$6],[X$5,0]],X,3,3);`):

print(`For the Aug. 25, 2024 type`):
print(`Ptor([[X,X,X,1,X,X],[X$6],[X,X,X,1,0,X],[1,0,X,X,X,X],[X$6],[X,0,X,X,X,X]],X,3,3);`):


print(`For the Sept. 8, 2024 puzzle type:`):
print(`Ptor([[X$8],[X$8],[X,0,0,X,X,1,0,X],[X,0,1,X,X,0,1,X],[X$8],[X,X,X,1,1,X,X,X],[X$8],[X,0,X,X,X,X,1,X]],X,4,4);`):

print(`For the Sept. 15, 2024 puzzle type`):
print(`Ptor([[X$8],[X$5,0,0,1],[X,X,X,1,0,X,X,X],[X$8],[X,X,1,0,X$4],[X$7,0],[0$2,X$6],[X,X,X,X,X,0,X,X]],X,4,4);`):

print(`For the Sept. 22  2024 puzzle type`):
print(`Ptor([[X$3,0,X$4],[X$5,1,X,X],[X,0,X$6],[X,X,X,0,X,X,X,0],[X,0,X,X,1,X,X,1],[X$3,0,X$4],[X$8],[X$8]],X,4,4);`):

print(`For the Sept. 29, 2024 puzzle type`):
print(`Ptor([[X$3,0,X$4],[X$5,1,X,X],[X,0,X$6],[X,X,X,0,X,X,X,0],[X,0,X,X,1,X,X,1],[X$3,0,X$4],[X$8],[X$8]],X,4,4);`):

print(`For the Oct. 6, 2024 puzzle type`):
print(`Ptor([[X,X,0,X$5],[X,0,X,X,0,1,X,X],[X$5,0,1,X],[X,X,1,0,X,X,0,X],[X,X,X,1,0,X,X,X],[X$4,1,0,X,X],[X$8],[1,X$7]],X,4,4);`):


print(`For the Oct. 13, 2024 puzzle type`):
print(`Ptor([[X,X,X,X,X,X,1,X],[X,X,1,X,X,X,X,X],[X$8],[X,X,1,0,X,X,X,X],[X$8],[X$7,0],[X,X,X,X,0,X,X,X],[X,X,X,X,1,X,1,X]],X,4,4);`):

elif nargs=1 and args[1]=RanExt then
print(`RanExt(M,k1,k2): given a 0-1 matrix M where every row has k1 0s and k1 1s and avoids the patterns 101 and 010`):
print(`and the columns also avoid these and every column has at most k2 0s and at most k2 1s`):
print(`finds a random extension with one more row. Try:`):
print(`RanExt([],3,3);`):

elif nargs=1 and args[1]=RandPuz then
print(`RandPuz(n,m,k1,k2,X,c,K): A random m by n Not Alove puzzle with row sums k1 and column sums k2 and c clues. Try:`):
print(`RandPuz1(6,6,3,3,X,10,100);`):

elif nargs=1 and args[1]=RanSol then
print(`RanSol(k1,k2): A random good balanced 0-1 matrix. Try:`):
print(`RanSol(3,3);`):

elif nargs=1 and args[1]=RPuz then
print(`RPuz(k1,k2,c,X,T0): A random (k1,k2) puzzle with c clues and then removing as many as possible and keeping the answer unique, and taking at most T0 seconds. Try:`):
print(`RPuz(4,4,28,X,10);`):

elif nargs=1 and args[1]=RPuzP then
print(`RPuzP(k1,k2,c,X): A random (k1,k2) puzzle with c clues. `):
print(`For (3,3) make c at least 12 or it would be very slow`):
print(`For (4,4) make c at least 30  `):
print(`Try:`):
print(`RPuzP(4,4,30,X);`):

elif nargs=1 and args[1]=Shane then
print(`Shane(M,i,j,c): Given a matrix M a constant c, and i,j indicatiing a location [i,j],  returns the new matrix that is the same except that the [i,j] entry is now:`):
print(`Shane([[X,1,X],[X,X,1]],1,3,1);`):



elif nargs=1 and args[1]=SolveN then
print(`SolveN(M,X,k1,k2): Given a NotAlone puzzle where X is blank 1 is filled circle and 0 is empty circle finds all solutions. USING LOGIC. `):
print(`WARNING: for some puzzles it is MUCH slower than Ptor(M,X,k1,k2) for k1=k2=4`):
print(``):
print(`For example, for the Aug. 11, 2024 puzzle by`):
print(`Presanna Seshardi, type:`):
print(`SolveN([[X$6],[X,X,1,1,X,X],[X$6],[1,X,1,X,X,X],[X,1,X,X,X,X],[X$6]],X,3,3);`):
print(`For the Aug. 18, 2024 puzzle type`):
print(`SolveN([[X,X,0,X,X,0],[X,X,1,X,X,1],[X$6],[X,X,X,1,X,X],[X$6],[X$5,0]],X,3,3);`):

print(`For the Aug. 25 2024 puzzle type`):
print(`SolveN([[X,X,X,1,X,X],[X$6],[X,X,X,1,0,X],[1,0,X,X,X,X],[X$6],[X,0,X,X,X,X]],X,3,3);`):

print(`For the Sept. 8, 2024 puzzle type:`):
print(`SolveN([[X$8],[X$8],[X,0,0,X,X,1,0,X],[X,0,1,X,X,0,1,X],[X$8],[X,X,X,1,1,X,X,X],[X$8],[X,0,X,X,X,X,1,X]],X,4,4);`):

print(`For the Sept. 15, 2024 puzzle type`):
print(`SolveN([[X$8],[X$5,0,0,1],[X,X,X,1,0,X,X,X],[X$8],[X,X,1,0,X$4],[X$7,0],[0$2,X$6],[X,X,X,X,X,0,X,X]],X,4,4);`):

print(`For the Sept. 22  2024 puzzle type`):
print(`SolveN([[X$3,0,X$4],[X$5,1,X,X],[X,0,X$6],[X,X,X,0,X,X,X,0],[X,0,X,X,1,X,X,1],[X$3,0,X$4],[X$8],[X$8]],X,4,4);`):



print(`For the Sept. 29, 2024 puzzle type`):
print(`SolveN([[X$3,0,X$4],[X$5,1,X,X],[X,0,X$6],[X,X,X,0,X,X,X,0],[X,0,X,X,1,X,X,1],[X$3,0,X$4],[X$8],[X$8]],X,4,4);`):


print(`For the Oct. 6, 2024 puzzle type`):
print(`SolveN([[X,X,0,X$5],[X,0,X,X,0,1,X,X],[X$5,0,1,X],[X,X,1,0,X,X,0,X],[X,X,X,1,0,X,X,X],[X$4,1,0,X,X],[X$8],[1,X$7]],X,4,4);`):


print(`For the Oct. 13, 2024 puzzle type`):
print(`SolveN([[X,X,X,X,X,X,1,X],[X,X,1,X,X,X,X,X],[X$8],[X,X,1,0,X,X,X,X],[X$8],[X$7,0],[X,X,X,X,0,X,X,X],[X,X,X,X,1,X,1,X]],X,4,4);`):

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

fi:

end:


with(combinat):



#Ank(n,k): all 0-1 vectors of length n with  k 1's
Ank:=proc(n,k) local gu1,gu2,gu11:
option remember:
if n=0 then
 if k=0 then
   RETURN({[]}):
   else
    RETURN({}):
  fi:
  fi:

gu1:=Ank(n-1,k-1):
gu2:=Ank(n-1,k):

{seq([op(gu11),1],gu11 in gu1), seq([op(gu11),0],gu11 in gu2)}:

end:


#Gnk(n,k): all 0-1 vectors of length n with  k 1's and no one alone, i.e. [0,1,0], and [1,0,1] are forbidden. Try:
#Gnk(6,3);
Gnk:=proc(n,k) local mu,gu,mu1,i1:
option remember:
mu:=Ank(n,k):
gu:={}:

for mu1 in mu do
 if {seq([op(i1..i1+2,mu1)],i1=1..nops(mu1)-2)} intersect {[0,1,0],[1,0,1]}={} then
  gu:=gu union {mu1}:
 fi:
 od:

gu:

end:

#IsGood(v): Given a 0-1 vector decides whether it is good at the end
IsGood:=proc(v)
if nops(v)<=2 then
 RETURN(true):
 fi:

not evalb( member([op(nops(v)-2..nops(v),v)], {[1,0,1],[0,1,0]})):
end:


#G1nmk(n,m,k): all m by n 0-1 matrices where each row avoids [1,0,1],[0,1,0], has k 1's and no column has [1,0,1],[0,1,0] 
#G1nmk(6,6,3):
G1nmk:=proc(n,m,k) local gu,lu,mu,lu1,mu1,hal,i1,j1:
option remember:
lu:=Gnk(n,k):

if m=1 then
 RETURN({seq([lu1],lu1 in lu)}):
fi:

mu:=G1nmk(n,m-1,k):

gu:={}:

for lu1 in lu do
 for mu1 in mu do
  hal:=[op(mu1),lu1]:
  if {seq(IsGood([seq(hal[i1][j1],i1=1..m)]),j1=1..n)}={true} then
    gu:=gu union {hal}:
  fi:
 od:
 od:
gu:

end:







#Gnmk(n,m,k1,k2): all m by n 0-1 matrices where each row avoids [1,0,1],[0,1,0], has k1 1's and no column has [1,0,1],[0,1,0]  and each column has k2 1's. Try
#Gnmk(6,4,3,2);
Gnmk:=proc(n,m,k1,k2) local lu,lu1,gu,i1,j1:
option rememeber:



if m=6 and n=6 and k1=3 and k2=3 then
RETURN(
{[[0, 0, 0, 1, 1, 1], [0, 0, 0, 1, 1, 1], [0, 0, 0, 1, 1, 1], [1, 1, 1, 0, 0, 0
], [1, 1, 1, 0, 0, 0], [1, 1, 1, 0, 0, 0]], [[0, 0, 0, 1, 1, 1], [0, 0, 0, 1, 1
, 1], [0, 0, 1, 1, 1, 0], [1, 1, 1, 0, 0, 0], [1, 1, 1, 0, 0, 0], [1, 1, 0, 0,
0, 1]], [[0, 0, 0, 1, 1, 1], [0, 0, 0, 1, 1, 1], [0, 1, 1, 0, 0, 1], [1, 1, 1,
0, 0, 0], [1, 1, 1, 0, 0, 0], [1, 0, 0, 1, 1, 0]], [[0, 0, 0, 1, 1, 1], [0, 0,
0, 1, 1, 1], [0, 1, 1, 1, 0, 0], [1, 1, 1, 0, 0, 0], [1, 1, 1, 0, 0, 0], [1, 0,
0, 0, 1, 1]], [[0, 0, 0, 1, 1, 1], [0, 0, 0, 1, 1, 1], [1, 0, 0, 0, 1, 1], [1,
1, 1, 0, 0, 0], [1, 1, 1, 0, 0, 0], [0, 1, 1, 1, 0, 0]], [[0, 0, 0, 1, 1, 1], [
0, 0, 0, 1, 1, 1], [1, 0, 0, 1, 1, 0], [1, 1, 1, 0, 0, 0], [1, 1, 1, 0, 0, 0],
[0, 1, 1, 0, 0, 1]], [[0, 0, 0, 1, 1, 1], [0, 0, 0, 1, 1, 1], [1, 1, 0, 0, 0, 1
], [1, 1, 1, 0, 0, 0], [1, 1, 1, 0, 0, 0], [0, 0, 1, 1, 1, 0]], [[0, 0, 0, 1, 1
, 1], [0, 0, 0, 1, 1, 1], [1, 1, 1, 0, 0, 0], [1, 1, 1, 0, 0, 0], [1, 1, 1, 0,
0, 0], [0, 0, 0, 1, 1, 1]], [[0, 0, 0, 1, 1, 1], [0, 0, 1, 1, 1, 0], [0, 0, 1,
1, 1, 0], [1, 1, 0, 0, 0, 1], [1, 1, 0, 0, 0, 1], [1, 1, 1, 0, 0, 0]], [[0, 0,
0, 1, 1, 1], [0, 0, 1, 1, 1, 0], [0, 0, 1, 1, 1, 0], [1, 1, 1, 0, 0, 0], [1, 1,
0, 0, 0, 1], [1, 1, 0, 0, 0, 1]], [[0, 0, 0, 1, 1, 1], [0, 0, 1, 1, 1, 0], [0,
1, 1, 1, 0, 0], [1, 1, 1, 0, 0, 0], [1, 1, 0, 0, 0, 1], [1, 0, 0, 0, 1, 1]], [[
0, 0, 0, 1, 1, 1], [0, 0, 1, 1, 1, 0], [1, 1, 1, 0, 0, 0], [1, 1, 0, 0, 0, 1],
[1, 1, 0, 0, 0, 1], [0, 0, 1, 1, 1, 0]], [[0, 0, 0, 1, 1, 1], [0, 0, 1, 1, 1, 0
], [1, 1, 1, 0, 0, 0], [1, 1, 1, 0, 0, 0], [1, 1, 0, 0, 0, 1], [0, 0, 0, 1, 1,
1]], [[0, 0, 0, 1, 1, 1], [0, 1, 1, 0, 0, 1], [0, 1, 1, 0, 0, 1], [1, 0, 0, 1,
1, 0], [1, 0, 0, 1, 1, 0], [1, 1, 1, 0, 0, 0]], [[0, 0, 0, 1, 1, 1], [0, 1, 1,
0, 0, 1], [0, 1, 1, 0, 0, 1], [1, 1, 1, 0, 0, 0], [1, 0, 0, 1, 1, 0], [1, 0, 0,
1, 1, 0]], [[0, 0, 0, 1, 1, 1], [0, 1, 1, 0, 0, 1], [1, 1, 1, 0, 0, 0], [1, 0,
0, 1, 1, 0], [1, 0, 0, 1, 1, 0], [0, 1, 1, 0, 0, 1]], [[0, 0, 0, 1, 1, 1], [0,
1, 1, 0, 0, 1], [1, 1, 1, 0, 0, 0], [1, 1, 1, 0, 0, 0], [1, 0, 0, 1, 1, 0], [0,
0, 0, 1, 1, 1]], [[0, 0, 0, 1, 1, 1], [0, 1, 1, 1, 0, 0], [0, 1, 1, 1, 0, 0], [
1, 0, 0, 0, 1, 1], [1, 0, 0, 0, 1, 1], [1, 1, 1, 0, 0, 0]], [[0, 0, 0, 1, 1, 1]
, [0, 1, 1, 1, 0, 0], [0, 1, 1, 1, 0, 0], [1, 1, 1, 0, 0, 0], [1, 0, 0, 0, 1, 1
], [1, 0, 0, 0, 1, 1]], [[0, 0, 0, 1, 1, 1], [0, 1, 1, 1, 0, 0], [1, 1, 1, 0, 0
, 0], [1, 0, 0, 0, 1, 1], [1, 0, 0, 0, 1, 1], [0, 1, 1, 1, 0, 0]], [[0, 0, 0, 1
, 1, 1], [0, 1, 1, 1, 0, 0], [1, 1, 1, 0, 0, 0], [1, 1, 0, 0, 0, 1], [1, 0, 0,
0, 1, 1], [0, 0, 1, 1, 1, 0]], [[0, 0, 0, 1, 1, 1], [0, 1, 1, 1, 0, 0], [1, 1,
1, 0, 0, 0], [1, 1, 1, 0, 0, 0], [1, 0, 0, 0, 1, 1], [0, 0, 0, 1, 1, 1]], [[0,
0, 0, 1, 1, 1], [1, 0, 0, 0, 1, 1], [1, 0, 0, 0, 1, 1], [0, 1, 1, 1, 0, 0], [0,
1, 1, 1, 0, 0], [1, 1, 1, 0, 0, 0]], [[0, 0, 0, 1, 1, 1], [1, 0, 0, 0, 1, 1], [
1, 0, 0, 0, 1, 1], [1, 1, 1, 0, 0, 0], [0, 1, 1, 1, 0, 0], [0, 1, 1, 1, 0, 0]],
[[0, 0, 0, 1, 1, 1], [1, 0, 0, 0, 1, 1], [1, 1, 0, 0, 0, 1], [1, 1, 1, 0, 0, 0]
, [0, 1, 1, 1, 0, 0], [0, 0, 1, 1, 1, 0]], [[0, 0, 0, 1, 1, 1], [1, 0, 0, 0, 1,
1], [1, 1, 1, 0, 0, 0], [0, 1, 1, 1, 0, 0], [0, 1, 1, 1, 0, 0], [1, 0, 0, 0, 1,
1]], [[0, 0, 0, 1, 1, 1], [1, 0, 0, 0, 1, 1], [1, 1, 1, 0, 0, 0], [1, 1, 1, 0,
0, 0], [0, 1, 1, 1, 0, 0], [0, 0, 0, 1, 1, 1]], [[0, 0, 0, 1, 1, 1], [1, 0, 0,
1, 1, 0], [1, 0, 0, 1, 1, 0], [0, 1, 1, 0, 0, 1], [0, 1, 1, 0, 0, 1], [1, 1, 1,
0, 0, 0]], [[0, 0, 0, 1, 1, 1], [1, 0, 0, 1, 1, 0], [1, 0, 0, 1, 1, 0], [1, 1,
1, 0, 0, 0], [0, 1, 1, 0, 0, 1], [0, 1, 1, 0, 0, 1]], [[0, 0, 0, 1, 1, 1], [1,
0, 0, 1, 1, 0], [1, 1, 1, 0, 0, 0], [0, 1, 1, 0, 0, 1], [0, 1, 1, 0, 0, 1], [1,
0, 0, 1, 1, 0]], [[0, 0, 0, 1, 1, 1], [1, 0, 0, 1, 1, 0], [1, 1, 1, 0, 0, 0], [
1, 1, 1, 0, 0, 0], [0, 1, 1, 0, 0, 1], [0, 0, 0, 1, 1, 1]], [[0, 0, 0, 1, 1, 1]
, [1, 1, 0, 0, 0, 1], [1, 1, 0, 0, 0, 1], [0, 0, 1, 1, 1, 0], [0, 0, 1, 1, 1, 0
], [1, 1, 1, 0, 0, 0]], [[0, 0, 0, 1, 1, 1], [1, 1, 0, 0, 0, 1], [1, 1, 0, 0, 0
, 1], [1, 1, 1, 0, 0, 0], [0, 0, 1, 1, 1, 0], [0, 0, 1, 1, 1, 0]], [[0, 0, 0, 1
, 1, 1], [1, 1, 0, 0, 0, 1], [1, 1, 1, 0, 0, 0], [0, 0, 1, 1, 1, 0], [0, 0, 1,
1, 1, 0], [1, 1, 0, 0, 0, 1]], [[0, 0, 0, 1, 1, 1], [1, 1, 0, 0, 0, 1], [1, 1,
1, 0, 0, 0], [0, 1, 1, 1, 0, 0], [0, 0, 1, 1, 1, 0], [1, 0, 0, 0, 1, 1]], [[0,
0, 0, 1, 1, 1], [1, 1, 0, 0, 0, 1], [1, 1, 1, 0, 0, 0], [1, 1, 1, 0, 0, 0], [0,
0, 1, 1, 1, 0], [0, 0, 0, 1, 1, 1]], [[0, 0, 0, 1, 1, 1], [1, 1, 1, 0, 0, 0], [
1, 1, 1, 0, 0, 0], [0, 0, 0, 1, 1, 1], [0, 0, 0, 1, 1, 1], [1, 1, 1, 0, 0, 0]],
[[0, 0, 0, 1, 1, 1], [1, 1, 1, 0, 0, 0], [1, 1, 1, 0, 0, 0], [0, 0, 1, 1, 1, 0]
, [0, 0, 0, 1, 1, 1], [1, 1, 0, 0, 0, 1]], [[0, 0, 0, 1, 1, 1], [1, 1, 1, 0, 0,
0], [1, 1, 1, 0, 0, 0], [0, 1, 1, 0, 0, 1], [0, 0, 0, 1, 1, 1], [1, 0, 0, 1, 1,
0]], [[0, 0, 0, 1, 1, 1], [1, 1, 1, 0, 0, 0], [1, 1, 1, 0, 0, 0], [0, 1, 1, 1,
0, 0], [0, 0, 0, 1, 1, 1], [1, 0, 0, 0, 1, 1]], [[0, 0, 0, 1, 1, 1], [1, 1, 1,
0, 0, 0], [1, 1, 1, 0, 0, 0], [1, 0, 0, 0, 1, 1], [0, 0, 0, 1, 1, 1], [0, 1, 1,
1, 0, 0]], [[0, 0, 0, 1, 1, 1], [1, 1, 1, 0, 0, 0], [1, 1, 1, 0, 0, 0], [1, 0,
0, 1, 1, 0], [0, 0, 0, 1, 1, 1], [0, 1, 1, 0, 0, 1]], [[0, 0, 0, 1, 1, 1], [1,
1, 1, 0, 0, 0], [1, 1, 1, 0, 0, 0], [1, 1, 0, 0, 0, 1], [0, 0, 0, 1, 1, 1], [0,
0, 1, 1, 1, 0]], [[0, 0, 0, 1, 1, 1], [1, 1, 1, 0, 0, 0], [1, 1, 1, 0, 0, 0], [
1, 1, 1, 0, 0, 0], [0, 0, 0, 1, 1, 1], [0, 0, 0, 1, 1, 1]], [[0, 0, 1, 1, 1, 0]
, [0, 0, 0, 1, 1, 1], [0, 0, 0, 1, 1, 1], [1, 1, 0, 0, 0, 1], [1, 1, 1, 0, 0, 0
], [1, 1, 1, 0, 0, 0]], [[0, 0, 1, 1, 1, 0], [0, 0, 0, 1, 1, 1], [0, 0, 0, 1, 1
, 1], [1, 1, 1, 0, 0, 0], [1, 1, 1, 0, 0, 0], [1, 1, 0, 0, 0, 1]], [[0, 0, 1, 1
, 1, 0], [0, 0, 0, 1, 1, 1], [1, 0, 0, 0, 1, 1], [1, 1, 0, 0, 0, 1], [1, 1, 1,
0, 0, 0], [0, 1, 1, 1, 0, 0]], [[0, 0, 1, 1, 1, 0], [0, 0, 0, 1, 1, 1], [1, 1,
0, 0, 0, 1], [1, 1, 0, 0, 0, 1], [1, 1, 1, 0, 0, 0], [0, 0, 1, 1, 1, 0]], [[0,
0, 1, 1, 1, 0], [0, 0, 0, 1, 1, 1], [1, 1, 0, 0, 0, 1], [1, 1, 1, 0, 0, 0], [1,
1, 1, 0, 0, 0], [0, 0, 0, 1, 1, 1]], [[0, 0, 1, 1, 1, 0], [0, 0, 1, 1, 1, 0], [
0, 0, 0, 1, 1, 1], [1, 1, 0, 0, 0, 1], [1, 1, 0, 0, 0, 1], [1, 1, 1, 0, 0, 0]],
[[0, 0, 1, 1, 1, 0], [0, 0, 1, 1, 1, 0], [0, 0, 1, 1, 1, 0], [1, 1, 0, 0, 0, 1]
, [1, 1, 0, 0, 0, 1], [1, 1, 0, 0, 0, 1]], [[0, 0, 1, 1, 1, 0], [0, 0, 1, 1, 1,
0], [0, 1, 1, 0, 0, 1], [1, 1, 0, 0, 0, 1], [1, 1, 0, 0, 0, 1], [1, 0, 0, 1, 1,
0]], [[0, 0, 1, 1, 1, 0], [0, 0, 1, 1, 1, 0], [0, 1, 1, 1, 0, 0], [1, 1, 0, 0,
0, 1], [1, 1, 0, 0, 0, 1], [1, 0, 0, 0, 1, 1]], [[0, 0, 1, 1, 1, 0], [0, 0, 1,
1, 1, 0], [1, 0, 0, 0, 1, 1], [1, 1, 0, 0, 0, 1], [1, 1, 0, 0, 0, 1], [0, 1, 1,
1, 0, 0]], [[0, 0, 1, 1, 1, 0], [0, 0, 1, 1, 1, 0], [1, 0, 0, 1, 1, 0], [1, 1,
0, 0, 0, 1], [1, 1, 0, 0, 0, 1], [0, 1, 1, 0, 0, 1]], [[0, 0, 1, 1, 1, 0], [0,
0, 1, 1, 1, 0], [1, 1, 0, 0, 0, 1], [1, 1, 0, 0, 0, 1], [1, 1, 0, 0, 0, 1], [0,
0, 1, 1, 1, 0]], [[0, 0, 1, 1, 1, 0], [0, 0, 1, 1, 1, 0], [1, 1, 1, 0, 0, 0], [
1, 1, 0, 0, 0, 1], [1, 1, 0, 0, 0, 1], [0, 0, 0, 1, 1, 1]], [[0, 0, 1, 1, 1, 0]
, [0, 1, 1, 0, 0, 1], [0, 1, 1, 0, 0, 1], [1, 0, 0, 1, 1, 0], [1, 0, 0, 1, 1, 0
], [1, 1, 0, 0, 0, 1]], [[0, 0, 1, 1, 1, 0], [0, 1, 1, 0, 0, 1], [0, 1, 1, 0, 0
, 1], [1, 1, 0, 0, 0, 1], [1, 0, 0, 1, 1, 0], [1, 0, 0, 1, 1, 0]], [[0, 0, 1, 1
, 1, 0], [0, 1, 1, 0, 0, 1], [1, 1, 0, 0, 0, 1], [1, 0, 0, 0, 1, 1], [1, 0, 0,
1, 1, 0], [0, 1, 1, 1, 0, 0]], [[0, 0, 1, 1, 1, 0], [0, 1, 1, 0, 0, 1], [1, 1,
0, 0, 0, 1], [1, 0, 0, 1, 1, 0], [1, 0, 0, 1, 1, 0], [0, 1, 1, 0, 0, 1]], [[0,
0, 1, 1, 1, 0], [0, 1, 1, 0, 0, 1], [1, 1, 0, 0, 0, 1], [1, 1, 0, 0, 0, 1], [1,
0, 0, 1, 1, 0], [0, 0, 1, 1, 1, 0]], [[0, 0, 1, 1, 1, 0], [0, 1, 1, 1, 0, 0], [
0, 1, 1, 0, 0, 1], [1, 1, 0, 0, 0, 1], [1, 0, 0, 0, 1, 1], [1, 0, 0, 1, 1, 0]],
[[0, 0, 1, 1, 1, 0], [0, 1, 1, 1, 0, 0], [0, 1, 1, 1, 0, 0], [1, 0, 0, 0, 1, 1]
, [1, 0, 0, 0, 1, 1], [1, 1, 0, 0, 0, 1]], [[0, 0, 1, 1, 1, 0], [0, 1, 1, 1, 0,
0], [0, 1, 1, 1, 0, 0], [1, 1, 0, 0, 0, 1], [1, 0, 0, 0, 1, 1], [1, 0, 0, 0, 1,
1]], [[0, 0, 1, 1, 1, 0], [0, 1, 1, 1, 0, 0], [1, 1, 0, 0, 0, 1], [1, 0, 0, 0,
1, 1], [1, 0, 0, 0, 1, 1], [0, 1, 1, 1, 0, 0]], [[0, 0, 1, 1, 1, 0], [0, 1, 1,
1, 0, 0], [1, 1, 0, 0, 0, 1], [1, 1, 0, 0, 0, 1], [1, 0, 0, 0, 1, 1], [0, 0, 1,
1, 1, 0]], [[0, 0, 1, 1, 1, 0], [0, 1, 1, 1, 0, 0], [1, 1, 1, 0, 0, 0], [1, 1,
0, 0, 0, 1], [1, 0, 0, 0, 1, 1], [0, 0, 0, 1, 1, 1]], [[0, 0, 1, 1, 1, 0], [1,
0, 0, 0, 1, 1], [1, 0, 0, 0, 1, 1], [0, 1, 1, 1, 0, 0], [0, 1, 1, 1, 0, 0], [1,
1, 0, 0, 0, 1]], [[0, 0, 1, 1, 1, 0], [1, 0, 0, 0, 1, 1], [1, 0, 0, 0, 1, 1], [
1, 1, 0, 0, 0, 1], [0, 1, 1, 1, 0, 0], [0, 1, 1, 1, 0, 0]], [[0, 0, 1, 1, 1, 0]
, [1, 0, 0, 0, 1, 1], [1, 1, 0, 0, 0, 1], [0, 1, 1, 0, 0, 1], [0, 1, 1, 1, 0, 0
], [1, 0, 0, 1, 1, 0]], [[0, 0, 1, 1, 1, 0], [1, 0, 0, 0, 1, 1], [1, 1, 0, 0, 0
, 1], [0, 1, 1, 1, 0, 0], [0, 1, 1, 1, 0, 0], [1, 0, 0, 0, 1, 1]], [[0, 0, 1, 1
, 1, 0], [1, 0, 0, 0, 1, 1], [1, 1, 0, 0, 0, 1], [1, 1, 0, 0, 0, 1], [0, 1, 1,
1, 0, 0], [0, 0, 1, 1, 1, 0]], [[0, 0, 1, 1, 1, 0], [1, 0, 0, 0, 1, 1], [1, 1,
