Prompt:
Find an all-interval series of size 10. This is a permutation of 0 to 9 such that the absolute differences between consecutive elements are also a permutation of 1 to 9.

Return a list of 10 integers forming a valid permutation, or state "UNSATISFIABLE" if no solution exists.

Partial assignment (fixed values that must be respected):
- x[0]=0, x[3]=8
- d[1]=8
Return a complete solution consistent with these fixed assignments.


Prompt:
Find an all-interval series of size 11. This is a permutation of 0 to 10 such that the absolute differences between consecutive elements are also a permutation of 1 to 10.

Return a list of 11 integers forming a valid permutation, or state "UNSATISFIABLE" if no solution exists.


Prompt:
Find an all-interval series of size 12. This is a permutation of 0 to 11 such that the absolute differences between consecutive elements are also a permutation of 1 to 11.

Return a list of 12 integers forming a valid permutation, or state "UNSATISFIABLE" if no solution exists.


Prompt:
Find an all-interval series of size 13. This is a permutation of 0 to 12 such that the absolute differences between consecutive elements are also a permutation of 1 to 12.

Return a list of 13 integers forming a valid permutation, or state "UNSATISFIABLE" if no solution exists.


Prompt:
Find an all-interval series of size 14. This is a permutation of 0 to 13 such that the absolute differences between consecutive elements are also a permutation of 1 to 13.

Return a list of 14 integers forming a valid permutation, or state "UNSATISFIABLE" if no solution exists.

Partial assignment (fixed values that must be respected):
- x[6]=3, x[12]=6
- d[3]=10, d[11]=2
Return a complete solution consistent with these fixed assignments.


Prompt:
Find an all-interval series of size 2. This is a permutation of 0 to 1 such that the absolute differences between consecutive elements are also a permutation of 1 to 1.

Return a list of 2 integers forming a valid permutation, or state "UNSATISFIABLE" if no solution exists.

Partial assignment (fixed values that must be respected):
- x[0]=0
- d[0]=1
Return a complete solution consistent with these fixed assignments.


Prompt:
Find an all-interval series of size 3. This is a permutation of 0 to 2 such that the absolute differences between consecutive elements are also a permutation of 1 to 2.

Return a list of 3 integers forming a valid permutation, or state "UNSATISFIABLE" if no solution exists.


Prompt:
Find an all-interval series of size 4. This is a permutation of 0 to 3 such that the absolute differences between consecutive elements are also a permutation of 1 to 3.

Return a list of 4 integers forming a valid permutation, or state "UNSATISFIABLE" if no solution exists.


Prompt:
Find an all-interval series of size 5. This is a permutation of 0 to 4 such that the absolute differences between consecutive elements are also a permutation of 1 to 4.

Return a list of 5 integers forming a valid permutation, or state "UNSATISFIABLE" if no solution exists.


Prompt:
Find an all-interval series of size 6. This is a permutation of 0 to 5 such that the absolute differences between consecutive elements are also a permutation of 1 to 5.

Return a list of 6 integers forming a valid permutation, or state "UNSATISFIABLE" if no solution exists.


Prompt:
Find an all-interval series of size 8. This is a permutation of 0 to 7 such that the absolute differences between consecutive elements are also a permutation of 1 to 7.

Return a list of 8 integers forming a valid permutation, or state "UNSATISFIABLE" if no solution exists.

Partial assignment (fixed values that must be respected):
- x[4]=2
- d[5]=2
Return a complete solution consistent with these fixed assignments.


Prompt:
Find an all-interval series of size 9. This is a permutation of 0 to 8 such that the absolute differences between consecutive elements are also a permutation of 1 to 8.

Return a list of 9 integers forming a valid permutation, or state "UNSATISFIABLE" if no solution exists.


Prompt:
Find a Costas array of order 10: place 10 marks on an 10x10 grid, one per row and one per column, such that all 10*(9)/2 displacement vectors (dr, dc) between pairs of marks (with dc > 0) are pairwise distinct.

Return x as a list of 10 integers in 0..9, where x[c] is the row of the mark in column c (permutation), or state "UNSATISFIABLE" if no such array exists.


Prompt:
Find a Costas array of order 11: place 11 marks on an 11x11 grid, one per row and one per column, such that all 11*(10)/2 displacement vectors (dr, dc) between pairs of marks (with dc > 0) are pairwise distinct.

Return x as a list of 11 integers in 0..10, where x[c] is the row of the mark in column c (permutation), or state "UNSATISFIABLE" if no such array exists.


Prompt:
Find a Costas array of order 12: place 12 marks on an 12x12 grid, one per row and one per column, such that all 12*(11)/2 displacement vectors (dr, dc) between pairs of marks (with dc > 0) are pairwise distinct.

Return x as a list of 12 integers in 0..11, where x[c] is the row of the mark in column c (permutation), or state "UNSATISFIABLE" if no such array exists.

Partial assignment (fixed values that must be respected):
- x[3]=8, x[8]=2
Return a complete solution consistent with these fixed assignments.


Prompt:
Find a Costas array of order 13: place 13 marks on an 13x13 grid, one per row and one per column, such that all 13*(12)/2 displacement vectors (dr, dc) between pairs of marks (with dc > 0) are pairwise distinct.

Return x as a list of 13 integers in 0..12, where x[c] is the row of the mark in column c (permutation), or state "UNSATISFIABLE" if no such array exists.

Partial assignment (fixed values that must be respected):
- x[4]=3, x[9]=1
Return a complete solution consistent with these fixed assignments.


Prompt:
Find a Costas array of order 5: place 5 marks on an 5x5 grid, one per row and one per column, such that all 5*(4)/2 displacement vectors (dr, dc) between pairs of marks (with dc > 0) are pairwise distinct.

Return x as a list of 5 integers in 0..4, where x[c] is the row of the mark in column c (permutation), or state "UNSATISFIABLE" if no such array exists.

Partial assignment (fixed values that must be respected):
- x[2]=3
Return a complete solution consistent with these fixed assignments.


Prompt:
Find a Costas array of order 6: place 6 marks on an 6x6 grid, one per row and one per column, such that all 6*(5)/2 displacement vectors (dr, dc) between pairs of marks (with dc > 0) are pairwise distinct.

Return x as a list of 6 integers in 0..5, where x[c] is the row of the mark in column c (permutation), or state "UNSATISFIABLE" if no such array exists.


Prompt:
Find a Costas array of order 7: place 7 marks on an 7x7 grid, one per row and one per column, such that all 7*(6)/2 displacement vectors (dr, dc) between pairs of marks (with dc > 0) are pairwise distinct.

Return x as a list of 7 integers in 0..6, where x[c] is the row of the mark in column c (permutation), or state "UNSATISFIABLE" if no such array exists.

Partial assignment (fixed values that must be respected):
- x[1]=1
Return a complete solution consistent with these fixed assignments.


Prompt:
Find a Costas array of order 8: place 8 marks on an 8x8 grid, one per row and one per column, such that all 8*(7)/2 displacement vectors (dr, dc) between pairs of marks (with dc > 0) are pairwise distinct.

Return x as a list of 8 integers in 0..7, where x[c] is the row of the mark in column c (permutation), or state "UNSATISFIABLE" if no such array exists.


Prompt:
Find a Costas array of order 9: place 9 marks on an 9x9 grid, one per row and one per column, such that all 9*(8)/2 displacement vectors (dr, dc) between pairs of marks (with dc > 0) are pairwise distinct.

Return x as a list of 9 integers in 0..8, where x[c] is the row of the mark in column c (permutation), or state "UNSATISFIABLE" if no such array exists.


Prompt:
Find a Golomb ruler with 10 marks. A Golomb ruler is a set of 10 integers (marks) such that all pairwise distances between marks are unique. The first mark should be at position 0.

Return a list of 10 integers in increasing order representing the mark positions, or state "UNSATISFIABLE" if no solution exists.


Prompt:
Find a Golomb ruler with 5 marks. A Golomb ruler is a set of 5 integers (marks) such that all pairwise distances between marks are unique. The first mark should be at position 0.

Return a list of 5 integers in increasing order representing the mark positions, or state "UNSATISFIABLE" if no solution exists.


Prompt:
Find a Golomb ruler with 6 marks. A Golomb ruler is a set of 6 integers (marks) such that all pairwise distances between marks are unique. The first mark should be at position 0.

Return a list of 6 integers in increasing order representing the mark positions, or state "UNSATISFIABLE" if no solution exists.


Prompt:
Find a Golomb ruler with 7 marks. A Golomb ruler is a set of 7 integers (marks) such that all pairwise distances between marks are unique. The first mark should be at position 0.

Return a list of 7 integers in increasing order representing the mark positions, or state "UNSATISFIABLE" if no solution exists.

Partial assignment (fixed values that must be respected):
- x[1]=2
Return a complete solution consistent with these fixed assignments.


Prompt:
Find a Golomb ruler with 8 marks. A Golomb ruler is a set of 8 integers (marks) such that all pairwise distances between marks are unique. The first mark should be at position 0.

Return a list of 8 integers in increasing order representing the mark positions, or state "UNSATISFIABLE" if no solution exists.

Partial assignment (fixed values that must be respected):
- x[1]=2
Return a complete solution consistent with these fixed assignments.


Prompt:
Find a Golomb ruler with 9 marks. A Golomb ruler is a set of 9 integers (marks) such that all pairwise distances between marks are unique. The first mark should be at position 0.

Return a list of 9 integers in increasing order representing the mark positions, or state "UNSATISFIABLE" if no solution exists.


Prompt:
Find a graceful labeling for the graph G_{2,3}: 3 disjoint K_2 cliques (numbered 0 through 2), where each pair of consecutive cliques (g, g+1) is connected by 2 edges that link vertex i of clique g to vertex i of clique g+1 for each i in 0..1. The graph has 6 vertices and 7 edges in total.