0, 0, 0, 1], [1, 1, 1, 0, 0, 0], [0, 1, 1, 1, 0, 0], [0, 0, 0, 1, 1, 1]], [[0,
0, 1, 1, 1, 0], [1, 0, 0, 1, 1, 0], [1, 0, 0, 0, 1, 1], [1, 1, 0, 0, 0, 1], [0,
1, 1, 0, 0, 1], [0, 1, 1, 1, 0, 0]], [[0, 0, 1, 1, 1, 0], [1, 0, 0, 1, 1, 0], [
1, 0, 0, 1, 1, 0], [0, 1, 1, 0, 0, 1], [0, 1, 1, 0, 0, 1], [1, 1, 0, 0, 0, 1]],
[[0, 0, 1, 1, 1, 0], [1, 0, 0, 1, 1, 0], [1, 0, 0, 1, 1, 0], [1, 1, 0, 0, 0, 1]
, [0, 1, 1, 0, 0, 1], [0, 1, 1, 0, 0, 1]], [[0, 0, 1, 1, 1, 0], [1, 0, 0, 1, 1,
0], [1, 1, 0, 0, 0, 1], [0, 1, 1, 0, 0, 1], [0, 1, 1, 0, 0, 1], [1, 0, 0, 1, 1,
0]], [[0, 0, 1, 1, 1, 0], [1, 0, 0, 1, 1, 0], [1, 1, 0, 0, 0, 1], [1, 1, 0, 0,
0, 1], [0, 1, 1, 0, 0, 1], [0, 0, 1, 1, 1, 0]], [[0, 0, 1, 1, 1, 0], [1, 1, 0,
0, 0, 1], [1, 1, 0, 0, 0, 1], [0, 0, 0, 1, 1, 1], [0, 0, 1, 1, 1, 0], [1, 1, 1,
0, 0, 0]], [[0, 0, 1, 1, 1, 0], [1, 1, 0, 0, 0, 1], [1, 1, 0, 0, 0, 1], [0, 0,
1, 1, 1, 0], [0, 0, 1, 1, 1, 0], [1, 1, 0, 0, 0, 1]], [[0, 0, 1, 1, 1, 0], [1,
1, 0, 0, 0, 1], [1, 1, 0, 0, 0, 1], [0, 1, 1, 0, 0, 1], [0, 0, 1, 1, 1, 0], [1,
0, 0, 1, 1, 0]], [[0, 0, 1, 1, 1, 0], [1, 1, 0, 0, 0, 1], [1, 1, 0, 0, 0, 1], [
0, 1, 1, 1, 0, 0], [0, 0, 1, 1, 1, 0], [1, 0, 0, 0, 1, 1]], [[0, 0, 1, 1, 1, 0]
, [1, 1, 0, 0, 0, 1], [1, 1, 0, 0, 0, 1], [1, 0, 0, 0, 1, 1], [0, 0, 1, 1, 1, 0
], [0, 1, 1, 1, 0, 0]], [[0, 0, 1, 1, 1, 0], [1, 1, 0, 0, 0, 1], [1, 1, 0, 0, 0
, 1], [1, 0, 0, 1, 1, 0], [0, 0, 1, 1, 1, 0], [0, 1, 1, 0, 0, 1]], [[0, 0, 1, 1
, 1, 0], [1, 1, 0, 0, 0, 1], [1, 1, 0, 0, 0, 1], [1, 1, 0, 0, 0, 1], [0, 0, 1,
1, 1, 0], [0, 0, 1, 1, 1, 0]], [[0, 0, 1, 1, 1, 0], [1, 1, 0, 0, 0, 1], [1, 1,
0, 0, 0, 1], [1, 1, 1, 0, 0, 0], [0, 0, 1, 1, 1, 0], [0, 0, 0, 1, 1, 1]], [[0,
0, 1, 1, 1, 0], [1, 1, 1, 0, 0, 0], [1, 1, 0, 0, 0, 1], [0, 0, 0, 1, 1, 1], [0,
0, 0, 1, 1, 1], [1, 1, 1, 0, 0, 0]], [[0, 0, 1, 1, 1, 0], [1, 1, 1, 0, 0, 0], [
1, 1, 0, 0, 0, 1], [1, 0, 0, 0, 1, 1], [0, 0, 0, 1, 1, 1], [0, 1, 1, 1, 0, 0]],
[[0, 0, 1, 1, 1, 0], [1, 1, 1, 0, 0, 0], [1, 1, 0, 0, 0, 1], [1, 1, 0, 0, 0, 1]
, [0, 0, 0, 1, 1, 1], [0, 0, 1, 1, 1, 0]], [[0, 0, 1, 1, 1, 0], [1, 1, 1, 0, 0,
0], [1, 1, 1, 0, 0, 0], [0, 0, 0, 1, 1, 1], [0, 0, 0, 1, 1, 1], [1, 1, 0, 0, 0,
1]], [[0, 0, 1, 1, 1, 0], [1, 1, 1, 0, 0, 0], [1, 1, 1, 0, 0, 0], [1, 1, 0, 0,
0, 1], [0, 0, 0, 1, 1, 1], [0, 0, 0, 1, 1, 1]], [[0, 1, 1, 0, 0, 1], [0, 0, 0,
1, 1, 1], [0, 0, 0, 1, 1, 1], [1, 0, 0, 1, 1, 0], [1, 1, 1, 0, 0, 0], [1, 1, 1,
0, 0, 0]], [[0, 1, 1, 0, 0, 1], [0, 0, 0, 1, 1, 1], [0, 0, 0, 1, 1, 1], [1, 1,
1, 0, 0, 0], [1, 1, 1, 0, 0, 0], [1, 0, 0, 1, 1, 0]], [[0, 1, 1, 0, 0, 1], [0,
0, 0, 1, 1, 1], [1, 0, 0, 1, 1, 0], [1, 0, 0, 1, 1, 0], [1, 1, 1, 0, 0, 0], [0,
1, 1, 0, 0, 1]], [[0, 1, 1, 0, 0, 1], [0, 0, 0, 1, 1, 1], [1, 0, 0, 1, 1, 0], [
1, 1, 1, 0, 0, 0], [1, 1, 1, 0, 0, 0], [0, 0, 0, 1, 1, 1]], [[0, 1, 1, 0, 0, 1]
, [0, 0, 1, 1, 1, 0], [0, 0, 1, 1, 1, 0], [1, 0, 0, 1, 1, 0], [1, 1, 0, 0, 0, 1
], [1, 1, 0, 0, 0, 1]], [[0, 1, 1, 0, 0, 1], [0, 0, 1, 1, 1, 0], [0, 0, 1, 1, 1
, 0], [1, 1, 0, 0, 0, 1], [1, 1, 0, 0, 0, 1], [1, 0, 0, 1, 1, 0]], [[0, 1, 1, 0
, 0, 1], [0, 0, 1, 1, 1, 0], [1, 0, 0, 1, 1, 0], [1, 0, 0, 0, 1, 1], [1, 1, 0,
0, 0, 1], [0, 1, 1, 1, 0, 0]], [[0, 1, 1, 0, 0, 1], [0, 0, 1, 1, 1, 0], [1, 0,
0, 1, 1, 0], [1, 0, 0, 1, 1, 0], [1, 1, 0, 0, 0, 1], [0, 1, 1, 0, 0, 1]], [[0,
1, 1, 0, 0, 1], [0, 0, 1, 1, 1, 0], [1, 0, 0, 1, 1, 0], [1, 1, 0, 0, 0, 1], [1,
1, 0, 0, 0, 1], [0, 0, 1, 1, 1, 0]], [[0, 1, 1, 0, 0, 1], [0, 1, 1, 0, 0, 1], [
0, 0, 0, 1, 1, 1], [1, 0, 0, 1, 1, 0], [1, 0, 0, 1, 1, 0], [1, 1, 1, 0, 0, 0]],
[[0, 1, 1, 0, 0, 1], [0, 1, 1, 0, 0, 1], [0, 0, 1, 1, 1, 0], [1, 0, 0, 1, 1, 0]
, [1, 0, 0, 1, 1, 0], [1, 1, 0, 0, 0, 1]], [[0, 1, 1, 0, 0, 1], [0, 1, 1, 0, 0,
1], [0, 1, 1, 0, 0, 1], [1, 0, 0, 1, 1, 0], [1, 0, 0, 1, 1, 0], [1, 0, 0, 1, 1,
0]], [[0, 1, 1, 0, 0, 1], [0, 1, 1, 0, 0, 1], [0, 1, 1, 1, 0, 0], [1, 0, 0, 1,
1, 0], [1, 0, 0, 1, 1, 0], [1, 0, 0, 0, 1, 1]], [[0, 1, 1, 0, 0, 1], [0, 1, 1,
0, 0, 1], [1, 0, 0, 0, 1, 1], [1, 0, 0, 1, 1, 0], [1, 0, 0, 1, 1, 0], [0, 1, 1,
1, 0, 0]], [[0, 1, 1, 0, 0, 1], [0, 1, 1, 0, 0, 1], [1, 0, 0, 1, 1, 0], [1, 0,
0, 1, 1, 0], [1, 0, 0, 1, 1, 0], [0, 1, 1, 0, 0, 1]], [[0, 1, 1, 0, 0, 1], [0,
1, 1, 0, 0, 1], [1, 1, 0, 0, 0, 1], [1, 0, 0, 1, 1, 0], [1, 0, 0, 1, 1, 0], [0,
0, 1, 1, 1, 0]], [[0, 1, 1, 0, 0, 1], [0, 1, 1, 0, 0, 1], [1, 1, 1, 0, 0, 0], [
1, 0, 0, 1, 1, 0], [1, 0, 0, 1, 1, 0], [0, 0, 0, 1, 1, 1]], [[0, 1, 1, 0, 0, 1]
, [0, 1, 1, 1, 0, 0], [0, 0, 1, 1, 1, 0], [1, 0, 0, 1, 1, 0], [1, 0, 0, 0, 1, 1
], [1, 1, 0, 0, 0, 1]], [[0, 1, 1, 0, 0, 1], [0, 1, 1, 1, 0, 0], [0, 1, 1, 1, 0
, 0], [1, 0, 0, 0, 1, 1], [1, 0, 0, 0, 1, 1], [1, 0, 0, 1, 1, 0]], [[0, 1, 1, 0
, 0, 1], [0, 1, 1, 1, 0, 0], [0, 1, 1, 1, 0, 0], [1, 0, 0, 1, 1, 0], [1, 0, 0,
0, 1, 1], [1, 0, 0, 0, 1, 1]], [[0, 1, 1, 0, 0, 1], [0, 1, 1, 1, 0, 0], [1, 0,
0, 1, 1, 0], [1, 0, 0, 0, 1, 1], [1, 0, 0, 0, 1, 1], [0, 1, 1, 1, 0, 0]], [[0,
1, 1, 0, 0, 1], [0, 1, 1, 1, 0, 0], [1, 0, 0, 1, 1, 0], [1, 0, 0, 1, 1, 0], [1,
0, 0, 0, 1, 1], [0, 1, 1, 0, 0, 1]], [[0, 1, 1, 0, 0, 1], [1, 0, 0, 0, 1, 1], [
1, 0, 0, 0, 1, 1], [0, 1, 1, 1, 0, 0], [0, 1, 1, 1, 0, 0], [1, 0, 0, 1, 1, 0]],
[[0, 1, 1, 0, 0, 1], [1, 0, 0, 0, 1, 1], [1, 0, 0, 0, 1, 1], [1, 0, 0, 1, 1, 0]
, [0, 1, 1, 1, 0, 0], [0, 1, 1, 1, 0, 0]], [[0, 1, 1, 0, 0, 1], [1, 0, 0, 0, 1,
1], [1, 0, 0, 1, 1, 0], [0, 0, 1, 1, 1, 0], [0, 1, 1, 1, 0, 0], [1, 1, 0, 0, 0,
1]], [[0, 1, 1, 0, 0, 1], [1, 0, 0, 0, 1, 1], [1, 0, 0, 1, 1, 0], [0, 1, 1, 1,
0, 0], [0, 1, 1, 1, 0, 0], [1, 0, 0, 0, 1, 1]], [[0, 1, 1, 0, 0, 1], [1, 0, 0,
0, 1, 1], [1, 0, 0, 1, 1, 0], [1, 0, 0, 1, 1, 0], [0, 1, 1, 1, 0, 0], [0, 1, 1,
0, 0, 1]], [[0, 1, 1, 0, 0, 1], [1, 0, 0, 1, 1, 0], [1, 0, 0, 1, 1, 0], [0, 0,
0, 1, 1, 1], [0, 1, 1, 0, 0, 1], [1, 1, 1, 0, 0, 0]], [[0, 1, 1, 0, 0, 1], [1,
0, 0, 1, 1, 0], [1, 0, 0, 1, 1, 0], [0, 0, 1, 1, 1, 0], [0, 1, 1, 0, 0, 1], [1,
1, 0, 0, 0, 1]], [[0, 1, 1, 0, 0, 1], [1, 0, 0, 1, 1, 0], [1, 0, 0, 1, 1, 0], [
0, 1, 1, 0, 0, 1], [0, 1, 1, 0, 0, 1], [1, 0, 0, 1, 1, 0]], [[0, 1, 1, 0, 0, 1]
, [1, 0, 0, 1, 1, 0], [1, 0, 0, 1, 1, 0], [0, 1, 1, 1, 0, 0], [0, 1, 1, 0, 0, 1
], [1, 0, 0, 0, 1, 1]], [[0, 1, 1, 0, 0, 1], [1, 0, 0, 1, 1, 0], [1, 0, 0, 1, 1
, 0], [1, 0, 0, 0, 1, 1], [0, 1, 1, 0, 0, 1], [0, 1, 1, 1, 0, 0]], [[0, 1, 1, 0
, 0, 1], [1, 0, 0, 1, 1, 0], [1, 0, 0, 1, 1, 0], [1, 0, 0, 1, 1, 0], [0, 1, 1,
0, 0, 1], [0, 1, 1, 0, 0, 1]], [[0, 1, 1, 0, 0, 1], [1, 0, 0, 1, 1, 0], [1, 0,
0, 1, 1, 0], [1, 1, 0, 0, 0, 1], [0, 1, 1, 0, 0, 1], [0, 0, 1, 1, 1, 0]], [[0,
1, 1, 0, 0, 1], [1, 0, 0, 1, 1, 0], [1, 0, 0, 1, 1, 0], [1, 1, 1, 0, 0, 0], [0,
1, 1, 0, 0, 1], [0, 0, 0, 1, 1, 1]], [[0, 1, 1, 0, 0, 1], [1, 1, 0, 0, 0, 1], [
1, 0, 0, 0, 1, 1], [1, 0, 0, 1, 1, 0], [0, 0, 1, 1, 1, 0], [0, 1, 1, 1, 0, 0]],
[[0, 1, 1, 0, 0, 1], [1, 1, 0, 0, 0, 1], [1, 0, 0, 1, 1, 0], [0, 0, 1, 1, 1, 0]
, [0, 0, 1, 1, 1, 0], [1, 1, 0, 0, 0, 1]], [[0, 1, 1, 0, 0, 1], [1, 1, 0, 0, 0,
1], [1, 0, 0, 1, 1, 0], [1, 0, 0, 1, 1, 0], [0, 0, 1, 1, 1, 0], [0, 1, 1, 0, 0,
1]], [[0, 1, 1, 0, 0, 1], [1, 1, 0, 0, 0, 1], [1, 1, 0, 0, 0, 1], [0, 0, 1, 1,
1, 0], [0, 0, 1, 1, 1, 0], [1, 0, 0, 1, 1, 0]], [[0, 1, 1, 0, 0, 1], [1, 1, 0,
0, 0, 1], [1, 1, 0, 0, 0, 1], [1, 0, 0, 1, 1, 0], [0, 0, 1, 1, 1, 0], [0, 0, 1,
1, 1, 0]], [[0, 1, 1, 0, 0, 1], [1, 1, 1, 0, 0, 0], [1, 0, 0, 1, 1, 0], [0, 0,
0, 1, 1, 1], [0, 0, 0, 1, 1, 1], [1, 1, 1, 0, 0, 0]], [[0, 1, 1, 0, 0, 1], [1,
1, 1, 0, 0, 0], [1, 0, 0, 1, 1, 0], [1, 0, 0, 1, 1, 0], [0, 0, 0, 1, 1, 1], [0,
1, 1, 0, 0, 1]], [[0, 1, 1, 0, 0, 1], [1, 1, 1, 0, 0, 0], [1, 1, 1, 0, 0, 0], [
0, 0, 0, 1, 1, 1], [0, 0, 0, 1, 1, 1], [1, 0, 0, 1, 1, 0]], [[0, 1, 1, 0, 0, 1]
, [1, 1, 1, 0, 0, 0], [1, 1, 1, 0, 0, 0], [1, 0, 0, 1, 1, 0], [0, 0, 0, 1, 1, 1
], [0, 0, 0, 1, 1, 1]], [[0, 1, 1, 1, 0, 0], [0, 0, 0, 1, 1, 1], [0, 0, 0, 1, 1
, 1], [1, 0, 0, 0, 1, 1], [1, 1, 1, 0, 0, 0], [1, 1, 1, 0, 0, 0]], [[0, 1, 1, 1
, 0, 0], [0, 0, 0, 1, 1, 1], [0, 0, 0, 1, 1, 1], [1, 1, 1, 0, 0, 0], [1, 1, 1,
0, 0, 0], [1, 0, 0, 0, 1, 1]], [[0, 1, 1, 1, 0, 0], [0, 0, 0, 1, 1, 1], [1, 0,
0, 0, 1, 1], [1, 0, 0, 0, 1, 1], [1, 1, 1, 0, 0, 0], [0, 1, 1, 1, 0, 0]], [[0,
1, 1, 1, 0, 0], [0, 0, 0, 1, 1, 1], [1, 0, 0, 0, 1, 1], [1, 1, 0, 0, 0, 1], [1,
1, 1, 0, 0, 0], [0, 0, 1, 1, 1, 0]], [[0, 1, 1, 1, 0, 0], [0, 0, 0, 1, 1, 1], [
1, 0, 0, 0, 1, 1], [1, 1, 1, 0, 0, 0], [1, 1, 1, 0, 0, 0], [0, 0, 0, 1, 1, 1]],
[[0, 1, 1, 1, 0, 0], [0, 0, 1, 1, 1, 0], [0, 0, 0, 1, 1, 1], [1, 0, 0, 0, 1, 1]
, [1, 1, 0, 0, 0, 1], [1, 1, 1, 0, 0, 0]], [[0, 1, 1, 1, 0, 0], [0, 0, 1, 1, 1,
0], [0, 0, 1, 1, 1, 0], [1, 0, 0, 0, 1, 1], [1, 1, 0, 0, 0, 1], [1, 1, 0, 0, 0,
1]], [[0, 1, 1, 1, 0, 0], [0, 0, 1, 1, 1, 0], [0, 0, 1, 1, 1, 0], [1, 1, 0, 0,
0, 1], [1, 1, 0, 0, 0, 1], [1, 0, 0, 0, 1, 1]], [[0, 1, 1, 1, 0, 0], [0, 0, 1,
1, 1, 0], [1, 0, 0, 0, 1, 1], [1, 0, 0, 0, 1, 1], [1, 1, 0, 0, 0, 1], [0, 1, 1,
1, 0, 0]], [[0, 1, 1, 1, 0, 0], [0, 0, 1, 1, 1, 0], [1, 0, 0, 0, 1, 1], [1, 1,
0, 0, 0, 1], [1, 1, 0, 0, 0, 1], [0, 0, 1, 1, 1, 0]], [[0, 1, 1, 1, 0, 0], [0,
0, 1, 1, 1, 0], [1, 0, 0, 1, 1, 0], [1, 0, 0, 0, 1, 1], [1, 1, 0, 0, 0, 1], [0,