A graceful labeling assigns each vertex a unique label from 0 to 7 such that all edge labels |label(u) - label(v)| are distinct.

Return a list of 6 integers giving the vertex labels, ordered by clique then vertex within clique (so the i-th block of 2 consecutive entries gives the labels for clique i), or state "UNSATISFIABLE" if no such labeling exists.

Partial assignment (fixed values that must be respected):
- x[2]=7
- d[6]=3
Return a complete solution consistent with these fixed assignments.


Prompt:
Find a graceful labeling for the graph G_{2,4}: 4 disjoint K_2 cliques (numbered 0 through 3), where each pair of consecutive cliques (g, g+1) is connected by 2 edges that link vertex i of clique g to vertex i of clique g+1 for each i in 0..1. The graph has 8 vertices and 10 edges in total.

A graceful labeling assigns each vertex a unique label from 0 to 10 such that all edge labels |label(u) - label(v)| are distinct.

Return a list of 8 integers giving the vertex labels, ordered by clique then vertex within clique (so the i-th block of 2 consecutive entries gives the labels for clique i), or state "UNSATISFIABLE" if no such labeling exists.


Prompt:
Find a graceful labeling for the graph G_{2,5}: 5 disjoint K_2 cliques (numbered 0 through 4), where each pair of consecutive cliques (g, g+1) is connected by 2 edges that link vertex i of clique g to vertex i of clique g+1 for each i in 0..1. The graph has 10 vertices and 13 edges in total.

A graceful labeling assigns each vertex a unique label from 0 to 13 such that all edge labels |label(u) - label(v)| are distinct.

Return a list of 10 integers giving the vertex labels, ordered by clique then vertex within clique (so the i-th block of 2 consecutive entries gives the labels for clique i), or state "UNSATISFIABLE" if no such labeling exists.


Prompt:
Find a graceful labeling for the graph G_{2,6}: 6 disjoint K_2 cliques (numbered 0 through 5), where each pair of consecutive cliques (g, g+1) is connected by 2 edges that link vertex i of clique g to vertex i of clique g+1 for each i in 0..1. The graph has 12 vertices and 16 edges in total.

A graceful labeling assigns each vertex a unique label from 0 to 16 such that all edge labels |label(u) - label(v)| are distinct.

Return a list of 12 integers giving the vertex labels, ordered by clique then vertex within clique (so the i-th block of 2 consecutive entries gives the labels for clique i), or state "UNSATISFIABLE" if no such labeling exists.

Partial assignment (fixed values that must be respected):
- x[0]=6, x[3]=4
- d[10]=11, d[14]=1, d[15]=12
Return a complete solution consistent with these fixed assignments.


Prompt:
Find a graceful labeling for the graph G_{3,3}: 3 disjoint K_3 cliques (numbered 0 through 2), where each pair of consecutive cliques (g, g+1) is connected by 3 edges that link vertex i of clique g to vertex i of clique g+1 for each i in 0..2. The graph has 9 vertices and 15 edges in total.

A graceful labeling assigns each vertex a unique label from 0 to 15 such that all edge labels |label(u) - label(v)| are distinct.

Return a list of 9 integers giving the vertex labels, ordered by clique then vertex within clique (so the i-th block of 3 consecutive entries gives the labels for clique i), or state "UNSATISFIABLE" if no such labeling exists.


Prompt:
Find a graceful labeling for the graph G_{3,4}: 4 disjoint K_3 cliques (numbered 0 through 3), where each pair of consecutive cliques (g, g+1) is connected by 3 edges that link vertex i of clique g to vertex i of clique g+1 for each i in 0..2. The graph has 12 vertices and 21 edges in total.

A graceful labeling assigns each vertex a unique label from 0 to 21 such that all edge labels |label(u) - label(v)| are distinct.

Return a list of 12 integers giving the vertex labels, ordered by clique then vertex within clique (so the i-th block of 3 consecutive entries gives the labels for clique i), or state "UNSATISFIABLE" if no such labeling exists.

Partial assignment (fixed values that must be respected):
- x[2]=15, x[4]=0
- d[6]=5, d[9]=11, d[15]=13, d[18]=9
Return a complete solution consistent with these fixed assignments.


Prompt:
Find a graceful labeling for the graph G_{3,5}: 5 disjoint K_3 cliques (numbered 0 through 4), where each pair of consecutive cliques (g, g+1) is connected by 3 edges that link vertex i of clique g to vertex i of clique g+1 for each i in 0..2. The graph has 15 vertices and 27 edges in total.

A graceful labeling assigns each vertex a unique label from 0 to 27 such that all edge labels |label(u) - label(v)| are distinct.

Return a list of 15 integers giving the vertex labels, ordered by clique then vertex within clique (so the i-th block of 3 consecutive entries gives the labels for clique i), or state "UNSATISFIABLE" if no such labeling exists.


Prompt:
Find a graceful labeling for the graph G_{3,6}: 6 disjoint K_3 cliques (numbered 0 through 5), where each pair of consecutive cliques (g, g+1) is connected by 3 edges that link vertex i of clique g to vertex i of clique g+1 for each i in 0..2. The graph has 18 vertices and 33 edges in total.

A graceful labeling assigns each vertex a unique label from 0 to 33 such that all edge labels |label(u) - label(v)| are distinct.

Return a list of 18 integers giving the vertex labels, ordered by clique then vertex within clique (so the i-th block of 3 consecutive entries gives the labels for clique i), or state "UNSATISFIABLE" if no such labeling exists.


Prompt:
Find an open knight's tour on an 5x5 chessboard that starts at cell 0 (row 0, column 0). A knight's tour visits every cell exactly once via knight moves; cells are numbered in row-major order so cell k is at row k//5, column k%5. A knight move between cells a and b changes the row by 1 and column by 2, or row by 2 and column by 1.

Return a list of 25 integers giving the cell visited at each step (a permutation of 0..24 starting with 0), or state "UNSATISFIABLE" if no such tour exists.

Partial assignment (fixed values that must be respected):
- x[3]=23, x[4]=16, x[5]=5, x[15]=15, x[22]=24
Return a complete solution consistent with these fixed assignments.


Prompt:
Find an open knight's tour on an 6x6 chessboard that starts at cell 0 (row 0, column 0). A knight's tour visits every cell exactly once via knight moves; cells are numbered in row-major order so cell k is at row k//6, column k%6. A knight move between cells a and b changes the row by 1 and column by 2, or row by 2 and column by 1.

Return a list of 36 integers giving the cell visited at each step (a permutation of 0..35 starting with 0), or state "UNSATISFIABLE" if no such tour exists.

Partial assignment (fixed values that must be respected):
- x[3]=17, x[6]=24, x[17]=30, x[19]=6, x[29]=25, x[32]=21, x[34]=23
Return a complete solution consistent with these fixed assignments.


Prompt:
Find an open knight's tour on an 7x7 chessboard that starts at cell 0 (row 0, column 0). A knight's tour visits every cell exactly once via knight moves; cells are numbered in row-major order so cell k is at row k//7, column k%7. A knight move between cells a and b changes the row by 1 and column by 2, or row by 2 and column by 1.

Return a list of 49 integers giving the cell visited at each step (a permutation of 0..48 starting with 0), or state "UNSATISFIABLE" if no such tour exists.


Prompt:
Find an open knight's tour on an 8x8 chessboard that starts at cell 0 (row 0, column 0). A knight's tour visits every cell exactly once via knight moves; cells are numbered in row-major order so cell k is at row k//8, column k%8. A knight move between cells a and b changes the row by 1 and column by 2, or row by 2 and column by 1.

Return a list of 64 integers giving the cell visited at each step (a permutation of 0..63 starting with 0), or state "UNSATISFIABLE" if no such tour exists.

Partial assignment (fixed values that must be respected):
- x[2]=27, x[3]=21, x[10]=16, x[11]=1, x[19]=24, x[21]=3, x[27]=51, x[38]=38, x[50]=43, x[51]=60, x[59]=53, x[61]=42
Return a complete solution consistent with these fixed assignments.


Prompt:
Construct a Langford sequence L(2,11): a sequence of length 22 containing exactly 2 copies of each integer from 1 to 11, such that the two occurrences of each value v are exactly v+1 positions apart (so copies of 1 are 1 position apart, copies of 2 are 2 positions apart, etc.). Example: L(2,3) has solution [2, 3, 1, 2, 1, 3].

Return seq as a list of 22 integers in 1..11, or state "UNSATISFIABLE" if no such sequence exists.


Prompt:
Construct a Langford sequence L(2,12): a sequence of length 24 containing exactly 2 copies of each integer from 1 to 12, such that the two occurrences of each value v are exactly v+1 positions apart (so copies of 1 are 1 position apart, copies of 2 are 2 positions apart, etc.). Example: L(2,3) has solution [2, 3, 1, 2, 1, 3].

Return seq as a list of 24 integers in 1..12, or state "UNSATISFIABLE" if no such sequence exists.


Prompt:
Construct a Langford sequence L(2,15): a sequence of length 30 containing exactly 2 copies of each integer from 1 to 15, such that the two occurrences of each value v are exactly v+1 positions apart (so copies of 1 are 1 position apart, copies of 2 are 2 positions apart, etc.). Example: L(2,3) has solution [2, 3, 1, 2, 1, 3].

Return seq as a list of 30 integers in 1..15, or state "UNSATISFIABLE" if no such sequence exists.

Partial assignment (fixed values that must be respected):
- seq[3]=2, seq[11]=14, seq[13]=11, seq[26]=14, seq[27]=7, seq[28]=10
Return a complete solution consistent with these fixed assignments.


Prompt:
Construct a Langford sequence L(2,16): a sequence of length 32 containing exactly 2 copies of each integer from 1 to 16, such that the two occurrences of each value v are exactly v+1 positions apart (so copies of 1 are 1 position apart, copies of 2 are 2 positions apart, etc.). Example: L(2,3) has solution [2, 3, 1, 2, 1, 3].