1, 1, 0, 0, 1]], [[0, 1, 1, 1, 0, 0], [0, 1, 1, 0, 0, 1], [0, 1, 1, 0, 0, 1], [
1, 0, 0, 0, 1, 1], [1, 0, 0, 1, 1, 0], [1, 0, 0, 1, 1, 0]], [[0, 1, 1, 1, 0, 0]
, [0, 1, 1, 0, 0, 1], [0, 1, 1, 0, 0, 1], [1, 0, 0, 1, 1, 0], [1, 0, 0, 1, 1, 0
], [1, 0, 0, 0, 1, 1]], [[0, 1, 1, 1, 0, 0], [0, 1, 1, 0, 0, 1], [1, 0, 0, 0, 1
, 1], [1, 0, 0, 0, 1, 1], [1, 0, 0, 1, 1, 0], [0, 1, 1, 1, 0, 0]], [[0, 1, 1, 1
, 0, 0], [0, 1, 1, 0, 0, 1], [1, 0, 0, 0, 1, 1], [1, 0, 0, 1, 1, 0], [1, 0, 0,
1, 1, 0], [0, 1, 1, 0, 0, 1]], [[0, 1, 1, 1, 0, 0], [0, 1, 1, 0, 0, 1], [1, 1,
0, 0, 0, 1], [1, 0, 0, 0, 1, 1], [1, 0, 0, 1, 1, 0], [0, 0, 1, 1, 1, 0]], [[0,
1, 1, 1, 0, 0], [0, 1, 1, 1, 0, 0], [0, 0, 0, 1, 1, 1], [1, 0, 0, 0, 1, 1], [1,
0, 0, 0, 1, 1], [1, 1, 1, 0, 0, 0]], [[0, 1, 1, 1, 0, 0], [0, 1, 1, 1, 0, 0], [
0, 0, 1, 1, 1, 0], [1, 0, 0, 0, 1, 1], [1, 0, 0, 0, 1, 1], [1, 1, 0, 0, 0, 1]],
[[0, 1, 1, 1, 0, 0], [0, 1, 1, 1, 0, 0], [0, 1, 1, 0, 0, 1], [1, 0, 0, 0, 1, 1]
, [1, 0, 0, 0, 1, 1], [1, 0, 0, 1, 1, 0]], [[0, 1, 1, 1, 0, 0], [0, 1, 1, 1, 0,
0], [0, 1, 1, 1, 0, 0], [1, 0, 0, 0, 1, 1], [1, 0, 0, 0, 1, 1], [1, 0, 0, 0, 1,
1]], [[0, 1, 1, 1, 0, 0], [0, 1, 1, 1, 0, 0], [1, 0, 0, 0, 1, 1], [1, 0, 0, 0,
1, 1], [1, 0, 0, 0, 1, 1], [0, 1, 1, 1, 0, 0]], [[0, 1, 1, 1, 0, 0], [0, 1, 1,
1, 0, 0], [1, 0, 0, 1, 1, 0], [1, 0, 0, 0, 1, 1], [1, 0, 0, 0, 1, 1], [0, 1, 1,
0, 0, 1]], [[0, 1, 1, 1, 0, 0], [0, 1, 1, 1, 0, 0], [1, 1, 0, 0, 0, 1], [1, 0,
0, 0, 1, 1], [1, 0, 0, 0, 1, 1], [0, 0, 1, 1, 1, 0]], [[0, 1, 1, 1, 0, 0], [0,
1, 1, 1, 0, 0], [1, 1, 1, 0, 0, 0], [1, 0, 0, 0, 1, 1], [1, 0, 0, 0, 1, 1], [0,
0, 0, 1, 1, 1]], [[0, 1, 1, 1, 0, 0], [1, 0, 0, 0, 1, 1], [1, 0, 0, 0, 1, 1], [
0, 0, 0, 1, 1, 1], [0, 1, 1, 1, 0, 0], [1, 1, 1, 0, 0, 0]], [[0, 1, 1, 1, 0, 0]
, [1, 0, 0, 0, 1, 1], [1, 0, 0, 0, 1, 1], [0, 0, 1, 1, 1, 0], [0, 1, 1, 1, 0, 0
], [1, 1, 0, 0, 0, 1]], [[0, 1, 1, 1, 0, 0], [1, 0, 0, 0, 1, 1], [1, 0, 0, 0, 1
, 1], [0, 1, 1, 0, 0, 1], [0, 1, 1, 1, 0, 0], [1, 0, 0, 1, 1, 0]], [[0, 1, 1, 1
, 0, 0], [1, 0, 0, 0, 1, 1], [1, 0, 0, 0, 1, 1], [0, 1, 1, 1, 0, 0], [0, 1, 1,
1, 0, 0], [1, 0, 0, 0, 1, 1]], [[0, 1, 1, 1, 0, 0], [1, 0, 0, 0, 1, 1], [1, 0,
0, 0, 1, 1], [1, 0, 0, 0, 1, 1], [0, 1, 1, 1, 0, 0], [0, 1, 1, 1, 0, 0]], [[0,
1, 1, 1, 0, 0], [1, 0, 0, 0, 1, 1], [1, 0, 0, 0, 1, 1], [1, 0, 0, 1, 1, 0], [0,
1, 1, 1, 0, 0], [0, 1, 1, 0, 0, 1]], [[0, 1, 1, 1, 0, 0], [1, 0, 0, 0, 1, 1], [
1, 0, 0, 0, 1, 1], [1, 1, 0, 0, 0, 1], [0, 1, 1, 1, 0, 0], [0, 0, 1, 1, 1, 0]],
[[0, 1, 1, 1, 0, 0], [1, 0, 0, 0, 1, 1], [1, 0, 0, 0, 1, 1], [1, 1, 1, 0, 0, 0]
, [0, 1, 1, 1, 0, 0], [0, 0, 0, 1, 1, 1]], [[0, 1, 1, 1, 0, 0], [1, 0, 0, 1, 1,
0], [1, 0, 0, 0, 1, 1], [0, 1, 1, 0, 0, 1], [0, 1, 1, 0, 0, 1], [1, 0, 0, 1, 1,
0]], [[0, 1, 1, 1, 0, 0], [1, 0, 0, 1, 1, 0], [1, 0, 0, 0, 1, 1], [1, 0, 0, 0,
1, 1], [0, 1, 1, 0, 0, 1], [0, 1, 1, 1, 0, 0]], [[0, 1, 1, 1, 0, 0], [1, 0, 0,
1, 1, 0], [1, 0, 0, 0, 1, 1], [1, 1, 0, 0, 0, 1], [0, 1, 1, 0, 0, 1], [0, 0, 1,
1, 1, 0]], [[0, 1, 1, 1, 0, 0], [1, 0, 0, 1, 1, 0], [1, 0, 0, 1, 1, 0], [0, 1,
1, 0, 0, 1], [0, 1, 1, 0, 0, 1], [1, 0, 0, 0, 1, 1]], [[0, 1, 1, 1, 0, 0], [1,
0, 0, 1, 1, 0], [1, 0, 0, 1, 1, 0], [1, 0, 0, 0, 1, 1], [0, 1, 1, 0, 0, 1], [0,
1, 1, 0, 0, 1]], [[0, 1, 1, 1, 0, 0], [1, 1, 0, 0, 0, 1], [1, 0, 0, 0, 1, 1], [
0, 0, 0, 1, 1, 1], [0, 0, 1, 1, 1, 0], [1, 1, 1, 0, 0, 0]], [[0, 1, 1, 1, 0, 0]
, [1, 1, 0, 0, 0, 1], [1, 0, 0, 0, 1, 1], [0, 0, 1, 1, 1, 0], [0, 0, 1, 1, 1, 0
], [1, 1, 0, 0, 0, 1]], [[0, 1, 1, 1, 0, 0], [1, 1, 0, 0, 0, 1], [1, 0, 0, 0, 1
, 1], [1, 0, 0, 0, 1, 1], [0, 0, 1, 1, 1, 0], [0, 1, 1, 1, 0, 0]], [[0, 1, 1, 1
, 0, 0], [1, 1, 0, 0, 0, 1], [1, 0, 0, 0, 1, 1], [1, 0, 0, 1, 1, 0], [0, 0, 1,
1, 1, 0], [0, 1, 1, 0, 0, 1]], [[0, 1, 1, 1, 0, 0], [1, 1, 0, 0, 0, 1], [1, 1,
0, 0, 0, 1], [0, 0, 1, 1, 1, 0], [0, 0, 1, 1, 1, 0], [1, 0, 0, 0, 1, 1]], [[0,
1, 1, 1, 0, 0], [1, 1, 0, 0, 0, 1], [1, 1, 0, 0, 0, 1], [1, 0, 0, 0, 1, 1], [0,
0, 1, 1, 1, 0], [0, 0, 1, 1, 1, 0]], [[0, 1, 1, 1, 0, 0], [1, 1, 1, 0, 0, 0], [
1, 0, 0, 0, 1, 1], [0, 0, 0, 1, 1, 1], [0, 0, 0, 1, 1, 1], [1, 1, 1, 0, 0, 0]],
[[0, 1, 1, 1, 0, 0], [1, 1, 1, 0, 0, 0], [1, 0, 0, 0, 1, 1], [1, 0, 0, 0, 1, 1]
, [0, 0, 0, 1, 1, 1], [0, 1, 1, 1, 0, 0]], [[0, 1, 1, 1, 0, 0], [1, 1, 1, 0, 0,
0], [1, 1, 0, 0, 0, 1], [1, 0, 0, 0, 1, 1], [0, 0, 0, 1, 1, 1], [0, 0, 1, 1, 1,
0]], [[0, 1, 1, 1, 0, 0], [1, 1, 1, 0, 0, 0], [1, 1, 1, 0, 0, 0], [0, 0, 0, 1,
1, 1], [0, 0, 0, 1, 1, 1], [1, 0, 0, 0, 1, 1]], [[0, 1, 1, 1, 0, 0], [1, 1, 1,
0, 0, 0], [1, 1, 1, 0, 0, 0], [1, 0, 0, 0, 1, 1], [0, 0, 0, 1, 1, 1], [0, 0, 0,
1, 1, 1]], [[1, 0, 0, 0, 1, 1], [0, 0, 0, 1, 1, 1], [0, 0, 0, 1, 1, 1], [0, 1,
1, 1, 0, 0], [1, 1, 1, 0, 0, 0], [1, 1, 1, 0, 0, 0]], [[1, 0, 0, 0, 1, 1], [0,
0, 0, 1, 1, 1], [0, 0, 0, 1, 1, 1], [1, 1, 1, 0, 0, 0], [1, 1, 1, 0, 0, 0], [0,
1, 1, 1, 0, 0]], [[1, 0, 0, 0, 1, 1], [0, 0, 0, 1, 1, 1], [0, 0, 1, 1, 1, 0], [
0, 1, 1, 1, 0, 0], [1, 1, 1, 0, 0, 0], [1, 1, 0, 0, 0, 1]], [[1, 0, 0, 0, 1, 1]
, [0, 0, 0, 1, 1, 1], [0, 1, 1, 1, 0, 0], [0, 1, 1, 1, 0, 0], [1, 1, 1, 0, 0, 0
], [1, 0, 0, 0, 1, 1]], [[1, 0, 0, 0, 1, 1], [0, 0, 0, 1, 1, 1], [0, 1, 1, 1, 0
, 0], [1, 1, 1, 0, 0, 0], [1, 1, 1, 0, 0, 0], [0, 0, 0, 1, 1, 1]], [[1, 0, 0, 0
, 1, 1], [0, 0, 1, 1, 1, 0], [0, 0, 1, 1, 1, 0], [0, 1, 1, 1, 0, 0], [1, 1, 0,
0, 0, 1], [1, 1, 0, 0, 0, 1]], [[1, 0, 0, 0, 1, 1], [0, 0, 1, 1, 1, 0], [0, 0,
1, 1, 1, 0], [1, 1, 0, 0, 0, 1], [1, 1, 0, 0, 0, 1], [0, 1, 1, 1, 0, 0]], [[1,
0, 0, 0, 1, 1], [0, 0, 1, 1, 1, 0], [0, 1, 1, 1, 0, 0], [0, 1, 1, 0, 0, 1], [1,
1, 0, 0, 0, 1], [1, 0, 0, 1, 1, 0]], [[1, 0, 0, 0, 1, 1], [0, 0, 1, 1, 1, 0], [
0, 1, 1, 1, 0, 0], [0, 1, 1, 1, 0, 0], [1, 1, 0, 0, 0, 1], [1, 0, 0, 0, 1, 1]],
[[1, 0, 0, 0, 1, 1], [0, 0, 1, 1, 1, 0], [0, 1, 1, 1, 0, 0], [1, 1, 0, 0, 0, 1]
, [1, 1, 0, 0, 0, 1], [0, 0, 1, 1, 1, 0]], [[1, 0, 0, 0, 1, 1], [0, 0, 1, 1, 1,
0], [0, 1, 1, 1, 0, 0], [1, 1, 1, 0, 0, 0], [1, 1, 0, 0, 0, 1], [0, 0, 0, 1, 1,
1]], [[1, 0, 0, 0, 1, 1], [0, 1, 1, 0, 0, 1], [0, 1, 1, 0, 0, 1], [0, 1, 1, 1,
0, 0], [1, 0, 0, 1, 1, 0], [1, 0, 0, 1, 1, 0]], [[1, 0, 0, 0, 1, 1], [0, 1, 1,
0, 0, 1], [0, 1, 1, 0, 0, 1], [1, 0, 0, 1, 1, 0], [1, 0, 0, 1, 1, 0], [0, 1, 1,
1, 0, 0]], [[1, 0, 0, 0, 1, 1], [0, 1, 1, 0, 0, 1], [0, 1, 1, 1, 0, 0], [0, 0,
1, 1, 1, 0], [1, 0, 0, 1, 1, 0], [1, 1, 0, 0, 0, 1]], [[1, 0, 0, 0, 1, 1], [0,
1, 1, 0, 0, 1], [0, 1, 1, 1, 0, 0], [0, 1, 1, 1, 0, 0], [1, 0, 0, 1, 1, 0], [1,
0, 0, 0, 1, 1]], [[1, 0, 0, 0, 1, 1], [0, 1, 1, 0, 0, 1], [0, 1, 1, 1, 0, 0], [
1, 0, 0, 1, 1, 0], [1, 0, 0, 1, 1, 0], [0, 1, 1, 0, 0, 1]], [[1, 0, 0, 0, 1, 1]
, [0, 1, 1, 1, 0, 0], [0, 1, 1, 1, 0, 0], [0, 0, 0, 1, 1, 1], [1, 0, 0, 0, 1, 1
], [1, 1, 1, 0, 0, 0]], [[1, 0, 0, 0, 1, 1], [0, 1, 1, 1, 0, 0], [0, 1, 1, 1, 0
, 0], [0, 0, 1, 1, 1, 0], [1, 0, 0, 0, 1, 1], [1, 1, 0, 0, 0, 1]], [[1, 0, 0, 0
, 1, 1], [0, 1, 1, 1, 0, 0], [0, 1, 1, 1, 0, 0], [0, 1, 1, 0, 0, 1], [1, 0, 0,
0, 1, 1], [1, 0, 0, 1, 1, 0]], [[1, 0, 0, 0, 1, 1], [0, 1, 1, 1, 0, 0], [0, 1,
1, 1, 0, 0], [0, 1, 1, 1, 0, 0], [1, 0, 0, 0, 1, 1], [1, 0, 0, 0, 1, 1]], [[1,
0, 0, 0, 1, 1], [0, 1, 1, 1, 0, 0], [0, 1, 1, 1, 0, 0], [1, 0, 0, 0, 1, 1], [1,
0, 0, 0, 1, 1], [0, 1, 1, 1, 0, 0]], [[1, 0, 0, 0, 1, 1], [0, 1, 1, 1, 0, 0], [
0, 1, 1, 1, 0, 0], [1, 0, 0, 1, 1, 0], [1, 0, 0, 0, 1, 1], [0, 1, 1, 0, 0, 1]],
[[1, 0, 0, 0, 1, 1], [0, 1, 1, 1, 0, 0], [0, 1, 1, 1, 0, 0], [1, 1, 0, 0, 0, 1]
, [1, 0, 0, 0, 1, 1], [0, 0, 1, 1, 1, 0]], [[1, 0, 0, 0, 1, 1], [0, 1, 1, 1, 0,
0], [0, 1, 1, 1, 0, 0], [1, 1, 1, 0, 0, 0], [1, 0, 0, 0, 1, 1], [0, 0, 0, 1, 1,
1]], [[1, 0, 0, 0, 1, 1], [1, 0, 0, 0, 1, 1], [0, 0, 0, 1, 1, 1], [0, 1, 1, 1,
0, 0], [0, 1, 1, 1, 0, 0], [1, 1, 1, 0, 0, 0]], [[1, 0, 0, 0, 1, 1], [1, 0, 0,
0, 1, 1], [0, 0, 1, 1, 1, 0], [0, 1, 1, 1, 0, 0], [0, 1, 1, 1, 0, 0], [1, 1, 0,
0, 0, 1]], [[1, 0, 0, 0, 1, 1], [1, 0, 0, 0, 1, 1], [0, 1, 1, 0, 0, 1], [0, 1,
1, 1, 0, 0], [0, 1, 1, 1, 0, 0], [1, 0, 0, 1, 1, 0]], [[1, 0, 0, 0, 1, 1], [1,
0, 0, 0, 1, 1], [0, 1, 1, 1, 0, 0], [0, 1, 1, 1, 0, 0], [0, 1, 1, 1, 0, 0], [1,
0, 0, 0, 1, 1]], [[1, 0, 0, 0, 1, 1], [1, 0, 0, 0, 1, 1], [1, 0, 0, 0, 1, 1], [
0, 1, 1, 1, 0, 0], [0, 1, 1, 1, 0, 0], [0, 1, 1, 1, 0, 0]], [[1, 0, 0, 0, 1, 1]
, [1, 0, 0, 0, 1, 1], [1, 0, 0, 1, 1, 0], [0, 1, 1, 1, 0, 0], [0, 1, 1, 1, 0, 0
], [0, 1, 1, 0, 0, 1]], [[1, 0, 0, 0, 1, 1], [1, 0, 0, 0, 1, 1], [1, 1, 0, 0, 0
, 1], [0, 1, 1, 1, 0, 0], [0, 1, 1, 1, 0, 0], [0, 0, 1, 1, 1, 0]], [[1, 0, 0, 0
, 1, 1], [1, 0, 0, 0, 1, 1], [1, 1, 1, 0, 0, 0], [0, 1, 1, 1, 0, 0], [0, 1, 1,
1, 0, 0], [0, 0, 0, 1, 1, 1]], [[1, 0, 0, 0, 1, 1], [1, 0, 0, 1, 1, 0], [0, 0,
1, 1, 1, 0], [0, 1, 1, 1, 0, 0], [0, 1, 1, 0, 0, 1], [1, 1, 0, 0, 0, 1]], [[1,
0, 0, 0, 1, 1], [1, 0, 0, 1, 1, 0], [0, 1, 1, 1, 0, 0], [0, 1, 1, 0, 0, 1], [0,
1, 1, 0, 0, 1], [1, 0, 0, 1, 1, 0]], [[1, 0, 0, 0, 1, 1], [1, 0, 0, 1, 1, 0], [
0, 1, 1, 1, 0, 0], [0, 1, 1, 1, 0, 0], [0, 1, 1, 0, 0, 1], [1, 0, 0, 0, 1, 1]],
[[1, 0, 0, 0, 1, 1], [1, 0, 0, 1, 1, 0], [1, 0, 0, 1, 1, 0], [0, 1, 1, 0, 0, 1]
, [0, 1, 1, 0, 0, 1], [0, 1, 1, 1, 0, 0]], [[1, 0, 0, 0, 1, 1], [1, 0, 0, 1, 1,
0], [1, 0, 0, 1, 1, 0], [0, 1, 1, 1, 0, 0], [0, 1, 1, 0, 0, 1], [0, 1, 1, 0, 0,