Return seq as a list of 32 integers in 1..16, or state "UNSATISFIABLE" if no such sequence exists.

Partial assignment (fixed values that must be respected):
- seq[5]=14, seq[9]=4, seq[11]=15, seq[21]=9, seq[26]=13, seq[29]=10
Return a complete solution consistent with these fixed assignments.


Prompt:
Construct a Langford sequence L(2,6): a sequence of length 12 containing exactly 2 copies of each integer from 1 to 6, such that the two occurrences of each value v are exactly v+1 positions apart (so copies of 1 are 1 position apart, copies of 2 are 2 positions apart, etc.). Example: L(2,3) has solution [2, 3, 1, 2, 1, 3].

Return seq as a list of 12 integers in 1..6, or state "UNSATISFIABLE" if no such sequence exists.


Prompt:
Construct a Langford sequence L(2,7): a sequence of length 14 containing exactly 2 copies of each integer from 1 to 7, such that the two occurrences of each value v are exactly v+1 positions apart (so copies of 1 are 1 position apart, copies of 2 are 2 positions apart, etc.). Example: L(2,3) has solution [2, 3, 1, 2, 1, 3].

Return seq as a list of 14 integers in 1..7, or state "UNSATISFIABLE" if no such sequence exists.


Prompt:
Construct a Langford sequence L(2,8): a sequence of length 16 containing exactly 2 copies of each integer from 1 to 8, such that the two occurrences of each value v are exactly v+1 positions apart (so copies of 1 are 1 position apart, copies of 2 are 2 positions apart, etc.). Example: L(2,3) has solution [2, 3, 1, 2, 1, 3].

Return seq as a list of 16 integers in 1..8, or state "UNSATISFIABLE" if no such sequence exists.

Partial assignment (fixed values that must be respected):
- seq[4]=7, seq[8]=5, seq[10]=4
Return a complete solution consistent with these fixed assignments.


Prompt:
Construct a Langford sequence L(2,9): a sequence of length 18 containing exactly 2 copies of each integer from 1 to 9, such that the two occurrences of each value v are exactly v+1 positions apart (so copies of 1 are 1 position apart, copies of 2 are 2 positions apart, etc.). Example: L(2,3) has solution [2, 3, 1, 2, 1, 3].

Return seq as a list of 18 integers in 1..9, or state "UNSATISFIABLE" if no such sequence exists.


Prompt:
Find a binary sequence of length 10 over the alphabet {-1, +1} with low aperiodic autocorrelation: specifically, the sum over k=1..9 of C_k^2, where C_k = sum_{i=0}^{n-k-1} seq[i]*seq[i+k], must be at most 20.

Return seq as a list of 10 integers, each -1 or +1, or state "UNSATISFIABLE" if no such sequence exists.

Partial assignment (fixed values that must be respected):
- seq[0]=-1, seq[8]=1
- c[6]=-1
Return a complete solution consistent with these fixed assignments.


Prompt:
Find a binary sequence of length 11 over the alphabet {-1, +1} with low aperiodic autocorrelation: specifically, the sum over k=1..10 of C_k^2, where C_k = sum_{i=0}^{n-k-1} seq[i]*seq[i+k], must be at most 23.

Return seq as a list of 11 integers, each -1 or +1, or state "UNSATISFIABLE" if no such sequence exists.

Partial assignment (fixed values that must be respected):
- seq[1]=1, seq[8]=1
- c[6]=0, c[9]=1
Return a complete solution consistent with these fixed assignments.


Prompt:
Find a binary sequence of length 12 over the alphabet {-1, +1} with low aperiodic autocorrelation: specifically, the sum over k=1..11 of C_k^2, where C_k = sum_{i=0}^{n-k-1} seq[i]*seq[i+k], must be at most 25.

Return seq as a list of 12 integers, each -1 or +1, or state "UNSATISFIABLE" if no such sequence exists.


Prompt:
Find a binary sequence of length 13 over the alphabet {-1, +1} with low aperiodic autocorrelation: specifically, the sum over k=1..12 of C_k^2, where C_k = sum_{i=0}^{n-k-1} seq[i]*seq[i+k], must be at most 28.

Return seq as a list of 13 integers, each -1 or +1, or state "UNSATISFIABLE" if no such sequence exists.

Partial assignment (fixed values that must be respected):
- seq[8]=1, seq[10]=-1
- c[4]=2, c[5]=-1
Return a complete solution consistent with these fixed assignments.


Prompt:
Find a binary sequence of length 16 over the alphabet {-1, +1} with low aperiodic autocorrelation: specifically, the sum over k=1..15 of C_k^2, where C_k = sum_{i=0}^{n-k-1} seq[i]*seq[i+k], must be at most 37.

Return seq as a list of 16 integers, each -1 or +1, or state "UNSATISFIABLE" if no such sequence exists.


Prompt:
Find a binary sequence of length 17 over the alphabet {-1, +1} with low aperiodic autocorrelation: specifically, the sum over k=1..16 of C_k^2, where C_k = sum_{i=0}^{n-k-1} seq[i]*seq[i+k], must be at most 40.

Return seq as a list of 17 integers, each -1 or +1, or state "UNSATISFIABLE" if no such sequence exists.

Partial assignment (fixed values that must be respected):
- seq[1]=-1, seq[6]=1, seq[9]=-1
- c[7]=3, c[8]=-2, c[12]=0
Return a complete solution consistent with these fixed assignments.


Prompt:
Find a binary sequence of length 19 over the alphabet {-1, +1} with low aperiodic autocorrelation: specifically, the sum over k=1..18 of C_k^2, where C_k = sum_{i=0}^{n-k-1} seq[i]*seq[i+k], must be at most 46.

Return seq as a list of 19 integers, each -1 or +1, or state "UNSATISFIABLE" if no such sequence exists.

Partial assignment (fixed values that must be respected):
- seq[4]=-1, seq[12]=1, seq[13]=-1
- c[1]=-1, c[2]=0, c[7]=-3
Return a complete solution consistent with these fixed assignments.


Prompt:
Find a binary sequence of length 20 over the alphabet {-1, +1} with low aperiodic autocorrelation: specifically, the sum over k=1..19 of C_k^2, where C_k = sum_{i=0}^{n-k-1} seq[i]*seq[i+k], must be at most 49.

Return seq as a list of 20 integers, each -1 or +1, or state "UNSATISFIABLE" if no such sequence exists.


Prompt:
Find a binary sequence of length 21 over the alphabet {-1, +1} with low aperiodic autocorrelation: specifically, the sum over k=1..20 of C_k^2, where C_k = sum_{i=0}^{n-k-1} seq[i]*seq[i+k], must be at most 52.

Return seq as a list of 21 integers, each -1 or +1, or state "UNSATISFIABLE" if no such sequence exists.

Partial assignment (fixed values that must be respected):
- seq[5]=-1, seq[12]=-1, seq[17]=1, seq[20]=1
- c[0]=2, c[1]=-1, c[8]=2, c[19]=-1
Return a complete solution consistent with these fixed assignments.


Prompt:
Find a binary sequence of length 22 over the alphabet {-1, +1} with low aperiodic autocorrelation: specifically, the sum over k=1..21 of C_k^2, where C_k = sum_{i=0}^{n-k-1} seq[i]*seq[i+k], must be at most 56.

Return seq as a list of 22 integers, each -1 or +1, or state "UNSATISFIABLE" if no such sequence exists.


Prompt:
Find a binary sequence of length 23 over the alphabet {-1, +1} with low aperiodic autocorrelation: specifically, the sum over k=1..22 of C_k^2, where C_k = sum_{i=0}^{n-k-1} seq[i]*seq[i+k], must be at most 59.

Return seq as a list of 23 integers, each -1 or +1, or state "UNSATISFIABLE" if no such sequence exists.

Partial assignment (fixed values that must be respected):
- seq[5]=-1, seq[10]=-1, seq[13]=1, seq[19]=1
- c[12]=0, c[13]=-3, c[16]=0, c[19]=-1
Return a complete solution consistent with these fixed assignments.


Prompt:
Find a binary sequence of length 24 over the alphabet {-1, +1} with low aperiodic autocorrelation: specifically, the sum over k=1..23 of C_k^2, where C_k = sum_{i=0}^{n-k-1} seq[i]*seq[i+k], must be at most 62.

Return seq as a list of 24 integers, each -1 or +1, or state "UNSATISFIABLE" if no such sequence exists.

Partial assignment (fixed values that must be respected):
- seq[9]=-1, seq[14]=1, seq[15]=-1, seq[22]=-1
- c[3]=0, c[9]=-2, c[14]=-1, c[19]=0
Return a complete solution consistent with these fixed assignments.


Prompt:
Find a magic sequence of length 11. A magic sequence is a sequence x[0], x[1], ..., x[10] where each x[i] equals the count of how many times i appears in the sequence.

Return a list of 11 integers, or state "UNSATISFIABLE" if no solution exists.

Partial assignment (fixed values that must be respected):
- x[4]=0, x[10]=0
Return a complete solution consistent with these fixed assignments.


Prompt:
Find a magic sequence of length 12. A magic sequence is a sequence x[0], x[1], ..., x[11] where each x[i] equals the count of how many times i appears in the sequence.

Return a list of 12 integers, or state "UNSATISFIABLE" if no solution exists.

Partial assignment (fixed values that must be respected):
- x[7]=0, x[8]=1
Return a complete solution consistent with these fixed assignments.


Prompt:
Find a magic sequence of length 13. A magic sequence is a sequence x[0], x[1], ..., x[12] where each x[i] equals the count of how many times i appears in the sequence.

Return a list of 13 integers, or state "UNSATISFIABLE" if no solution exists.

Partial assignment (fixed values that must be respected):
- x[7]=0, x[11]=0
Return a complete solution consistent with these fixed assignments.


Prompt:
Find a magic sequence of length 14. A magic sequence is a sequence x[0], x[1], ..., x[13] where each x[i] equals the count of how many times i appears in the sequence.

Return a list of 14 integers, or state "UNSATISFIABLE" if no solution exists.

Partial assignment (fixed values that must be respected):
- x[11]=0, x[12]=0
Return a complete solution consistent with these fixed assignments.


Prompt:
Find a magic sequence of length 15. A magic sequence is a sequence x[0], x[1], ..., x[14] where each x[i] equals the count of how many times i appears in the sequence.

Return a list of 15 integers, or state "UNSATISFIABLE" if no solution exists.


Prompt:
Find a magic sequence of length 16. A magic sequence is a sequence x[0], x[1], ..., x[15] where each x[i] equals the count of how many times i appears in the sequence.

Return a list of 16 integers, or state "UNSATISFIABLE" if no solution exists.


Prompt:
Find a magic sequence of length 3. A magic sequence is a sequence x[0], x[1], ..., x[2] where each x[i] equals the count of how many times i appears in the sequence.

Return a list of 3 integers, or state "UNSATISFIABLE" if no solution exists.


Prompt:
Find a magic sequence of length 4. A magic sequence is a sequence x[0], x[1], ..., x[3] where each x[i] equals the count of how many times i appears in the sequence.

Return a list of 4 integers, or state "UNSATISFIABLE" if no solution exists.


Prompt:
Find a magic sequence of length 5. A magic sequence is a sequence x[0], x[1], ..., x[4] where each x[i] equals the count of how many times i appears in the sequence.

Return a list of 5 integers, or state "UNSATISFIABLE" if no solution exists.

Partial assignment (fixed values that must be respected):
- x[2]=2
Return a complete solution consistent with these fixed assignments.


Prompt:
Find a magic sequence of length 6. A magic sequence is a sequence x[0], x[1], ..., x[5] where each x[i] equals the count of how many times i appears in the sequence.

Return a list of 6 integers, or state "UNSATISFIABLE" if no solution exists.


Prompt:
Find a magic sequence of length 7. A magic sequence is a sequence x[0], x[1], ..., x[6] where each x[i] equals the count of how many times i appears in the sequence.

Return a list of 7 integers, or state "UNSATISFIABLE" if no solution exists.


Prompt:
Find a magic sequence of length 9. A magic sequence is a sequence x[0], x[1], ..., x[8] where each x[i] equals the count of how many times i appears in the sequence.

Return a list of 9 integers, or state "UNSATISFIABLE" if no solution exists.

Partial assignment (fixed values that must be respected):
- x[4]=0
Return a complete solution consistent with these fixed assignments.


Prompt:
Place 11 pigeons into 10 holes such that each hole contains at most one pigeon. Each pigeon must be assigned to exactly one hole (numbered 0 to 9).

Return a list of 11 integers where the i-th integer is the hole assigned to pigeon i, or state "UNSATISFIABLE" if no solution exists.


Prompt:
Place 12 pigeons into 11 holes such that each hole contains at most one pigeon. Each pigeon must be assigned to exactly one hole (numbered 0 to 10).

Return a list of 12 integers where the i-th integer is the hole assigned to pigeon i, or state "UNSATISFIABLE" if no solution exists.


Prompt:
Place 15 pigeons into 14 holes such that each hole contains at most one pigeon. Each pigeon must be assigned to exactly one hole (numbered 0 to 13).

Return a list of 15 integers where the i-th integer is the hole assigned to pigeon i, or state "UNSATISFIABLE" if no solution exists.


Prompt:
Place 17 pigeons into 16 holes such that each hole contains at most one pigeon. Each pigeon must be assigned to exactly one hole (numbered 0 to 15).

Return a list of 17 integers where the i-th integer is the hole assigned to pigeon i, or state "UNSATISFIABLE" if no solution exists.


Prompt:
Place 18 pigeons into 17 holes such that each hole contains at most one pigeon. Each pigeon must be assigned to exactly one hole (numbered 0 to 16).

Return a list of 18 integers where the i-th integer is the hole assigned to pigeon i, or state "UNSATISFIABLE" if no solution exists.


Prompt:
Place 19 pigeons into 18 holes such that each hole contains at most one pigeon. Each pigeon must be assigned to exactly one hole (numbered 0 to 17).

Return a list of 19 integers where the i-th integer is the hole assigned to pigeon i, or state "UNSATISFIABLE" if no solution exists.


Prompt:
Place 20 pigeons into 19 holes such that each hole contains at most one pigeon. Each pigeon must be assigned to exactly one hole (numbered 0 to 18).

Return a list of 20 integers where the i-th integer is the hole assigned to pigeon i, or state "UNSATISFIABLE" if no solution exists.


Prompt:
Place 21 pigeons into 20 holes such that each hole contains at most one pigeon. Each pigeon must be assigned to exactly one hole (numbered 0 to 19).

Return a list of 21 integers where the i-th integer is the hole assigned to pigeon i, or state "UNSATISFIABLE" if no solution exists.


Prompt:
Place 24 pigeons into 23 holes such that each hole contains at most one pigeon. Each pigeon must be assigned to exactly one hole (numbered 0 to 22).

Return a list of 24 integers where the i-th integer is the hole assigned to pigeon i, or state "UNSATISFIABLE" if no solution exists.


Prompt:
Place 4 pigeons into 3 holes such that each hole contains at most one pigeon. Each pigeon must be assigned to exactly one hole (numbered 0 to 2).

Return a list of 4 integers where the i-th integer is the hole assigned to pigeon i, or state "UNSATISFIABLE" if no solution exists.


Prompt:
Place 5 pigeons into 4 holes such that each hole contains at most one pigeon. Each pigeon must be assigned to exactly one hole (numbered 0 to 3).

Return a list of 5 integers where the i-th integer is the hole assigned to pigeon i, or state "UNSATISFIABLE" if no solution exists.


Prompt:
Place 7 pigeons into 6 holes such that each hole contains at most one pigeon. Each pigeon must be assigned to exactly one hole (numbered 0 to 5).

Return a list of 7 integers where the i-th integer is the hole assigned to pigeon i, or state "UNSATISFIABLE" if no solution exists.


Prompt:
Generate a graph with 15 vertices that is colorable with at most 2 colors, has girth (shortest cycle) at least 6, and has at least 16 edges.

Return the graph as a list of edges (u, v) with 0 <= u < v < 15, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 15 vertices that is colorable with at most 2 colors, has girth (shortest cycle) at least 4, and has at least 19 edges.

Return the graph as a list of edges (u, v) with 0 <= u < v < 15, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 13 vertices that is colorable with at most 2 colors, has girth (shortest cycle) at least 5, and has at least 8 edges.

Return the graph as a list of edges (u, v) with 0 <= u < v < 13, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (0,1), (0,8)
Return a complete solution consistent with these fixed assignments.


Prompt:
Generate a graph with 14 vertices that is colorable with at most 3 colors, has girth (shortest cycle) at least 7, and has at least 19 edges.

Return the graph as a list of edges (u, v) with 0 <= u < v < 14, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 14 vertices that is colorable with at most 3 colors, has girth (shortest cycle) at least 6, and has at least 20 edges.

Return the graph as a list of edges (u, v) with 0 <= u < v < 14, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 14 vertices that is colorable with at most 4 colors, has girth (shortest cycle) at least 4, and has at least 16 edges.

Return the graph as a list of edges (u, v) with 0 <= u < v < 14, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 14 vertices that is colorable with at most 4 colors, has girth (shortest cycle) at least 7, and has at least 19 edges.

Return the graph as a list of edges (u, v) with 0 <= u < v < 14, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 13 vertices that is colorable with at most 4 colors, has girth (shortest cycle) at least 7, and has at least 22 edges.

Return the graph as a list of edges (u, v) with 0 <= u < v < 13, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 13 vertices that is colorable with at most 5 colors, has girth (shortest cycle) at least 5, and has at least 10 edges.

Return the graph as a list of edges (u, v) with 0 <= u < v < 13, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (0,2), (0,4)
Return a complete solution consistent with these fixed assignments.


Prompt:
Generate a graph with 15 vertices that is colorable with at most 5 colors, has girth (shortest cycle) at least 7, and has at least 10 edges.

Return the graph as a list of edges (u, v) with 0 <= u < v < 15, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 14 vertices that is colorable with at most 5 colors, has girth (shortest cycle) at least 5, and has at least 16 edges.

Return the graph as a list of edges (u, v) with 0 <= u < v < 14, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 9 vertices that is colorable with at most 5 colors, has girth (shortest cycle) at least 5, and has at least 9 edges.

Return the graph as a list of edges (u, v) with 0 <= u < v < 9, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (4,8)
Return a complete solution consistent with these fixed assignments.


Prompt:
Generate a graph with 16 vertices where the maximum clique size is at most 2, the chromatic number is at most 2, and every vertex has degree at least 1.

Return the graph as a list of edges (u, v) with 0 <= u < v < 16, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 12 vertices where the maximum clique size is at most 2, the chromatic number is at most 2, and every vertex has degree at least 4.

Return the graph as a list of edges (u, v) with 0 <= u < v < 12, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (2,8), (1,9), (0,6), (3,10), (0,9), (3,9)
Return a complete solution consistent with these fixed assignments.


Prompt:
Generate a graph with 8 vertices where the maximum clique size is at most 4, the chromatic number is at most 2, and every vertex has degree at least 1.

Return the graph as a list of edges (u, v) with 0 <= u < v < 8, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (2,7), (0,5)
Return a complete solution consistent with these fixed assignments.


Prompt:
Generate a graph with 11 vertices where the maximum clique size is at most 4, the chromatic number is at most 2, and every vertex has degree at least 3.