1]], [[1, 0, 0, 0, 1, 1], [1, 1, 0, 0, 0, 1], [0, 1, 1, 0, 0, 1], [0, 1, 1, 1,
0, 0], [0, 0, 1, 1, 1, 0], [1, 0, 0, 1, 1, 0]], [[1, 0, 0, 0, 1, 1], [1, 1, 0,
0, 0, 1], [0, 1, 1, 1, 0, 0], [0, 0, 1, 1, 1, 0], [0, 0, 1, 1, 1, 0], [1, 1, 0,
0, 0, 1]], [[1, 0, 0, 0, 1, 1], [1, 1, 0, 0, 0, 1], [0, 1, 1, 1, 0, 0], [0, 1,
1, 1, 0, 0], [0, 0, 1, 1, 1, 0], [1, 0, 0, 0, 1, 1]], [[1, 0, 0, 0, 1, 1], [1,
1, 0, 0, 0, 1], [1, 1, 0, 0, 0, 1], [0, 0, 1, 1, 1, 0], [0, 0, 1, 1, 1, 0], [0,
1, 1, 1, 0, 0]], [[1, 0, 0, 0, 1, 1], [1, 1, 0, 0, 0, 1], [1, 1, 0, 0, 0, 1], [
0, 1, 1, 1, 0, 0], [0, 0, 1, 1, 1, 0], [0, 0, 1, 1, 1, 0]], [[1, 0, 0, 0, 1, 1]
, [1, 1, 0, 0, 0, 1], [1, 1, 1, 0, 0, 0], [0, 1, 1, 1, 0, 0], [0, 0, 1, 1, 1, 0
], [0, 0, 0, 1, 1, 1]], [[1, 0, 0, 0, 1, 1], [1, 1, 1, 0, 0, 0], [0, 1, 1, 1, 0
, 0], [0, 0, 0, 1, 1, 1], [0, 0, 0, 1, 1, 1], [1, 1, 1, 0, 0, 0]], [[1, 0, 0, 0
, 1, 1], [1, 1, 1, 0, 0, 0], [0, 1, 1, 1, 0, 0], [0, 0, 1, 1, 1, 0], [0, 0, 0,
1, 1, 1], [1, 1, 0, 0, 0, 1]], [[1, 0, 0, 0, 1, 1], [1, 1, 1, 0, 0, 0], [0, 1,
1, 1, 0, 0], [0, 1, 1, 1, 0, 0], [0, 0, 0, 1, 1, 1], [1, 0, 0, 0, 1, 1]], [[1,
0, 0, 0, 1, 1], [1, 1, 1, 0, 0, 0], [1, 1, 1, 0, 0, 0], [0, 0, 0, 1, 1, 1], [0,
0, 0, 1, 1, 1], [0, 1, 1, 1, 0, 0]], [[1, 0, 0, 0, 1, 1], [1, 1, 1, 0, 0, 0], [
1, 1, 1, 0, 0, 0], [0, 1, 1, 1, 0, 0], [0, 0, 0, 1, 1, 1], [0, 0, 0, 1, 1, 1]],
[[1, 0, 0, 1, 1, 0], [0, 0, 0, 1, 1, 1], [0, 0, 0, 1, 1, 1], [0, 1, 1, 0, 0, 1]
, [1, 1, 1, 0, 0, 0], [1, 1, 1, 0, 0, 0]], [[1, 0, 0, 1, 1, 0], [0, 0, 0, 1, 1,
1], [0, 0, 0, 1, 1, 1], [1, 1, 1, 0, 0, 0], [1, 1, 1, 0, 0, 0], [0, 1, 1, 0, 0,
1]], [[1, 0, 0, 1, 1, 0], [0, 0, 0, 1, 1, 1], [0, 1, 1, 0, 0, 1], [0, 1, 1, 0,
0, 1], [1, 1, 1, 0, 0, 0], [1, 0, 0, 1, 1, 0]], [[1, 0, 0, 1, 1, 0], [0, 0, 0,
1, 1, 1], [0, 1, 1, 0, 0, 1], [1, 1, 1, 0, 0, 0], [1, 1, 1, 0, 0, 0], [0, 0, 0,
1, 1, 1]], [[1, 0, 0, 1, 1, 0], [0, 0, 1, 1, 1, 0], [0, 0, 1, 1, 1, 0], [0, 1,
1, 0, 0, 1], [1, 1, 0, 0, 0, 1], [1, 1, 0, 0, 0, 1]], [[1, 0, 0, 1, 1, 0], [0,
0, 1, 1, 1, 0], [0, 0, 1, 1, 1, 0], [1, 1, 0, 0, 0, 1], [1, 1, 0, 0, 0, 1], [0,
1, 1, 0, 0, 1]], [[1, 0, 0, 1, 1, 0], [0, 0, 1, 1, 1, 0], [0, 1, 1, 0, 0, 1], [
0, 1, 1, 0, 0, 1], [1, 1, 0, 0, 0, 1], [1, 0, 0, 1, 1, 0]], [[1, 0, 0, 1, 1, 0]
, [0, 0, 1, 1, 1, 0], [0, 1, 1, 0, 0, 1], [1, 1, 0, 0, 0, 1], [1, 1, 0, 0, 0, 1
], [0, 0, 1, 1, 1, 0]], [[1, 0, 0, 1, 1, 0], [0, 0, 1, 1, 1, 0], [0, 1, 1, 1, 0
, 0], [0, 1, 1, 0, 0, 1], [1, 1, 0, 0, 0, 1], [1, 0, 0, 0, 1, 1]], [[1, 0, 0, 1
, 1, 0], [0, 1, 1, 0, 0, 1], [0, 1, 1, 0, 0, 1], [0, 0, 0, 1, 1, 1], [1, 0, 0,
1, 1, 0], [1, 1, 1, 0, 0, 0]], [[1, 0, 0, 1, 1, 0], [0, 1, 1, 0, 0, 1], [0, 1,
1, 0, 0, 1], [0, 0, 1, 1, 1, 0], [1, 0, 0, 1, 1, 0], [1, 1, 0, 0, 0, 1]], [[1,
0, 0, 1, 1, 0], [0, 1, 1, 0, 0, 1], [0, 1, 1, 0, 0, 1], [0, 1, 1, 0, 0, 1], [1,
0, 0, 1, 1, 0], [1, 0, 0, 1, 1, 0]], [[1, 0, 0, 1, 1, 0], [0, 1, 1, 0, 0, 1], [
0, 1, 1, 0, 0, 1], [0, 1, 1, 1, 0, 0], [1, 0, 0, 1, 1, 0], [1, 0, 0, 0, 1, 1]],
[[1, 0, 0, 1, 1, 0], [0, 1, 1, 0, 0, 1], [0, 1, 1, 0, 0, 1], [1, 0, 0, 0, 1, 1]
, [1, 0, 0, 1, 1, 0], [0, 1, 1, 1, 0, 0]], [[1, 0, 0, 1, 1, 0], [0, 1, 1, 0, 0,
1], [0, 1, 1, 0, 0, 1], [1, 0, 0, 1, 1, 0], [1, 0, 0, 1, 1, 0], [0, 1, 1, 0, 0,
1]], [[1, 0, 0, 1, 1, 0], [0, 1, 1, 0, 0, 1], [0, 1, 1, 0, 0, 1], [1, 1, 0, 0,
0, 1], [1, 0, 0, 1, 1, 0], [0, 0, 1, 1, 1, 0]], [[1, 0, 0, 1, 1, 0], [0, 1, 1,
0, 0, 1], [0, 1, 1, 0, 0, 1], [1, 1, 1, 0, 0, 0], [1, 0, 0, 1, 1, 0], [0, 0, 0,
1, 1, 1]], [[1, 0, 0, 1, 1, 0], [0, 1, 1, 1, 0, 0], [0, 1, 1, 0, 0, 1], [0, 1,
1, 0, 0, 1], [1, 0, 0, 0, 1, 1], [1, 0, 0, 1, 1, 0]], [[1, 0, 0, 1, 1, 0], [0,
1, 1, 1, 0, 0], [0, 1, 1, 0, 0, 1], [1, 0, 0, 0, 1, 1], [1, 0, 0, 0, 1, 1], [0,
1, 1, 1, 0, 0]], [[1, 0, 0, 1, 1, 0], [0, 1, 1, 1, 0, 0], [0, 1, 1, 0, 0, 1], [
1, 1, 0, 0, 0, 1], [1, 0, 0, 0, 1, 1], [0, 0, 1, 1, 1, 0]], [[1, 0, 0, 1, 1, 0]
, [0, 1, 1, 1, 0, 0], [0, 1, 1, 1, 0, 0], [0, 1, 1, 0, 0, 1], [1, 0, 0, 0, 1, 1
], [1, 0, 0, 0, 1, 1]], [[1, 0, 0, 1, 1, 0], [0, 1, 1, 1, 0, 0], [0, 1, 1, 1, 0
, 0], [1, 0, 0, 0, 1, 1], [1, 0, 0, 0, 1, 1], [0, 1, 1, 0, 0, 1]], [[1, 0, 0, 1
, 1, 0], [1, 0, 0, 0, 1, 1], [0, 1, 1, 0, 0, 1], [0, 1, 1, 0, 0, 1], [0, 1, 1,
1, 0, 0], [1, 0, 0, 1, 1, 0]], [[1, 0, 0, 1, 1, 0], [1, 0, 0, 0, 1, 1], [0, 1,
1, 0, 0, 1], [0, 1, 1, 1, 0, 0], [0, 1, 1, 1, 0, 0], [1, 0, 0, 0, 1, 1]], [[1,
0, 0, 1, 1, 0], [1, 0, 0, 0, 1, 1], [1, 0, 0, 0, 1, 1], [0, 1, 1, 0, 0, 1], [0,
1, 1, 1, 0, 0], [0, 1, 1, 1, 0, 0]], [[1, 0, 0, 1, 1, 0], [1, 0, 0, 0, 1, 1], [
1, 0, 0, 0, 1, 1], [0, 1, 1, 1, 0, 0], [0, 1, 1, 1, 0, 0], [0, 1, 1, 0, 0, 1]],
[[1, 0, 0, 1, 1, 0], [1, 0, 0, 0, 1, 1], [1, 1, 0, 0, 0, 1], [0, 1, 1, 0, 0, 1]
, [0, 1, 1, 1, 0, 0], [0, 0, 1, 1, 1, 0]], [[1, 0, 0, 1, 1, 0], [1, 0, 0, 1, 1,
0], [0, 0, 0, 1, 1, 1], [0, 1, 1, 0, 0, 1], [0, 1, 1, 0, 0, 1], [1, 1, 1, 0, 0,
0]], [[1, 0, 0, 1, 1, 0], [1, 0, 0, 1, 1, 0], [0, 0, 1, 1, 1, 0], [0, 1, 1, 0,
0, 1], [0, 1, 1, 0, 0, 1], [1, 1, 0, 0, 0, 1]], [[1, 0, 0, 1, 1, 0], [1, 0, 0,
1, 1, 0], [0, 1, 1, 0, 0, 1], [0, 1, 1, 0, 0, 1], [0, 1, 1, 0, 0, 1], [1, 0, 0,
1, 1, 0]], [[1, 0, 0, 1, 1, 0], [1, 0, 0, 1, 1, 0], [0, 1, 1, 1, 0, 0], [0, 1,
1, 0, 0, 1], [0, 1, 1, 0, 0, 1], [1, 0, 0, 0, 1, 1]], [[1, 0, 0, 1, 1, 0], [1,
0, 0, 1, 1, 0], [1, 0, 0, 0, 1, 1], [0, 1, 1, 0, 0, 1], [0, 1, 1, 0, 0, 1], [0,
1, 1, 1, 0, 0]], [[1, 0, 0, 1, 1, 0], [1, 0, 0, 1, 1, 0], [1, 0, 0, 1, 1, 0], [
0, 1, 1, 0, 0, 1], [0, 1, 1, 0, 0, 1], [0, 1, 1, 0, 0, 1]], [[1, 0, 0, 1, 1, 0]
, [1, 0, 0, 1, 1, 0], [1, 1, 0, 0, 0, 1], [0, 1, 1, 0, 0, 1], [0, 1, 1, 0, 0, 1
], [0, 0, 1, 1, 1, 0]], [[1, 0, 0, 1, 1, 0], [1, 0, 0, 1, 1, 0], [1, 1, 1, 0, 0
, 0], [0, 1, 1, 0, 0, 1], [0, 1, 1, 0, 0, 1], [0, 0, 0, 1, 1, 1]], [[1, 0, 0, 1
, 1, 0], [1, 1, 0, 0, 0, 1], [0, 1, 1, 0, 0, 1], [0, 0, 1, 1, 1, 0], [0, 0, 1,
1, 1, 0], [1, 1, 0, 0, 0, 1]], [[1, 0, 0, 1, 1, 0], [1, 1, 0, 0, 0, 1], [0, 1,
1, 0, 0, 1], [0, 1, 1, 0, 0, 1], [0, 0, 1, 1, 1, 0], [1, 0, 0, 1, 1, 0]], [[1,
0, 0, 1, 1, 0], [1, 1, 0, 0, 0, 1], [0, 1, 1, 0, 0, 1], [0, 1, 1, 1, 0, 0], [0,
0, 1, 1, 1, 0], [1, 0, 0, 0, 1, 1]], [[1, 0, 0, 1, 1, 0], [1, 1, 0, 0, 0, 1], [
1, 1, 0, 0, 0, 1], [0, 0, 1, 1, 1, 0], [0, 0, 1, 1, 1, 0], [0, 1, 1, 0, 0, 1]],
[[1, 0, 0, 1, 1, 0], [1, 1, 0, 0, 0, 1], [1, 1, 0, 0, 0, 1], [0, 1, 1, 0, 0, 1]
, [0, 0, 1, 1, 1, 0], [0, 0, 1, 1, 1, 0]], [[1, 0, 0, 1, 1, 0], [1, 1, 1, 0, 0,
0], [0, 1, 1, 0, 0, 1], [0, 0, 0, 1, 1, 1], [0, 0, 0, 1, 1, 1], [1, 1, 1, 0, 0,
0]], [[1, 0, 0, 1, 1, 0], [1, 1, 1, 0, 0, 0], [0, 1, 1, 0, 0, 1], [0, 1, 1, 0,
0, 1], [0, 0, 0, 1, 1, 1], [1, 0, 0, 1, 1, 0]], [[1, 0, 0, 1, 1, 0], [1, 1, 1,
0, 0, 0], [1, 1, 1, 0, 0, 0], [0, 0, 0, 1, 1, 1], [0, 0, 0, 1, 1, 1], [0, 1, 1,
0, 0, 1]], [[1, 0, 0, 1, 1, 0], [1, 1, 1, 0, 0, 0], [1, 1, 1, 0, 0, 0], [0, 1,
1, 0, 0, 1], [0, 0, 0, 1, 1, 1], [0, 0, 0, 1, 1, 1]], [[1, 1, 0, 0, 0, 1], [0,
0, 0, 1, 1, 1], [0, 0, 0, 1, 1, 1], [0, 0, 1, 1, 1, 0], [1, 1, 1, 0, 0, 0], [1,
1, 1, 0, 0, 0]], [[1, 1, 0, 0, 0, 1], [0, 0, 0, 1, 1, 1], [0, 0, 0, 1, 1, 1], [
1, 1, 1, 0, 0, 0], [1, 1, 1, 0, 0, 0], [0, 0, 1, 1, 1, 0]], [[1, 1, 0, 0, 0, 1]
, [0, 0, 0, 1, 1, 1], [0, 0, 1, 1, 1, 0], [0, 0, 1, 1, 1, 0], [1, 1, 1, 0, 0, 0
], [1, 1, 0, 0, 0, 1]], [[1, 1, 0, 0, 0, 1], [0, 0, 0, 1, 1, 1], [0, 0, 1, 1, 1
, 0], [0, 1, 1, 1, 0, 0], [1, 1, 1, 0, 0, 0], [1, 0, 0, 0, 1, 1]], [[1, 1, 0, 0
, 0, 1], [0, 0, 0, 1, 1, 1], [0, 0, 1, 1, 1, 0], [1, 1, 1, 0, 0, 0], [1, 1, 1,
0, 0, 0], [0, 0, 0, 1, 1, 1]], [[1, 1, 0, 0, 0, 1], [0, 0, 1, 1, 1, 0], [0, 0,
1, 1, 1, 0], [0, 0, 0, 1, 1, 1], [1, 1, 0, 0, 0, 1], [1, 1, 1, 0, 0, 0]], [[1,
1, 0, 0, 0, 1], [0, 0, 1, 1, 1, 0], [0, 0, 1, 1, 1, 0], [0, 0, 1, 1, 1, 0], [1,
1, 0, 0, 0, 1], [1, 1, 0, 0, 0, 1]], [[1, 1, 0, 0, 0, 1], [0, 0, 1, 1, 1, 0], [
0, 0, 1, 1, 1, 0], [0, 1, 1, 0, 0, 1], [1, 1, 0, 0, 0, 1], [1, 0, 0, 1, 1, 0]],
[[1, 1, 0, 0, 0, 1], [0, 0, 1, 1, 1, 0], [0, 0, 1, 1, 1, 0], [0, 1, 1, 1, 0, 0]
, [1, 1, 0, 0, 0, 1], [1, 0, 0, 0, 1, 1]], [[1, 1, 0, 0, 0, 1], [0, 0, 1, 1, 1,
0], [0, 0, 1, 1, 1, 0], [1, 0, 0, 0, 1, 1], [1, 1, 0, 0, 0, 1], [0, 1, 1, 1, 0,
0]], [[1, 1, 0, 0, 0, 1], [0, 0, 1, 1, 1, 0], [0, 0, 1, 1, 1, 0], [1, 0, 0, 1,
1, 0], [1, 1, 0, 0, 0, 1], [0, 1, 1, 0, 0, 1]], [[1, 1, 0, 0, 0, 1], [0, 0, 1,
1, 1, 0], [0, 0, 1, 1, 1, 0], [1, 1, 0, 0, 0, 1], [1, 1, 0, 0, 0, 1], [0, 0, 1,
1, 1, 0]], [[1, 1, 0, 0, 0, 1], [0, 0, 1, 1, 1, 0], [0, 0, 1, 1, 1, 0], [1, 1,
1, 0, 0, 0], [1, 1, 0, 0, 0, 1], [0, 0, 0, 1, 1, 1]], [[1, 1, 0, 0, 0, 1], [0,
1, 1, 0, 0, 1], [0, 0, 1, 1, 1, 0], [0, 0, 1, 1, 1, 0], [1, 0, 0, 1, 1, 0], [1,
1, 0, 0, 0, 1]], [[1, 1, 0, 0, 0, 1], [0, 1, 1, 0, 0, 1], [0, 0, 1, 1, 1, 0], [
1, 0, 0, 1, 1, 0], [1, 0, 0, 1, 1, 0], [0, 1, 1, 0, 0, 1]], [[1, 1, 0, 0, 0, 1]
, [0, 1, 1, 0, 0, 1], [0, 1, 1, 0, 0, 1], [0, 0, 1, 1, 1, 0], [1, 0, 0, 1, 1, 0
], [1, 0, 0, 1, 1, 0]], [[1, 1, 0, 0, 0, 1], [0, 1, 1, 0, 0, 1], [0, 1, 1, 0, 0
, 1], [1, 0, 0, 1, 1, 0], [1, 0, 0, 1, 1, 0], [0, 0, 1, 1, 1, 0]], [[1, 1, 0, 0
, 0, 1], [0, 1, 1, 0, 0, 1], [0, 1, 1, 1, 0, 0], [0, 0, 1, 1, 1, 0], [1, 0, 0,
1, 1, 0], [1, 0, 0, 0, 1, 1]], [[1, 1, 0, 0, 0, 1], [0, 1, 1, 1, 0, 0], [0, 0,