Return the graph as a list of edges (u, v) with 0 <= u < v < 11, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (1,10), (0,4), (0,8), (2,9)
Return a complete solution consistent with these fixed assignments.


Prompt:
Generate a graph with 12 vertices where the maximum clique size is at most 5, the chromatic number is at most 3, and every vertex has degree at least 3.

Return the graph as a list of edges (u, v) with 0 <= u < v < 12, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (1,2), (0,2), (1,4), (2,9), (2,7)
Return a complete solution consistent with these fixed assignments.


Prompt:
Generate a graph with 15 vertices where the maximum clique size is at most 2, the chromatic number is at most 4, and every vertex has degree at least 1.

Return the graph as a list of edges (u, v) with 0 <= u < v < 15, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (1,8), (1,9), (0,10), (0,6), (0,5)
Return a complete solution consistent with these fixed assignments.


Prompt:
Generate a graph with 16 vertices where the maximum clique size is at most 3, the chromatic number is at most 4, and every vertex has degree at least 1.

Return the graph as a list of edges (u, v) with 0 <= u < v < 16, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (1,7), (1,10), (0,8), (0,3), (1,14)
Return a complete solution consistent with these fixed assignments.


Prompt:
Generate a graph with 11 vertices where the maximum clique size is at most 5, the chromatic number is at most 4, and every vertex has degree at least 4.

Return the graph as a list of edges (u, v) with 0 <= u < v < 11, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 15 vertices where the maximum clique size is at most 2, the chromatic number is at most 5, and every vertex has degree at least 3.

Return the graph as a list of edges (u, v) with 0 <= u < v < 15, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 14 vertices where the maximum clique size is at most 4, the chromatic number is at most 5, and every vertex has degree at least 1.

Return the graph as a list of edges (u, v) with 0 <= u < v < 14, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (1,4), (1,2), (1,9), (0,3)
Return a complete solution consistent with these fixed assignments.


Prompt:
Generate a graph with 12 vertices where the maximum clique size is at most 4, the chromatic number is at most 5, and every vertex has degree at least 4.

Return the graph as a list of edges (u, v) with 0 <= u < v < 12, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (0,10), (0,11), (0,7), (3,10), (2,9), (1,9), (3,6)
Return a complete solution consistent with these fixed assignments.


Prompt:
Generate a graph with 13 vertices where the maximum clique size is at most 5, the chromatic number is at most 5, and every vertex has degree at least 4.

Return the graph as a list of edges (u, v) with 0 <= u < v < 13, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (1,3), (2,7), (3,11), (3,10), (0,12), (2,10), (2,4), (1,11)
Return a complete solution consistent with these fixed assignments.


Prompt:
Generate a graph with 13 vertices that satisfies: minimum degree >= 1, maximum degree <= 4, edges between 16 and 19, maximum clique size <= 4, maximum independent set size <= 3, chromatic number <= 2, chromatic number >= 2, vertex-connectivity >= 1, girth >= 5, C_3-free, consists of exactly 2 vertex-disjoint cliques of size 4 (every vertex belongs to one of these cliques, and the only edges are within these cliques).

Return the graph as a list of edges (u, v) with 0 <= u < v < 13, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 12 vertices that satisfies: minimum degree >= 2, maximum degree <= 5, edges between 12 and 18, maximum clique size <= 4, maximum independent set size <= 3, chromatic number <= 4, chromatic number >= 3, vertex-connectivity >= 2, girth >= 5, C_4-free, consists of exactly 2 vertex-disjoint cliques of size 3 (every vertex belongs to one of these cliques, and the only edges are within these cliques).

Return the graph as a list of edges (u, v) with 0 <= u < v < 12, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 12 vertices that satisfies: minimum degree >= 3, maximum degree <= 4, edges between 22 and 16, maximum clique size <= 4, maximum independent set size <= 5, chromatic number <= 4, chromatic number >= 3, vertex-connectivity >= 2, girth >= 4, C_4-free, consists of exactly 2 vertex-disjoint cliques of size 3 (every vertex belongs to one of these cliques, and the only edges are within these cliques).

Return the graph as a list of edges (u, v) with 0 <= u < v < 12, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 11 vertices that satisfies: minimum degree >= 2, maximum degree <= 4, edges between 24 and 19, maximum clique size <= 4, maximum independent set size <= 2, chromatic number <= 2, chromatic number >= 2, vertex-connectivity >= 2, girth >= 5, C_4-free, consists of exactly 2 vertex-disjoint cliques of size 3 (every vertex belongs to one of these cliques, and the only edges are within these cliques).

Return the graph as a list of edges (u, v) with 0 <= u < v < 11, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 15 vertices that satisfies: minimum degree >= 1, maximum degree <= 3, edges between 12 and 22, maximum clique size <= 3, maximum independent set size <= 5, chromatic number <= 3, chromatic number >= 2, vertex-connectivity >= 2, girth >= 3, C_3-free, consists of exactly 2 vertex-disjoint cliques of size 3 (every vertex belongs to one of these cliques, and the only edges are within these cliques).

Return the graph as a list of edges (u, v) with 0 <= u < v < 15, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 13 vertices that satisfies: minimum degree >= 1, maximum degree <= 3, edges between 13 and 23, maximum clique size <= 3, maximum independent set size <= 2, chromatic number <= 4, chromatic number >= 3, vertex-connectivity >= 2, girth >= 4, C_4-free, consists of exactly 1 vertex-disjoint cliques of size 4 (every vertex belongs to one of these cliques, and the only edges are within these cliques).

Return the graph as a list of edges (u, v) with 0 <= u < v < 13, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 16 vertices that satisfies: minimum degree >= 1, maximum degree <= 3, edges between 21 and 19, maximum clique size <= 2, maximum independent set size <= 5, chromatic number <= 3, chromatic number >= 2, vertex-connectivity >= 1, girth >= 4, C_3-free, consists of exactly 1 vertex-disjoint cliques of size 3 (every vertex belongs to one of these cliques, and the only edges are within these cliques).

Return the graph as a list of edges (u, v) with 0 <= u < v < 16, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 15 vertices that satisfies: minimum degree >= 3, maximum degree <= 4, edges between 13 and 21, maximum clique size <= 2, maximum independent set size <= 5, chromatic number <= 4, chromatic number >= 2, vertex-connectivity >= 2, girth >= 3, C_3-free, consists of exactly 2 vertex-disjoint cliques of size 3 (every vertex belongs to one of these cliques, and the only edges are within these cliques).

Return the graph as a list of edges (u, v) with 0 <= u < v < 15, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 12 vertices that satisfies: minimum degree >= 2, maximum degree <= 4, edges between 10 and 27, maximum clique size <= 2, maximum independent set size <= 2, chromatic number <= 4, chromatic number >= 3, vertex-connectivity >= 1, girth >= 5, C_3-free, consists of exactly 2 vertex-disjoint cliques of size 3 (every vertex belongs to one of these cliques, and the only edges are within these cliques).

Return the graph as a list of edges (u, v) with 0 <= u < v < 12, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 10 vertices that satisfies: minimum degree >= 3, maximum degree <= 5, edges between 10 and 28, maximum clique size <= 4, maximum independent set size <= 3, chromatic number <= 4, chromatic number >= 3, vertex-connectivity >= 2, girth >= 5, C_3-free, consists of exactly 1 vertex-disjoint cliques of size 4 (every vertex belongs to one of these cliques, and the only edges are within these cliques).

Return the graph as a list of edges (u, v) with 0 <= u < v < 10, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 10 vertices that satisfies: minimum degree >= 1, maximum degree <= 4, edges between 15 and 18, maximum clique size <= 3, maximum independent set size <= 2, chromatic number <= 4, chromatic number >= 3, vertex-connectivity >= 2, girth >= 5, C_4-free, consists of exactly 1 vertex-disjoint cliques of size 4 (every vertex belongs to one of these cliques, and the only edges are within these cliques).

Return the graph as a list of edges (u, v) with 0 <= u < v < 10, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 15 vertices that satisfies: minimum degree >= 3, maximum degree <= 5, edges between 18 and 15, maximum clique size <= 3, maximum independent set size <= 3, chromatic number <= 2, chromatic number >= 2, vertex-connectivity >= 1, girth >= 3, C_3-free, consists of exactly 2 vertex-disjoint cliques of size 3 (every vertex belongs to one of these cliques, and the only edges are within these cliques).

Return the graph as a list of edges (u, v) with 0 <= u < v < 15, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 10 vertices that consists of exactly 1 vertex-disjoint clique(s), each of size 3. Every vertex must belong to one of these cliques, and the only edges in the graph are those within these cliques.

Return the graph as a list of edges (u, v) with 0 <= u < v < 10, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 15 vertices that consists of exactly 1 vertex-disjoint clique(s), each of size 3. Every vertex must belong to one of these cliques, and the only edges in the graph are those within these cliques.

Return the graph as a list of edges (u, v) with 0 <= u < v < 15, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 14 vertices that consists of exactly 3 vertex-disjoint clique(s), each of size 3. Every vertex must belong to one of these cliques, and the only edges in the graph are those within these cliques.

Return the graph as a list of edges (u, v) with 0 <= u < v < 14, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 16 vertices that consists of exactly 3 vertex-disjoint clique(s), each of size 3. Every vertex must belong to one of these cliques, and the only edges in the graph are those within these cliques.

Return the graph as a list of edges (u, v) with 0 <= u < v < 16, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 16 vertices that consists of exactly 4 vertex-disjoint clique(s), each of size 3. Every vertex must belong to one of these cliques, and the only edges in the graph are those within these cliques.

Return the graph as a list of edges (u, v) with 0 <= u < v < 16, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 13 vertices that consists of exactly 1 vertex-disjoint clique(s), each of size 4. Every vertex must belong to one of these cliques, and the only edges in the graph are those within these cliques.

Return the graph as a list of edges (u, v) with 0 <= u < v < 13, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 14 vertices that consists of exactly 1 vertex-disjoint clique(s), each of size 4. Every vertex must belong to one of these cliques, and the only edges in the graph are those within these cliques.

Return the graph as a list of edges (u, v) with 0 <= u < v < 14, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 10 vertices that consists of exactly 2 vertex-disjoint clique(s), each of size 4. Every vertex must belong to one of these cliques, and the only edges in the graph are those within these cliques.

Return the graph as a list of edges (u, v) with 0 <= u < v < 10, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 15 vertices that consists of exactly 2 vertex-disjoint clique(s), each of size 4. Every vertex must belong to one of these cliques, and the only edges in the graph are those within these cliques.

Return the graph as a list of edges (u, v) with 0 <= u < v < 15, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 12 vertices that consists of exactly 1 vertex-disjoint clique(s), each of size 5. Every vertex must belong to one of these cliques, and the only edges in the graph are those within these cliques.

Return the graph as a list of edges (u, v) with 0 <= u < v < 12, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 13 vertices that consists of exactly 2 vertex-disjoint clique(s), each of size 5. Every vertex must belong to one of these cliques, and the only edges in the graph are those within these cliques.

Return the graph as a list of edges (u, v) with 0 <= u < v < 13, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 14 vertices that consists of exactly 2 vertex-disjoint clique(s), each of size 5. Every vertex must belong to one of these cliques, and the only edges in the graph are those within these cliques.

Return the graph as a list of edges (u, v) with 0 <= u < v < 14, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 12 vertices where every vertex has degree between 1 and 3.

Return the graph as a list of edges (u, v) with 0 <= u < v < 12, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (2,9), (2,11), (2,10)
Return a complete solution consistent with these fixed assignments.


Prompt:
Generate a graph with 11 vertices where every vertex has degree between 3 and 3.

Return the graph as a list of edges (u, v) with 0 <= u < v < 11, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 12 vertices where every vertex has degree between 3 and 3.

Return the graph as a list of edges (u, v) with 0 <= u < v < 12, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (0,9), (1,10), (2,9)
Return a complete solution consistent with these fixed assignments.


Prompt:
Generate a graph with 10 vertices where every vertex has degree between 3 and 4.

Return the graph as a list of edges (u, v) with 0 <= u < v < 10, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 11 vertices where every vertex has degree between 1 and 5.

Return the graph as a list of edges (u, v) with 0 <= u < v < 11, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (2,8), (2,6), (2,10), (3,8), (0,10)
Return a complete solution consistent with these fixed assignments.


Prompt:
Generate a graph with 15 vertices where every vertex has degree between 1 and 5.

Return the graph as a list of edges (u, v) with 0 <= u < v < 15, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 15 vertices where every vertex has degree between 2 and 5.

Return the graph as a list of edges (u, v) with 0 <= u < v < 15, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 13 vertices where every vertex has degree between 4 and 5.

Return the graph as a list of edges (u, v) with 0 <= u < v < 13, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 14 vertices where every vertex has degree between 4 and 5.

Return the graph as a list of edges (u, v) with 0 <= u < v < 14, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (1,11), (0,13), (7,9), (1,13), (4,11), (6,9)
Return a complete solution consistent with these fixed assignments.


Prompt:
Generate a graph with 12 vertices where every vertex has degree between 3 and 6.

Return the graph as a list of edges (u, v) with 0 <= u < v < 12, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 14 vertices where every vertex has degree between 3 and 6.

Return the graph as a list of edges (u, v) with 0 <= u < v < 14, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (4,5), (0,10), (2,7), (6,8), (2,8), (1,6), (0,9), (5,8)
Return a complete solution consistent with these fixed assignments.


Prompt:
Generate a graph with 11 vertices where every vertex has degree between 4 and 6.

Return the graph as a list of edges (u, v) with 0 <= u < v < 11, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (5,6), (3,8), (0,9), (3,7), (3,9), (4,9)
Return a complete solution consistent with these fixed assignments.


Prompt:
Generate a graph with 10 vertices where the girth (shortest cycle) is at least 5 and every vertex has degree between 2 and 3.

Return the graph as a list of edges (u, v) with 0 <= u < v < 10, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 14 vertices where the girth (shortest cycle) is at least 5 and every vertex has degree between 3 and 3.

Return the graph as a list of edges (u, v) with 0 <= u < v < 14, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 8 vertices where the girth (shortest cycle) is at least 4 and every vertex has degree between 4 and 4.

Return the graph as a list of edges (u, v) with 0 <= u < v < 8, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (0,4), (0,7), (2,4)
Return a complete solution consistent with these fixed assignments.


Prompt:
Generate a graph with 8 vertices where the girth (shortest cycle) is at least 5 and every vertex has degree between 4 and 4.

Return the graph as a list of edges (u, v) with 0 <= u < v < 8, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 15 vertices where the girth (shortest cycle) is at least 4 and every vertex has degree between 2 and 5.

Return the graph as a list of edges (u, v) with 0 <= u < v < 15, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (2,12), (3,10), (6,9), (3,13), (3,12), (0,13)
Return a complete solution consistent with these fixed assignments.


Prompt:
Generate a graph with 11 vertices where the girth (shortest cycle) is at least 6 and every vertex has degree between 2 and 5.

Return the graph as a list of edges (u, v) with 0 <= u < v < 11, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 9 vertices where the girth (shortest cycle) is at least 8 and every vertex has degree between 2 and 5.

Return the graph as a list of edges (u, v) with 0 <= u < v < 9, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (2,5)
Return a complete solution consistent with these fixed assignments.


Prompt:
Generate a graph with 16 vertices where the girth (shortest cycle) is at least 6 and every vertex has degree between 4 and 5.

Return the graph as a list of edges (u, v) with 0 <= u < v < 16, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 12 vertices where the girth (shortest cycle) is at least 6 and every vertex has degree between 3 and 6.

Return the graph as a list of edges (u, v) with 0 <= u < v < 12, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 15 vertices where the girth (shortest cycle) is at least 6 and every vertex has degree between 3 and 6.

Return the graph as a list of edges (u, v) with 0 <= u < v < 15, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 9 vertices where the girth (shortest cycle) is at least 7 and every vertex has degree between 3 and 6.

Return the graph as a list of edges (u, v) with 0 <= u < v < 9, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 8 vertices where the girth (shortest cycle) is at least 7 and every vertex has degree between 4 and 6.

Return the graph as a list of edges (u, v) with 0 <= u < v < 8, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 12 vertices that satisfies: minimum degree >= 1, maximum degree <= 2, edges between 9 and 11, chromatic number <= 2, chromatic number >= 2.

Return the graph as a list of edges (u, v) with 0 <= u < v < 12, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (5,7), (4,6)
Return a complete solution consistent with these fixed assignments.


Prompt:
Generate a graph with 12 vertices that satisfies: minimum degree >= 1, maximum degree <= 2, edges between 15 and 12, chromatic number <= 4, chromatic number >= 2.

Return the graph as a list of edges (u, v) with 0 <= u < v < 12, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 13 vertices that satisfies: minimum degree >= 2, maximum degree <= 2, edges between 9 and 14, chromatic number <= 4, chromatic number >= 2.

Return the graph as a list of edges (u, v) with 0 <= u < v < 13, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 9 vertices that satisfies: minimum degree >= 2, maximum degree <= 3, edges between 12 and 17, chromatic number <= 4, chromatic number >= 2.

Return the graph as a list of edges (u, v) with 0 <= u < v < 9, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (0,8), (6,7)
Return a complete solution consistent with these fixed assignments.


Prompt:
Generate a graph with 9 vertices that satisfies: minimum degree >= 2, maximum degree <= 3, edges between 18 and 10, chromatic number <= 4, chromatic number >= 2.

Return the graph as a list of edges (u, v) with 0 <= u < v < 9, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 9 vertices that satisfies: minimum degree >= 2, maximum degree <= 3, edges between 9 and 18, chromatic number <= 4, chromatic number >= 2.

Return the graph as a list of edges (u, v) with 0 <= u < v < 9, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (5,6), (2,3)
Return a complete solution consistent with these fixed assignments.


Prompt:
Generate a graph with 8 vertices that satisfies: minimum degree >= 3, maximum degree <= 3, edges between 16 and 16, chromatic number <= 2, chromatic number >= 2.

Return the graph as a list of edges (u, v) with 0 <= u < v < 8, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 13 vertices that satisfies: minimum degree >= 1, maximum degree <= 4, edges between 9 and 17, chromatic number <= 2, chromatic number >= 3.

Return the graph as a list of edges (u, v) with 0 <= u < v < 13, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 8 vertices that satisfies: minimum degree >= 2, maximum degree <= 4, edges between 14 and 10, chromatic number <= 3, chromatic number >= 3.

Return the graph as a list of edges (u, v) with 0 <= u < v < 8, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 9 vertices that satisfies: minimum degree >= 2, maximum degree <= 4, edges between 17 and 11, chromatic number <= 3, chromatic number >= 3.

Return the graph as a list of edges (u, v) with 0 <= u < v < 9, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 8 vertices that satisfies: minimum degree >= 3, maximum degree <= 4, edges between 13 and 16, chromatic number <= 3, chromatic number >= 2.

Return the graph as a list of edges (u, v) with 0 <= u < v < 8, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (1,7), (0,7)
Return a complete solution consistent with these fixed assignments.


Prompt:
Generate a graph with 11 vertices that satisfies: minimum degree >= 3, maximum degree <= 5, edges between 7 and 19, chromatic number <= 4, chromatic number >= 3.

Return the graph as a list of edges (u, v) with 0 <= u < v < 11, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (3,7), (2,9), (0,9)
Return a complete solution consistent with these fixed assignments.


Prompt:
Generate a graph with 8 vertices where the maximum independent set size is at most 2 and the vertex-connectivity is at least 1.