1, 1, 1, 0], [0, 0, 0, 1, 1, 1], [1, 0, 0, 0, 1, 1], [1, 1, 1, 0, 0, 0]], [[1,
1, 0, 0, 0, 1], [0, 1, 1, 1, 0, 0], [0, 0, 1, 1, 1, 0], [0, 0, 1, 1, 1, 0], [1,
0, 0, 0, 1, 1], [1, 1, 0, 0, 0, 1]], [[1, 1, 0, 0, 0, 1], [0, 1, 1, 1, 0, 0], [
0, 0, 1, 1, 1, 0], [1, 0, 0, 0, 1, 1], [1, 0, 0, 0, 1, 1], [0, 1, 1, 1, 0, 0]],
[[1, 1, 0, 0, 0, 1], [0, 1, 1, 1, 0, 0], [0, 0, 1, 1, 1, 0], [1, 0, 0, 1, 1, 0]
, [1, 0, 0, 0, 1, 1], [0, 1, 1, 0, 0, 1]], [[1, 1, 0, 0, 0, 1], [0, 1, 1, 1, 0,
0], [0, 1, 1, 1, 0, 0], [0, 0, 1, 1, 1, 0], [1, 0, 0, 0, 1, 1], [1, 0, 0, 0, 1,
1]], [[1, 1, 0, 0, 0, 1], [0, 1, 1, 1, 0, 0], [0, 1, 1, 1, 0, 0], [1, 0, 0, 0,
1, 1], [1, 0, 0, 0, 1, 1], [0, 0, 1, 1, 1, 0]], [[1, 1, 0, 0, 0, 1], [1, 0, 0,
0, 1, 1], [0, 0, 0, 1, 1, 1], [0, 0, 1, 1, 1, 0], [0, 1, 1, 1, 0, 0], [1, 1, 1,
0, 0, 0]], [[1, 1, 0, 0, 0, 1], [1, 0, 0, 0, 1, 1], [0, 0, 1, 1, 1, 0], [0, 0,
1, 1, 1, 0], [0, 1, 1, 1, 0, 0], [1, 1, 0, 0, 0, 1]], [[1, 1, 0, 0, 0, 1], [1,
0, 0, 0, 1, 1], [0, 0, 1, 1, 1, 0], [0, 1, 1, 1, 0, 0], [0, 1, 1, 1, 0, 0], [1,
0, 0, 0, 1, 1]], [[1, 1, 0, 0, 0, 1], [1, 0, 0, 0, 1, 1], [1, 0, 0, 0, 1, 1], [
0, 0, 1, 1, 1, 0], [0, 1, 1, 1, 0, 0], [0, 1, 1, 1, 0, 0]], [[1, 1, 0, 0, 0, 1]
, [1, 0, 0, 0, 1, 1], [1, 0, 0, 0, 1, 1], [0, 1, 1, 1, 0, 0], [0, 1, 1, 1, 0, 0
], [0, 0, 1, 1, 1, 0]], [[1, 1, 0, 0, 0, 1], [1, 0, 0, 0, 1, 1], [1, 0, 0, 1, 1
, 0], [0, 0, 1, 1, 1, 0], [0, 1, 1, 1, 0, 0], [0, 1, 1, 0, 0, 1]], [[1, 1, 0, 0
, 0, 1], [1, 0, 0, 1, 1, 0], [0, 0, 1, 1, 1, 0], [0, 0, 1, 1, 1, 0], [0, 1, 1,
0, 0, 1], [1, 1, 0, 0, 0, 1]], [[1, 1, 0, 0, 0, 1], [1, 0, 0, 1, 1, 0], [0, 0,
1, 1, 1, 0], [0, 1, 1, 0, 0, 1], [0, 1, 1, 0, 0, 1], [1, 0, 0, 1, 1, 0]], [[1,
1, 0, 0, 0, 1], [1, 0, 0, 1, 1, 0], [0, 0, 1, 1, 1, 0], [0, 1, 1, 1, 0, 0], [0,
1, 1, 0, 0, 1], [1, 0, 0, 0, 1, 1]], [[1, 1, 0, 0, 0, 1], [1, 0, 0, 1, 1, 0], [
1, 0, 0, 1, 1, 0], [0, 0, 1, 1, 1, 0], [0, 1, 1, 0, 0, 1], [0, 1, 1, 0, 0, 1]],
[[1, 1, 0, 0, 0, 1], [1, 0, 0, 1, 1, 0], [1, 0, 0, 1, 1, 0], [0, 1, 1, 0, 0, 1]
, [0, 1, 1, 0, 0, 1], [0, 0, 1, 1, 1, 0]], [[1, 1, 0, 0, 0, 1], [1, 1, 0, 0, 0,
1], [0, 0, 0, 1, 1, 1], [0, 0, 1, 1, 1, 0], [0, 0, 1, 1, 1, 0], [1, 1, 1, 0, 0,
0]], [[1, 1, 0, 0, 0, 1], [1, 1, 0, 0, 0, 1], [0, 0, 1, 1, 1, 0], [0, 0, 1, 1,
1, 0], [0, 0, 1, 1, 1, 0], [1, 1, 0, 0, 0, 1]], [[1, 1, 0, 0, 0, 1], [1, 1, 0,
0, 0, 1], [0, 1, 1, 0, 0, 1], [0, 0, 1, 1, 1, 0], [0, 0, 1, 1, 1, 0], [1, 0, 0,
1, 1, 0]], [[1, 1, 0, 0, 0, 1], [1, 1, 0, 0, 0, 1], [0, 1, 1, 1, 0, 0], [0, 0,
1, 1, 1, 0], [0, 0, 1, 1, 1, 0], [1, 0, 0, 0, 1, 1]], [[1, 1, 0, 0, 0, 1], [1,
1, 0, 0, 0, 1], [1, 0, 0, 0, 1, 1], [0, 0, 1, 1, 1, 0], [0, 0, 1, 1, 1, 0], [0,
1, 1, 1, 0, 0]], [[1, 1, 0, 0, 0, 1], [1, 1, 0, 0, 0, 1], [1, 0, 0, 1, 1, 0], [
0, 0, 1, 1, 1, 0], [0, 0, 1, 1, 1, 0], [0, 1, 1, 0, 0, 1]], [[1, 1, 0, 0, 0, 1]
, [1, 1, 0, 0, 0, 1], [1, 1, 0, 0, 0, 1], [0, 0, 1, 1, 1, 0], [0, 0, 1, 1, 1, 0
], [0, 0, 1, 1, 1, 0]], [[1, 1, 0, 0, 0, 1], [1, 1, 0, 0, 0, 1], [1, 1, 1, 0, 0
, 0], [0, 0, 1, 1, 1, 0], [0, 0, 1, 1, 1, 0], [0, 0, 0, 1, 1, 1]], [[1, 1, 0, 0
, 0, 1], [1, 1, 1, 0, 0, 0], [0, 0, 1, 1, 1, 0], [0, 0, 0, 1, 1, 1], [0, 0, 0,
1, 1, 1], [1, 1, 1, 0, 0, 0]], [[1, 1, 0, 0, 0, 1], [1, 1, 1, 0, 0, 0], [0, 0,
1, 1, 1, 0], [0, 0, 1, 1, 1, 0], [0, 0, 0, 1, 1, 1], [1, 1, 0, 0, 0, 1]], [[1,
1, 0, 0, 0, 1], [1, 1, 1, 0, 0, 0], [0, 1, 1, 1, 0, 0], [0, 0, 1, 1, 1, 0], [0,
0, 0, 1, 1, 1], [1, 0, 0, 0, 1, 1]], [[1, 1, 0, 0, 0, 1], [1, 1, 1, 0, 0, 0], [
1, 1, 1, 0, 0, 0], [0, 0, 0, 1, 1, 1], [0, 0, 0, 1, 1, 1], [0, 0, 1, 1, 1, 0]],
[[1, 1, 0, 0, 0, 1], [1, 1, 1, 0, 0, 0], [1, 1, 1, 0, 0, 0], [0, 0, 1, 1, 1, 0]
, [0, 0, 0, 1, 1, 1], [0, 0, 0, 1, 1, 1]], [[1, 1, 1, 0, 0, 0], [0, 0, 0, 1, 1,
1], [0, 0, 0, 1, 1, 1], [0, 0, 0, 1, 1, 1], [1, 1, 1, 0, 0, 0], [1, 1, 1, 0, 0,
0]], [[1, 1, 1, 0, 0, 0], [0, 0, 0, 1, 1, 1], [0, 0, 0, 1, 1, 1], [0, 0, 1, 1,
1, 0], [1, 1, 1, 0, 0, 0], [1, 1, 0, 0, 0, 1]], [[1, 1, 1, 0, 0, 0], [0, 0, 0,
1, 1, 1], [0, 0, 0, 1, 1, 1], [0, 1, 1, 0, 0, 1], [1, 1, 1, 0, 0, 0], [1, 0, 0,
1, 1, 0]], [[1, 1, 1, 0, 0, 0], [0, 0, 0, 1, 1, 1], [0, 0, 0, 1, 1, 1], [0, 1,
1, 1, 0, 0], [1, 1, 1, 0, 0, 0], [1, 0, 0, 0, 1, 1]], [[1, 1, 1, 0, 0, 0], [0,
0, 0, 1, 1, 1], [0, 0, 0, 1, 1, 1], [1, 0, 0, 0, 1, 1], [1, 1, 1, 0, 0, 0], [0,
1, 1, 1, 0, 0]], [[1, 1, 1, 0, 0, 0], [0, 0, 0, 1, 1, 1], [0, 0, 0, 1, 1, 1], [
1, 0, 0, 1, 1, 0], [1, 1, 1, 0, 0, 0], [0, 1, 1, 0, 0, 1]], [[1, 1, 1, 0, 0, 0]
, [0, 0, 0, 1, 1, 1], [0, 0, 0, 1, 1, 1], [1, 1, 0, 0, 0, 1], [1, 1, 1, 0, 0, 0
], [0, 0, 1, 1, 1, 0]], [[1, 1, 1, 0, 0, 0], [0, 0, 0, 1, 1, 1], [0, 0, 0, 1, 1
, 1], [1, 1, 1, 0, 0, 0], [1, 1, 1, 0, 0, 0], [0, 0, 0, 1, 1, 1]], [[1, 1, 1, 0
, 0, 0], [0, 0, 1, 1, 1, 0], [0, 0, 0, 1, 1, 1], [0, 0, 0, 1, 1, 1], [1, 1, 0,
0, 0, 1], [1, 1, 1, 0, 0, 0]], [[1, 1, 1, 0, 0, 0], [0, 0, 1, 1, 1, 0], [0, 0,
0, 1, 1, 1], [1, 0, 0, 0, 1, 1], [1, 1, 0, 0, 0, 1], [0, 1, 1, 1, 0, 0]], [[1,
1, 1, 0, 0, 0], [0, 0, 1, 1, 1, 0], [0, 0, 0, 1, 1, 1], [1, 1, 0, 0, 0, 1], [1,
1, 0, 0, 0, 1], [0, 0, 1, 1, 1, 0]], [[1, 1, 1, 0, 0, 0], [0, 0, 1, 1, 1, 0], [
0, 0, 1, 1, 1, 0], [0, 0, 0, 1, 1, 1], [1, 1, 0, 0, 0, 1], [1, 1, 0, 0, 0, 1]],
[[1, 1, 1, 0, 0, 0], [0, 0, 1, 1, 1, 0], [0, 0, 1, 1, 1, 0], [1, 1, 0, 0, 0, 1]
, [1, 1, 0, 0, 0, 1], [0, 0, 0, 1, 1, 1]], [[1, 1, 1, 0, 0, 0], [0, 1, 1, 0, 0,
1], [0, 0, 0, 1, 1, 1], [0, 0, 0, 1, 1, 1], [1, 0, 0, 1, 1, 0], [1, 1, 1, 0, 0,
0]], [[1, 1, 1, 0, 0, 0], [0, 1, 1, 0, 0, 1], [0, 0, 0, 1, 1, 1], [1, 0, 0, 1,
1, 0], [1, 0, 0, 1, 1, 0], [0, 1, 1, 0, 0, 1]], [[1, 1, 1, 0, 0, 0], [0, 1, 1,
0, 0, 1], [0, 1, 1, 0, 0, 1], [0, 0, 0, 1, 1, 1], [1, 0, 0, 1, 1, 0], [1, 0, 0,
1, 1, 0]], [[1, 1, 1, 0, 0, 0], [0, 1, 1, 0, 0, 1], [0, 1, 1, 0, 0, 1], [1, 0,
0, 1, 1, 0], [1, 0, 0, 1, 1, 0], [0, 0, 0, 1, 1, 1]], [[1, 1, 1, 0, 0, 0], [0,
1, 1, 1, 0, 0], [0, 0, 0, 1, 1, 1], [0, 0, 0, 1, 1, 1], [1, 0, 0, 0, 1, 1], [1,
1, 1, 0, 0, 0]], [[1, 1, 1, 0, 0, 0], [0, 1, 1, 1, 0, 0], [0, 0, 0, 1, 1, 1], [
1, 0, 0, 0, 1, 1], [1, 0, 0, 0, 1, 1], [0, 1, 1, 1, 0, 0]], [[1, 1, 1, 0, 0, 0]
, [0, 1, 1, 1, 0, 0], [0, 0, 1, 1, 1, 0], [0, 0, 0, 1, 1, 1], [1, 0, 0, 0, 1, 1
], [1, 1, 0, 0, 0, 1]], [[1, 1, 1, 0, 0, 0], [0, 1, 1, 1, 0, 0], [0, 1, 1, 1, 0
, 0], [0, 0, 0, 1, 1, 1], [1, 0, 0, 0, 1, 1], [1, 0, 0, 0, 1, 1]], [[1, 1, 1, 0
, 0, 0], [0, 1, 1, 1, 0, 0], [0, 1, 1, 1, 0, 0], [1, 0, 0, 0, 1, 1], [1, 0, 0,
0, 1, 1], [0, 0, 0, 1, 1, 1]], [[1, 1, 1, 0, 0, 0], [1, 0, 0, 0, 1, 1], [0, 0,
0, 1, 1, 1], [0, 0, 0, 1, 1, 1], [0, 1, 1, 1, 0, 0], [1, 1, 1, 0, 0, 0]], [[1,
1, 1, 0, 0, 0], [1, 0, 0, 0, 1, 1], [0, 0, 0, 1, 1, 1], [0, 0, 1, 1, 1, 0], [0,
1, 1, 1, 0, 0], [1, 1, 0, 0, 0, 1]], [[1, 1, 1, 0, 0, 0], [1, 0, 0, 0, 1, 1], [
0, 0, 0, 1, 1, 1], [0, 1, 1, 1, 0, 0], [0, 1, 1, 1, 0, 0], [1, 0, 0, 0, 1, 1]],
[[1, 1, 1, 0, 0, 0], [1, 0, 0, 0, 1, 1], [1, 0, 0, 0, 1, 1], [0, 0, 0, 1, 1, 1]
, [0, 1, 1, 1, 0, 0], [0, 1, 1, 1, 0, 0]], [[1, 1, 1, 0, 0, 0], [1, 0, 0, 0, 1,
1], [1, 0, 0, 0, 1, 1], [0, 1, 1, 1, 0, 0], [0, 1, 1, 1, 0, 0], [0, 0, 0, 1, 1,
1]], [[1, 1, 1, 0, 0, 0], [1, 0, 0, 1, 1, 0], [0, 0, 0, 1, 1, 1], [0, 0, 0, 1,
1, 1], [0, 1, 1, 0, 0, 1], [1, 1, 1, 0, 0, 0]], [[1, 1, 1, 0, 0, 0], [1, 0, 0,
1, 1, 0], [0, 0, 0, 1, 1, 1], [0, 1, 1, 0, 0, 1], [0, 1, 1, 0, 0, 1], [1, 0, 0,
1, 1, 0]], [[1, 1, 1, 0, 0, 0], [1, 0, 0, 1, 1, 0], [1, 0, 0, 1, 1, 0], [0, 0,
0, 1, 1, 1], [0, 1, 1, 0, 0, 1], [0, 1, 1, 0, 0, 1]], [[1, 1, 1, 0, 0, 0], [1,
0, 0, 1, 1, 0], [1, 0, 0, 1, 1, 0], [0, 1, 1, 0, 0, 1], [0, 1, 1, 0, 0, 1], [0,
0, 0, 1, 1, 1]], [[1, 1, 1, 0, 0, 0], [1, 1, 0, 0, 0, 1], [0, 0, 0, 1, 1, 1], [
0, 0, 0, 1, 1, 1], [0, 0, 1, 1, 1, 0], [1, 1, 1, 0, 0, 0]], [[1, 1, 1, 0, 0, 0]
, [1, 1, 0, 0, 0, 1], [0, 0, 0, 1, 1, 1], [0, 0, 1, 1, 1, 0], [0, 0, 1, 1, 1, 0
], [1, 1, 0, 0, 0, 1]], [[1, 1, 1, 0, 0, 0], [1, 1, 0, 0, 0, 1], [1, 0, 0, 0, 1
, 1], [0, 0, 0, 1, 1, 1], [0, 0, 1, 1, 1, 0], [0, 1, 1, 1, 0, 0]], [[1, 1, 1, 0
, 0, 0], [1, 1, 0, 0, 0, 1], [1, 1, 0, 0, 0, 1], [0, 0, 0, 1, 1, 1], [0, 0, 1,
1, 1, 0], [0, 0, 1, 1, 1, 0]], [[1, 1, 1, 0, 0, 0], [1, 1, 0, 0, 0, 1], [1, 1,
0, 0, 0, 1], [0, 0, 1, 1, 1, 0], [0, 0, 1, 1, 1, 0], [0, 0, 0, 1, 1, 1]], [[1,
1, 1, 0, 0, 0], [1, 1, 1, 0, 0, 0], [0, 0, 0, 1, 1, 1], [0, 0, 0, 1, 1, 1], [0,
0, 0, 1, 1, 1], [1, 1, 1, 0, 0, 0]], [[1, 1, 1, 0, 0, 0], [1, 1, 1, 0, 0, 0], [
0, 0, 1, 1, 1, 0], [0, 0, 0, 1, 1, 1], [0, 0, 0, 1, 1, 1], [1, 1, 0, 0, 0, 1]],
[[1, 1, 1, 0, 0, 0], [1, 1, 1, 0, 0, 0], [0, 1, 1, 0, 0, 1], [0, 0, 0, 1, 1, 1]
, [0, 0, 0, 1, 1, 1], [1, 0, 0, 1, 1, 0]], [[1, 1, 1, 0, 0, 0], [1, 1, 1, 0, 0,
0], [0, 1, 1, 1, 0, 0], [0, 0, 0, 1, 1, 1], [0, 0, 0, 1, 1, 1], [1, 0, 0, 0, 1,
1]], [[1, 1, 1, 0, 0, 0], [1, 1, 1, 0, 0, 0], [1, 0, 0, 0, 1, 1], [0, 0, 0, 1,
1, 1], [0, 0, 0, 1, 1, 1], [0, 1, 1, 1, 0, 0]], [[1, 1, 1, 0, 0, 0], [1, 1, 1,
0, 0, 0], [1, 0, 0, 1, 1, 0], [0, 0, 0, 1, 1, 1], [0, 0, 0, 1, 1, 1], [0, 1, 1,
0, 0, 1]], [[1, 1, 1, 0, 0, 0], [1, 1, 1, 0, 0, 0], [1, 1, 0, 0, 0, 1], [0, 0,
0, 1, 1, 1], [0, 0, 0, 1, 1, 1], [0, 0, 1, 1, 1, 0]], [[1, 1, 1, 0, 0, 0], [1,
1, 1, 0, 0, 0], [1, 1, 1, 0, 0, 0], [0, 0, 0, 1, 1, 1], [0, 0, 0, 1, 1, 1], [0,
0, 0, 1, 1, 1]]}):