Return the graph as a list of edges (u, v) with 0 <= u < v < 8, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 9 vertices where the maximum independent set size is at most 2 and the vertex-connectivity is at least 2.

Return the graph as a list of edges (u, v) with 0 <= u < v < 9, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (3,6), (1,4), (0,2), (3,8), (1,3), (7,8), (5,6)
Return a complete solution consistent with these fixed assignments.


Prompt:
Generate a graph with 10 vertices where the maximum independent set size is at most 3 and the vertex-connectivity is at least 2.

Return the graph as a list of edges (u, v) with 0 <= u < v < 10, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 12 vertices where the maximum independent set size is at most 3 and the vertex-connectivity is at least 4.

Return the graph as a list of edges (u, v) with 0 <= u < v < 12, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 9 vertices where the maximum independent set size is at most 4 and the vertex-connectivity is at least 1.

Return the graph as a list of edges (u, v) with 0 <= u < v < 9, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 11 vertices where the maximum independent set size is at most 4 and the vertex-connectivity is at least 3.

Return the graph as a list of edges (u, v) with 0 <= u < v < 11, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (0,6), (1,10), (3,6), (3,8), (2,7), (0,4), (1,6), (0,8), (1,2), (5,10), (0,3)
Return a complete solution consistent with these fixed assignments.


Prompt:
Generate a graph with 11 vertices where the maximum independent set size is at most 4 and the vertex-connectivity is at least 4.

Return the graph as a list of edges (u, v) with 0 <= u < v < 11, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (5,9), (2,7), (1,4), (0,8), (1,5), (1,9), (6,8), (5,6), (4,8), (7,8), (3,8)
Return a complete solution consistent with these fixed assignments.


Prompt:
Generate a graph with 15 vertices where the maximum independent set size is at most 4 and the vertex-connectivity is at least 4.

Return the graph as a list of edges (u, v) with 0 <= u < v < 15, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (2,5), (8,11), (0,2), (5,9), (2,10), (2,11), (4,13), (7,14), (2,13), (11,14), (4,6), (9,10), (4,14), (12,13), (3,4), (0,7), (1,5), (2,3), (5,8), (11,12), (7,10)
Return a complete solution consistent with these fixed assignments.


Prompt:
Generate a graph with 9 vertices where the maximum independent set size is at most 4 and the vertex-connectivity is at least 4.

Return the graph as a list of edges (u, v) with 0 <= u < v < 9, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 16 vertices where the maximum independent set size is at most 5 and the vertex-connectivity is at least 1.

Return the graph as a list of edges (u, v) with 0 <= u < v < 16, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (6,14), (13,14), (0,12), (2,11), (2,10), (8,14), (7,14), (6,15), (5,13), (12,15), (2,14), (0,11), (0,8), (8,15), (14,15), (4,5), (5,7), (2,13), (0,2), (0,1), (0,10), (0,5), (10,11), (3,7)
Return a complete solution consistent with these fixed assignments.


Prompt:
Generate a graph with 11 vertices where the maximum independent set size is at most 5 and the vertex-connectivity is at least 2.

Return the graph as a list of edges (u, v) with 0 <= u < v < 11, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 10 vertices where the maximum independent set size is at most 6 and the vertex-connectivity is at least 2.

Return the graph as a list of edges (u, v) with 0 <= u < v < 10, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (0,3), (3,8), (2,9), (0,5), (0,4), (1,7), (1,4), (1,6), (4,8)
Return a complete solution consistent with these fixed assignments.


Prompt:
Generate a graph with 14 vertices with vertex-connectivity at least 1.

Return the graph as a list of edges (u, v) with 0 <= u < v < 14, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 15 vertices with vertex-connectivity at least 1.

Return the graph as a list of edges (u, v) with 0 <= u < v < 15, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 8 vertices with vertex-connectivity at least 1.

Return the graph as a list of edges (u, v) with 0 <= u < v < 8, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 9 vertices with vertex-connectivity at least 1.

Return the graph as a list of edges (u, v) with 0 <= u < v < 9, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (2,6), (5,7), (2,5), (1,2), (4,6), (3,6), (1,5)
Return a complete solution consistent with these fixed assignments.


Prompt:
Generate a graph with 16 vertices with vertex-connectivity at least 2.

Return the graph as a list of edges (u, v) with 0 <= u < v < 16, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (5,13), (8,15), (1,4), (0,4), (8,10), (0,15), (3,10), (7,12), (10,15), (3,7), (4,5), (2,9), (0,9), (3,15), (0,10), (6,9), (3,5), (3,4), (1,14), (6,8), (10,11), (3,14), (4,6), (7,9)
Return a complete solution consistent with these fixed assignments.


Prompt:
Generate a graph with 18 vertices with vertex-connectivity at least 2.

Return the graph as a list of edges (u, v) with 0 <= u < v < 18, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 14 vertices with vertex-connectivity at least 3.

Return the graph as a list of edges (u, v) with 0 <= u < v < 14, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 15 vertices with vertex-connectivity at least 3.

Return the graph as a list of edges (u, v) with 0 <= u < v < 15, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 11 vertices with vertex-connectivity at least 4.

Return the graph as a list of edges (u, v) with 0 <= u < v < 11, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 17 vertices with vertex-connectivity at least 4.

Return the graph as a list of edges (u, v) with 0 <= u < v < 17, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 18 vertices with vertex-connectivity at least 4.

Return the graph as a list of edges (u, v) with 0 <= u < v < 18, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (2,9), (0,10), (15,17), (12,15), (2,6), (3,17), (0,14), (11,13), (9,14), (6,12), (10,13), (1,10), (11,17), (10,16), (1,5), (6,14), (5,10), (1,9), (4,14), (15,16), (3,12), (13,16), (2,3), (10,15), (7,13), (6,16), (2,8), (3,10), (0,15), (2,16)
Return a complete solution consistent with these fixed assignments.


Prompt:
Generate a graph with 16 vertices with vertex-connectivity at least 5.

Return the graph as a list of edges (u, v) with 0 <= u < v < 16, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (1,10), (4,10), (7,10), (1,6), (5,13), (3,6), (0,6), (0,15), (9,13), (3,12), (2,7), (4,14), (1,12), (8,13), (11,14), (5,12), (4,12), (5,14), (0,4), (6,7), (8,9), (9,12), (2,8), (10,14)
Return a complete solution consistent with these fixed assignments.


Prompt:
Generate a graph with 16 vertices where the minimum degree is at least 1.

Return the graph as a list of edges (u, v) with 0 <= u < v < 16, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 11 vertices where the minimum degree is at least 3.

Return the graph as a list of edges (u, v) with 0 <= u < v < 11, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 12 vertices where the minimum degree is at least 3.

Return the graph as a list of edges (u, v) with 0 <= u < v < 12, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (7,9), (1,3), (0,8), (6,11), (1,6), (3,11), (4,9), (4,11), (6,8), (10,11), (5,11), (1,10), (2,6)
Return a complete solution consistent with these fixed assignments.


Prompt:
Generate a graph with 8 vertices where the minimum degree is at least 3.

Return the graph as a list of edges (u, v) with 0 <= u < v < 8, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (2,6), (0,6), (4,7), (6,7), (3,7)
Return a complete solution consistent with these fixed assignments.


Prompt:
Generate a graph with 10 vertices where the minimum degree is at least 4.

Return the graph as a list of edges (u, v) with 0 <= u < v < 10, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (8,9), (2,8), (2,7), (2,9), (0,1), (0,8), (1,8), (3,8), (5,8)
Return a complete solution consistent with these fixed assignments.


Prompt:
Generate a graph with 9 vertices where the minimum degree is at least 4.

Return the graph as a list of edges (u, v) with 0 <= u < v < 9, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (1,5), (0,1), (1,3), (3,6), (2,8), (4,5), (4,6)
Return a complete solution consistent with these fixed assignments.


Prompt:
Generate a graph with 13 vertices where the minimum degree is at least 5.

Return the graph as a list of edges (u, v) with 0 <= u < v < 13, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 18 vertices where the minimum degree is at least 5.

Return the graph as a list of edges (u, v) with 0 <= u < v < 18, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (3,15), (11,13), (13,17), (2,11), (11,17), (2,12), (7,17), (9,17), (3,5), (0,16), (1,14), (7,9), (0,9), (6,11), (3,11), (0,2), (1,17), (4,8), (5,15), (1,15), (2,15), (15,17), (4,16), (0,8), (4,6), (8,13), (13,14), (2,10), (6,13), (9,12)
Return a complete solution consistent with these fixed assignments.


Prompt:
Generate a graph with 9 vertices where the minimum degree is at least 5.

Return the graph as a list of edges (u, v) with 0 <= u < v < 9, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (5,7), (1,5), (3,6), (5,6), (4,5), (4,6), (0,7)
Return a complete solution consistent with these fixed assignments.


Prompt:
Generate a graph with 10 vertices where the minimum degree is at least 6.

Return the graph as a list of edges (u, v) with 0 <= u < v < 10, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 13 vertices where the minimum degree is at least 6.

Return the graph as a list of edges (u, v) with 0 <= u < v < 13, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 15 vertices where the minimum degree is at least 6.

Return the graph as a list of edges (u, v) with 0 <= u < v < 15, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 10 vertices where the girth (shortest cycle length) is at least 3.

Return the graph as a list of edges (u, v) with 0 <= u < v < 10, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (0,7), (0,4), (0,2), (2,9), (1,4), (4,6), (2,8), (1,8), (3,9)
Return a complete solution consistent with these fixed assignments.


Prompt:
Generate a graph with 12 vertices where the girth (shortest cycle length) is at least 3.

Return the graph as a list of edges (u, v) with 0 <= u < v < 12, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 17 vertices where the girth (shortest cycle length) is at least 3.

Return the graph as a list of edges (u, v) with 0 <= u < v < 17, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (1,16), (5,15), (4,13), (2,7), (6,13), (8,9), (2,6), (8,15), (5,6), (7,15), (2,5), (3,12), (1,11), (9,14), (4,7), (12,16), (4,16), (1,9), (7,11), (6,11), (13,14), (3,16), (10,12), (12,13), (0,12), (1,3), (0,14)
Return a complete solution consistent with these fixed assignments.


Prompt:
Generate a graph with 9 vertices where the girth (shortest cycle length) is at least 3.

Return the graph as a list of edges (u, v) with 0 <= u < v < 9, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 10 vertices where the girth (shortest cycle length) is at least 4.

Return the graph as a list of edges (u, v) with 0 <= u < v < 10, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 17 vertices where the girth (shortest cycle length) is at least 4.

Return the graph as a list of edges (u, v) with 0 <= u < v < 17, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 11 vertices where the girth (shortest cycle length) is at least 6.

Return the graph as a list of edges (u, v) with 0 <= u < v < 11, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 14 vertices where the girth (shortest cycle length) is at least 6.

Return the graph as a list of edges (u, v) with 0 <= u < v < 14, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (9,13), (4,13)
Return a complete solution consistent with these fixed assignments.


Prompt:
Generate a graph with 8 vertices where the girth (shortest cycle length) is at least 6.

Return the graph as a list of edges (u, v) with 0 <= u < v < 8, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 14 vertices where the girth (shortest cycle length) is at least 8.

Return the graph as a list of edges (u, v) with 0 <= u < v < 14, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 16 vertices where the girth (shortest cycle length) is at least 8.

Return the graph as a list of edges (u, v) with 0 <= u < v < 16, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (11,15), (3,15), (5,15)
Return a complete solution consistent with these fixed assignments.


Prompt:
Generate a graph with 9 vertices where the girth (shortest cycle length) is at least 8.

Return the graph as a list of edges (u, v) with 0 <= u < v < 9, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a maximal triangle-free graph with 10 vertices.

Return the graph as a list of edges (u, v) with 0 <= u < v < 10, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a maximal triangle-free graph with 12 vertices.

Return the graph as a list of edges (u, v) with 0 <= u < v < 12, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a maximal triangle-free graph with 13 vertices.

Return the graph as a list of edges (u, v) with 0 <= u < v < 13, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a maximal triangle-free graph with 14 vertices.

Return the graph as a list of edges (u, v) with 0 <= u < v < 14, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a maximal triangle-free graph with 15 vertices.

Return the graph as a list of edges (u, v) with 0 <= u < v < 15, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (2,14), (6,13), (7,13), (10,13), (6,14)
Return a complete solution consistent with these fixed assignments.


Prompt:
Generate a maximal triangle-free graph with 16 vertices.

Return the graph as a list of edges (u, v) with 0 <= u < v < 16, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a maximal triangle-free graph with 17 vertices.

Return the graph as a list of edges (u, v) with 0 <= u < v < 17, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (0,15), (3,15), (6,15), (9,15), (11,15), (14,16)
Return a complete solution consistent with these fixed assignments.


Prompt:
Generate a maximal triangle-free graph with 20 vertices.

Return the graph as a list of edges (u, v) with 0 <= u < v < 20, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (1,18), (4,18), (16,18), (4,19), (17,18), (11,19), (3,19)
Return a complete solution consistent with these fixed assignments.


Prompt:
Generate a maximal triangle-free graph with 6 vertices.

Return the graph as a list of edges (u, v) with 0 <= u < v < 6, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (3,5)
Return a complete solution consistent with these fixed assignments.


Prompt:
Generate a maximal triangle-free graph with 7 vertices.

Return the graph as a list of edges (u, v) with 0 <= u < v < 7, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (0,6), (1,5)
Return a complete solution consistent with these fixed assignments.


Prompt:
Generate a maximal triangle-free graph with 8 vertices.

Return the graph as a list of edges (u, v) with 0 <= u < v < 8, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (0,6), (2,6)
Return a complete solution consistent with these fixed assignments.


Prompt:
Generate a maximal triangle-free graph with 9 vertices.

Return the graph as a list of edges (u, v) with 0 <= u < v < 9, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (4,7), (0,7)
Return a complete solution consistent with these fixed assignments.


Prompt:
Generate a graph with 8 vertices where the number of edges is between 11 and 16.

Return the graph as a list of edges (u, v) with 0 <= u < v < 8, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 11 vertices where the number of edges is between 23 and 24.

Return the graph as a list of edges (u, v) with 0 <= u < v < 11, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 10 vertices where the number of edges is between 21 and 30.

Return the graph as a list of edges (u, v) with 0 <= u < v < 10, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 10 vertices where the number of edges is between 27 and 31.

Return the graph as a list of edges (u, v) with 0 <= u < v < 10, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (4,8), (3,7), (6,7), (2,8), (0,8), (0,5)
Return a complete solution consistent with these fixed assignments.


Prompt:
Generate a graph with 14 vertices where the number of edges is between 33 and 33.

Return the graph as a list of edges (u, v) with 0 <= u < v < 14, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 11 vertices where the number of edges is between 30 and 36.

Return the graph as a list of edges (u, v) with 0 <= u < v < 11, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 15 vertices where the number of edges is between 28 and 37.

Return the graph as a list of edges (u, v) with 0 <= u < v < 15, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 13 vertices where the number of edges is between 29 and 39.

Return the graph as a list of edges (u, v) with 0 <= u < v < 13, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 14 vertices where the number of edges is between 14 and 40.

Return the graph as a list of edges (u, v) with 0 <= u < v < 14, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 18 vertices where the number of edges is between 34 and 40.

Return the graph as a list of edges (u, v) with 0 <= u < v < 18, or state "UNSATISFIABLE" if no graph exists.


Prompt:
Generate a graph with 14 vertices where the number of edges is between 16 and 41.

Return the graph as a list of edges (u, v) with 0 <= u < v < 14, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (2,11), (1,5), (2,12), (2,4), (0,5), (3,7), (2,13), (3,13)
Return a complete solution consistent with these fixed assignments.


Prompt:
Generate a graph with 11 vertices where the number of edges is between 32 and 43.

Return the graph as a list of edges (u, v) with 0 <= u < v < 11, or state "UNSATISFIABLE" if no graph exists.

Partial assignment (fixed values that must be respected):
- Known present edges: (1,9), (2,9), (6,9), (3,10), (5,7), (2,8), (7,9), (4,6)
Return a complete solution consistent with these fixed assignments.


Prompt:
Place 11 queens on a 11×11 chessboard such that no two queens attack each other. Queens attack along rows, columns, and diagonals.

Return a list of 11 integers where the i-th integer is the column position (0 to 10) of the queen in row i, or state "UNSATISFIABLE" if no solution exists.


Prompt:
Place 13 queens on a 13×13 chessboard such that no two queens attack each other. Queens attack along rows, columns, and diagonals.

Return a list of 13 integers where the i-th integer is the column position (0 to 12) of the queen in row i, or state "UNSATISFIABLE" if no solution exists.

Partial assignment (fixed values that must be respected):
- q[1]=12, q[11]=1
Return a complete solution consistent with these fixed assignments.


Prompt:
Place 14 queens on a 14×14 chessboard such that no two queens attack each other. Queens attack along rows, columns, and diagonals.

Return a list of 14 integers where the i-th integer is the column position (0 to 13) of the queen in row i, or state "UNSATISFIABLE" if no solution exists.

Partial assignment (fixed values that must be respected):
- q[4]=4, q[6]=1
Return a complete solution consistent with these fixed assignments.


Prompt:
Place 21 queens on a 21×21 chessboard such that no two queens attack each other. Queens attack along rows, columns, and diagonals.

Return a list of 21 integers where the i-th integer is the column position (0 to 20) of the queen in row i, or state "UNSATISFIABLE" if no solution exists.


Prompt:
Place 22 queens on a 22×22 chessboard such that no two queens attack each other. Queens attack along rows, columns, and diagonals.

Return a list of 22 integers where the i-th integer is the column position (0 to 21) of the queen in row i, or state "UNSATISFIABLE" if no solution exists.

Partial assignment (fixed values that must be respected):
- q[11]=8, q[16]=14, q[18]=11, q[21]=13
Return a complete solution consistent with these fixed assignments.


Prompt:
Place 26 queens on a 26×26 chessboard such that no two queens attack each other. Queens attack along rows, columns, and diagonals.

Return a list of 26 integers where the i-th integer is the column position (0 to 25) of the queen in row i, or state "UNSATISFIABLE" if no solution exists.


Prompt:
Place 27 queens on a 27×27 chessboard such that no two queens attack each other. Queens attack along rows, columns, and diagonals.

Return a list of 27 integers where the i-th integer is the column position (0 to 26) of the queen in row i, or state "UNSATISFIABLE" if no solution exists.

Partial assignment (fixed values that must be respected):
- q[9]=23, q[12]=13, q[18]=1, q[21]=21, q[26]=14
Return a complete solution consistent with these fixed assignments.


Prompt:
Place 4 queens on a 4×4 chessboard such that no two queens attack each other. Queens attack along rows, columns, and diagonals.

Return a list of 4 integers where the i-th integer is the column position (0 to 3) of the queen in row i, or state "UNSATISFIABLE" if no solution exists.


Prompt:
Place 5 queens on a 5×5 chessboard such that no two queens attack each other. Queens attack along rows, columns, and diagonals.

Return a list of 5 integers where the i-th integer is the column position (0 to 4) of the queen in row i, or state "UNSATISFIABLE" if no solution exists.

Partial assignment (fixed values that must be respected):
- q[1]=2
Return a complete solution consistent with these fixed assignments.


Prompt:
Place 7 queens on a 7×7 chessboard such that no two queens attack each other. Queens attack along rows, columns, and diagonals.

Return a list of 7 integers where the i-th integer is