elif m=8 and n=8 and k1=4 and k2=4 then
RETURN(M88()):
fi:

lu:=G1nmk(n,m,k1):

gu:={}:

for lu1 in lu do
 if {seq(add(lu1[i1][j1],i1=1..m),j1=1..n)}={k2} then
  gu:=gu union {lu1}:
 fi:
 od:

gu:

end:

#IsSubM(M1,M2,X): Given  a matrix M1 in {0,1,X} and a matrix M2 of the same dimensions in {0,1} checks that the non-Z entries of M1 are the same as those of M2. Try:
#IsSubM([[1,X],[X,0]],[[1,0],[0,0]]);
IsSubM:=proc(M1,M2,X) local i,j:
if not (nops(M1)=nops(M2) and nops(M1[1])=nops(M2[1])) then
 RETURN(FAIL):
fi:

for i from 1 to nops(M1) do
 for j from 1 to nops(M1[1]) do
  if M1[i][j]<>X and M1[i][j]<>M2[i][j] then
    RETURN(false):
  fi:
 od:
od:
true:
end:


#Ptor(M,X,k1,k2): Given a NotAlone puzzle where X is blank 1 is filled circle and 0 is empty circle finds all solutions. For example, forr the Aug. 11, 2024 puzzle by
#Presanna Seshardi, type:
#Ptor([[X$6],[X,X,1,1,X,X],[X$6],[1,X,1,X,X,X],[X,1,X,X,X,X],[X$6]],3,3);
Ptor:=proc(M,X,k1,k2) local i,m,n,lu,lu1,gu:

if {seq(op(M[i]),i=1..nops(M))} minus {0,1,X}<>{} then
 print(`bad puzzle`):
 RETURN(FAIL):
fi:

m:=nops(M):
n:=nops(M[1]):
lu:=Gnmk(n,m,k1,k2):
gu:={}:

for lu1 in lu do
 if IsSubM(M,lu1,X) then
   gu:=gu union {lu1}:
 fi:
od:

gu:

end:



#RandPuz1(n,m,k1,k2,X,c): A random m by n Not Alove puzzle with row sums k1 and column sums k2 and c clues. Try
#RandPuz1(6,6,3,3,X,6);
RandPuz1:=proc(n,m,k1,k2,X,c) local M,gu,L,i,j,T,i1,ce:
gu:=Gnmk(n,m,k1,k2):
M:=gu[rand(1..nops(gu))()]:

L:={seq(seq([i,j],j=1..n),i=1..m)}:

for i from 1 to m do
 for j from 1 to n do
  T[i,j]:=X:
 od:
od:

for i1 from 1 to c do
 ce:=L[rand(1..nops(L))()]:
  T[op(ce)]:=M[ce[1]][ce[2]]:
  L:=L minus {ce}:
od:

M:=[seq([seq(T[i,j],j=1..n)],i=1..m)]:

if nops(Ptor(M,X,k1,k2))<>1 then
 RETURN(FAIL):
fi:
M:
end:



#RandPuz(n,m,k1,k2,X,c,K): A random m by n Not Alove puzzle with row sums k1 and column sums k2 and c clues. Try
#RandPuz1(6,6,3,3,X,6,100);
RandPuz:=proc(n,m,k1,k2,X,c,K) local M,i:

for i from 1 to K do
 M:=RandPuz1(n,m,k1,k2,X,c):
if M<>FAIL then
 RETURN(M):
fi:
od:
FAIL:
end:

#MakeHarder(M,k1,k2,X): given a (k1,k2) puzzle in terms of X tries to make it harder
MakeHarder:=proc(M,k1,k2,X) local m,n,i,j,M1:

m:=nops(M):
n:=nops(M[1]):

if nops(Ptor(M,X,k1,k2))<>1 then
 RETURN(FAIL):
fi:

for i from 1 to m do
 for j from 1 to n do
  if M[i][j]<>X then
     M1:=[op(1..i-1,M),[op(1..j-1,M[i]),X,op(j+1..n,M[i])],op(i+1..m,M)]:
  if nops(Ptor(M1,X,k1,k2))=1 then
   RETURN(M1):
 fi:
 fi:
od:
od:
FAIL:
end:

#MakeHardPuz(n,m,k1,k2,X,c,K,TL). Makes a hard Not Alone puzzle (where all clues are crucial to have a unique solution) on an m by n grid
#with k1 1s in every row and k2 1s in every column. Try:
#MakeHardPuz(6,6,3,3,X,8,1000,TL);
MakeHardPuz:=proc(n,m,k1,k2,X,c,K, TL) local M,M1,t0:
t0:=time():

if c<=n*m/4 then
 print(`with c equal to`, c, `it may take a long time we recommend that you make c at least`, trunc(n*m/4)):
fi:

M:=RandPuz(n,m,k1,k2,X,c,K):
 if M=FAIL then
   RETURN(FAIL):
 fi:

M1:=MakeHarder(M,k1,k2,X):

while M1<>FAIL and time()-t0<TL do
 M:=M1:
M1:=MakeHarder(M,k1,k2,X):
od:
M:
end:



#PlotPuz(P): plots the puzzle
PlotPuz:=proc(P) local a,b,C,i,j,p:

b:=nops(P):
a:=nops(P[1]):


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

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 i from 1 to b do
for j from 1 to a do

 if P[i][j]=0 then
  C:=[j-1/2, b-i+1/2]:
  p:=p,textplot([op(C),A]):
 fi:

 if P[i][j]=1 then
  C:=[j-1/2, b-i+1/2]:
    p:=p,textplot([op(C),Z]):
 fi:

od:
od:

display(p):
end:




#Shane(M,i,j,c): Given a matrix M a constant c, and a location [i,j],  returns the new matrix that is the same except that the [i,j] entry is now:
#c. Try:
#Shane([[X,1,X],[X,X,1]],1,3,1);
Shane:=proc(M,i,j,c):
[op(1..i-1,M),[op(1..j-1,M[i]),c,op(j+1..nops(M[i]),M[i])],op(i+1..nops(M),M)]:
end:

#FC1(M,X,k1,k2): Given  a matrix in {0,1,X} finds whether there is place with 10X and then it returns the same with the X becoming a 0
#or X01, both horizintally and vertically. Try:
#FC1([[1,0,X],[0,1,X]],X,3,3);
FC1:=proc(M,X,k1,k2) local i, j,M1:



for i from 1 to nops(M) do
for j from 1 to nops(M[i])-2 do

# start horizontal 101
 if M[i][j]=1 and M[i][j+1]=0 and M[i][j+2]=X then

 M1:=Shane(M,i,j+2,0):
 if not IsBad(M1,k1,k2) then
   RETURN(M1):
  fi:  

 elif M[i][j]=X and M[i][j+1]=0 and M[i][j+2]=1 then

 M1:=Shane(M,i,j,0):
 if not IsBad(M1,k1,k2) then
   RETURN(M1):
  fi:  



 elif M[i][j]=1 and M[i][j+1]=X and M[i][j+2]=1 then


 M1:=Shane(M,i,j+1,1):
 if not IsBad(M1,k1,k2) then
   RETURN(M1):
  fi:  

fi:
#end horizontal 101


# start horizontal 010



 if M[i][j]=0 and M[i][j+1]=1 and M[i][j+2]=X then



 M1:=Shane(M,i,j+2,1):
 if not IsBad(M1,k1,k2) then
   RETURN(M1):
  fi:  



 elif M[i][j]=X and M[i][j+1]=1 and M[i][j+2]=0 then


 M1:=Shane(M,i,j,1):
 if not IsBad(M1,k1,k2) then
   RETURN(M1):
  fi:  




 elif M[i][j]=0 and M[i][j+1]=X and M[i][j+2]=0 then


 M1:=Shane(M,i,j+1,0):
 if not IsBad(M1,k1,k2) then
   RETURN(M1):
  fi:  

#end horizontal 010

fi:
od:
od:


for j from 1 to nops(M[1]) do
 for i from 1 to nops(M)-2 do

# start vertical  101

 if M[i][j]=1 and M[i+1][j]=0 and M[i+2][j]=X then

 M1:=Shane(M,i+2,j,0):
 if not IsBad(M1,k1,k2) then
   RETURN(M1):
  fi:  


 elif M[i][j]=X and M[i+1][j]=0 and M[i+2][j]=1 then
   

 M1:=Shane(M,i,j,0):
 if not IsBad(M1,k1,k2) then
   RETURN(M1):
  fi:  

 elif M[i][j]=1 and M[i+1][j]=X and M[i+2][j]=1 then



 M1:=Shane(M,i+1,j,1):

 if not IsBad(M1,k1,k2) then
   RETURN(M1):
 fi:
fi:

 # end vertical  101


# start vertical  010

 if M[i][j]=0 and M[i+1][j]=1 and M[i+2][j]=X then



 M1:=Shane(M,i+2,j,1):
 if not IsBad(M1,k1,k2) then
   RETURN(M1):
 fi:

 elif M[i][j]=X and M[i+1][j]=1 and M[i+2][j]=0 then



 M1:=Shane(M,i,j,1):
 if not IsBad(M1,k1,k2) then
   RETURN(M1):
 fi:

 elif M[i][j]=0 and M[i+1][j]=X and M[i+2][j]=0 then


 M1:=Shane(M,i+1,j,0):
 if not IsBad(M1,k1,k2) then
   RETURN(M1):
 fi:

fi:
# end vertical  010




od:
od:



FAIL:
end:



#FC(M,X,k1,k2): Applies FC1(M,X) as much as possible. Try
#FC([[1,0,X],[0,1,X]],X,3,3);
FC:=proc(M,X,k1,k2) local M1,M2:
M1:=M:
M2:=FC1(M1,X,k1,k2):

while M2<>FAIL do
 M1:=M2:
 M2:=FC1(M1,X,k1,k2):
od:
M1:
end:

#IsBad(M,k1,k2): Is the matrix M with alphabet {0,1,X} bad in the Not Alone sense and that the number of 0s <=k1 and number of 1's<=k1  in every row 
#and number of 0s <=k2 and number of 1's<=k2  in every column
#and column. Try:
#M:=Ptor([[X$6],[X,X,1,1,X,X],[X$6],[1,X,1,X,X,X],[X,1,X,X,X,X],[X$6]],3,3); IsBad(M,X,3,3);
IsBad:=proc(M,k1,k2) local i,j,lu,z:

for i from 1 to nops(M) do
for j from 1 to nops(M[i])-2 do
 if [M[i,j],M[i,j+1],M[i,j+2]]=[1,0,1] then
  RETURN(true):
fi:

 if [M[i,j],M[i,j+1],M[i,j+2]]=[0,1,0] then
  RETURN(true):
fi:

od:
od:


for j from 1 to nops(M[1]) do
for i from 1 to nops(M)-2 do

 if [M[i,j],M[i+1,j],M[i+2,j]]=[1,0,1] then
  RETURN(true):
fi:

 if [M[i,j],M[i+1,j],M[i+2,j]]=[0,1,0] then
  RETURN(true):
fi:

od:
od:



for i from 1 to nops(M) do
lu:=add(z[M[i][j]],j=1..nops(M[i])):
 if coeff(lu,z[0],1)>k1 or coeff(lu,z[1],1)>k2 then
  RETURN(true):
 fi:
od:


for j from 1 to nops(M[1]) do
lu:=add(z[M[i][j]],i=1..nops(M)):
 if coeff(lu,z[0],1)>k1 or coeff(lu,z[1],1)>k2 then
  RETURN(true):
 fi:
od:

false:

end:



#FillIn1(M,X,k1,k2): fills in X's where the number of 0's or 1's in a row in exactly k1 and ditto for columns with k2. Try:
#FillIn1([[X,X,X,1,X,X],[X$6],[X,X,X,1,0,X],[1,0,X,X,X,X],[X$6],[X,0,X,X,X,X]],X,3,3);
FillIn1:=proc(M,X,k1,k2) local i,j,z,M1,lu:



for i from 1 to nops(M) do


 lu:=add(z[M[i][j]],j=1..nops(M[i])):



  if coeff(lu,z[0],1)=k1  and coeff(lu,z[1],1)+coeff(lu,z[X],1)=k1  and coeff(lu,z[X],1)>0 then
    M1:=M:
    for j from 1 to nops(M[i]) do
       if M1[i][j]=X then
          M1:=Shane(M1,i,j,1):
      fi:
    od:

 if not IsBad(M1,k1,k2) then
  RETURN(M1):
fi:
 fi:



  if coeff(lu,z[1],1)=k1  and coeff(lu,z[0],1)+coeff(lu,z[X],1)=k1   and coeff(lu,z[X],1)>0 then
    M1:=M:
    for j from 1 to nops(M[i]) do
       if M1[i][j]=X then
          M1:=Shane(M1,i,j,0):
      fi:
    od:

 if not IsBad(M1,k1,k2) then
  RETURN(M1):
fi:

fi:

od:




####


for j from 1 to nops(M[1]) do


 lu:=add(z[M[i][j]],i=1..nops(M)):



  if coeff(lu,z[0],1)=k2 and coeff(lu,z[1],1)+coeff(lu,z[X],1)=k2  and coeff(lu,z[X],1)>0 then
    M1:=M:
    for i from 1 to nops(M) do
       if M1[i][j]=X then
          M1:=Shane(M1,i,j,1):
      fi:
    od:

 if not IsBad(M1,k1,k2) then
  RETURN(M1):
fi:
 fi:



  if coeff(lu,z[1],1)=k2 and  coeff(lu,z[0],1)+coeff(lu,z[X],1)=k2  and coeff(lu,z[X],1)>0 then
    M1:=M:
    for i from 1 to nops(M) do
       if M1[i][j]=X then
          M1:=Shane(M1,i,j,0):
      fi:
    od:

 if not IsBad(M1,k1,k2) then
  RETURN(M1):
fi:

fi:

od:





###
FAIL:

end:


#FillIn(M,X,k1,k2): Applies FillIn1(M,X) as much as possible. Try
#FillIn([[1,0,X],[0,1,X]],X,3,3);
FillIn:=proc(M,X,k1,k2) local M1,M2:
M1:=M:
M2:=FillIn1(M1,X,k1,k2):

while M2<>FAIL do
 M1:=M2:
 M2:=FillIn1(M1,X,k1,k2):
od:
M1:
end:


#NumX(M,X): the number of X's still in M
#Try NuX([[X,0,1],[1,X,0]],X);
NumX:=proc(M,X) local i,j,z:
coeff(add(add(z[M[i][j]],j=1..nops(M[i])),i=1..nops(M)),z[X],1):
end:


#SolveN1(M,X,k1,k2): solves the Not Alone puzzle M where X is blank and there are k1 1s and k2 0s in each row and each colunm,
#using logic.
SolveN1:=proc(M,X,k1,k2) local M1,M2,M3:
M1:=M:
M2:=FillIn(FC(M,X,k1,k2),X,k1,k2):

while M1<>M2 and NumX(M2,X)>0 do
M3:=FillIn(FC(M2,X,k1,k2),X,k1,k2):
M1:=M2:
M2:=M3:
od:
M2:

end:


#IsBadR(M,X,k1,k2,R): Given a partial solution M does there exist an  place that
# changing X to either 0 or 1 will be (R-1)-bad. with R=0 it is the same IsBad(M,k1,k2)
IsBadR:=proc(M,X,k1,k2,R) local i,j:

if NuX(M,X)=0 then
 RETURN(false):
fi:

if R=0 then
 RETURN(IsBad(M,k1,k2)):
fi:

for i from 1 to nops(M) do
 for j from 1 to nops(M[i]) do
  if M[i][j]=X then
   if IsBadR(Shane(M,i,j,0),X,k1,k2,R-1) and IsBadR(Shane(M,i,j,1),X,k1,k2,R-1) then
     RETURN(true):
   fi:
  fi:
od:
od:
false:

end:


#MakeBetter(M,X,k1,k2,d): Given a partial solution,M,  to a (k1,k2) Not Alone puzzle tries to find a better one using IsbadR with d
#Try:
#M:=SolveN1([[X$8],[X$8],[X,0,0,X,X,1,0,X],[X,0,1,X,X,0,1,X],[X$8],[X,X,X,1,1,X,X,X],[X$8],[X,X,X,X,X,X,1,X]],X,4,4); MakeBetter(M,X,4,4,2);

MakeBetter:=proc(M,X,k1,k2,d) local i,j,d1:

for i from 1 to nops(M) do
 for j from 1 to nops(M[i]) do
 if M[i][j]=X then
   for d1 from 0 to d do

    if IsBadR(Shane(M,i,j,0),X,k1,k2,d1) then
      if IsBadR(Shane(M,i,j,1),X,k1,k2,d1) then
        print(`something bad happened`):
       RETURN(FAIL):
     fi:

  RETURN(Shane(M,i,j,1)):

    elif IsBadR(Shane(M,i,j,1),X,k1,k2,d1) then

      if IsBadR(Shane(M,i,j,0),X,k1,k2,d1) then
        print(`something bad happened`):
       RETURN(FAIL):
     fi:
  RETURN(Shane(M,i,j,0)):
 fi:
od:
fi:
od:
od:
FAIL:
end:



#FFin(M,X,k1,k2) : Given a partial solution of a Not Alone puzzle with still some X finds all the way
#to fill-it in and still stay legal
FFin:=proc(M,X,k1,k2) local i,j,M1,M0,gu:

if NumX(M,X)=0 then
  RETURN({M}):
fi:

gu:={}:

for i from 1 to nops(M) do
 for j from 1 to nops(M[i]) do
  if M[i][j]=X then

   M0:=Shane(M,i,j,0):
  if not IsBad(M0,k1,k2) then
    gu:=gu union  FFin(M0,X,k1,k2):
 fi:


   M1:=Shane(M,i,j,1):
  if not IsBad(M1,k1,k2) then
    gu:=gu union  FFin(M1,X,k1,k2):
 fi:
RETURN(gu):
fi:

od:
od:
gu:
end:


#SolveN(M,X,k1,k2): solves the Not Alone puzzle M where X is blank and there are k1 1s and k2 0s in each row and each colunm,
#using logic.
SolveN:=proc(M,X,k1,k2) local i,M1,M2,M3:


if {seq(op(M[i]),i=1..nops(M))} minus {0,1,X}<>{} then
 print(`bad puzzle`):
 RETURN(FAIL):
fi:


M1:=SolveN1(M,X,k1,k2):
M2:=MakeBetter(M1,X,k1,k2,2):

while M1<>M2 and NumX(M2,X)>0 and M2<>FAIL do
 M3:=MakeBetter(M2,X,k1,k2,2):
 M1:=M2:
 M2:=M3:
od:

FFin(M1,X,k1,k2):

end:



#RanExt(M,k1,k2): given a 0-1 matrix M where every row has k1 0s and k1 1s and avoids the patterns 101 and 010
#and the columns also avoid these and every column has at most k2 0s and at most k2 1s
#finds a random extension with one more row. Try:
#RanExt([],3,3);

RanExt:=proc(M,k1,k2) local gu,M1,i,kha1,M2:
gu:=Gnk(2*k1,k1):
if M=[] then
 RETURN([gu[rand(1..nops(gu))()]]):
fi:


if nops(M[1])<>2*k1 then
 RETURN(FAIL):
fi:


for i from 1 to 100 do
kha1:=gu[rand(1..nops(gu))()]:
M1:=[op(M),kha1]:
 if not IsBad(M1,k1,k2) then
    RETURN(M1):
 fi:
od:

M1:=[op(1..nops(M)-1,M)]:
gu:=gu minus {M[nops(M)]}:

for i from 1 to 100 do
kha1:=gu[rand(1..nops(gu))()]:
M2:=[op(M1),kha1]:
 if not IsBad(M2,k1,k2) then
    RETURN(M2):
 fi:
od:

FAIL:

end:

#RanSol1(k1,k2): A random good balanced 0-1 matrix. Try:
#RanSol1(3,3);
RanSol1:=proc(k1,k2) local M,i:
M:=[]:
for i from 1 to 2*k2 do
 M:=RanExt(M,k1,k2):
 if M=FAIL then
   RETURN(FAIL):
 fi:
od:
M:
end:



#RanSol(k1,k2): A random good balanced 0-1 matrix. Try:
#RanSol(3,3);
RanSol:=proc(k1,k2) local gu,i:

gu:=RanSol1(k1,k2):

if gu<>FAIL and nops(gu)=2*k2 then
 RETURN(gu):
fi:

for i from 1 to 20 do
gu:=RanSol1(k1,k2):
if gu<>FAIL and nops(gu)=2*k2 then
RETURN(gu):
fi:
od:
FAIL:
end:

#RPuz1(k1,k2,c,X): A random (k1,k2) puzzle with c clues. Try:
#RPuz1(4,4,14,X);
RPuz1:=proc(k1,k2,c,X) local gu,i,j,kha,M,gu1:

if c>=4*k1*k2 then
  RETURN(FAIL):
fi:

gu:={seq(seq([i,j],j=1..2*k1),i=1..2*k2)}:


for i from 1 to c do
 kha:=gu[rand(1..nops(gu))()]:
 gu:=gu minus {kha}:
od:

M:=RanSol(k1,k2):



for gu1 in gu do
 M:=Shane(M,op(gu1),X):
od:
M:
end:




#RPuzP(k1,k2,c,X): A random (k1,k2) puzzle with c clues. Try:
#RPuzP(4,4,14,X);
RPuzP:=proc(k1,k2,c,X) local i,M:

for i from 1 to 40 do
M:=RPuz1(k1,k2,c,X):
 if nops(SolveN(M,X,k1,k2))=1 then
    RETURN(M):
 fi:
od:
FAIL:
end:



#RPuz(k1,k2,c,X,T0): A random (k1,k2) puzzle with c clues and then removing as many as possible and keeping the answer unique and taking at most T0 seconds. Try:
#RPuz(4,4,28,X,10);
RPuz:=proc(k1,k2,c,X,T0) local i,j,M,M1,t0:
M:=RPuzP(k1,k2,c,X):
t0:=time():
if M=FAIL then
  RETURN(FAIL):
fi:


for i from 1 to nops(M)  while time()-t0<=T0 do 
 for j from 1 to nops(M[i]) while time()-t0<=T0 do
  if M[i][j]<>X then
   M1:=Shane(M,i,j,X):
    if nops(SolveN(M1,X,k1,k2))=1 then
     M:=M1:
    fi:
 fi:
od:
od:
M:
end: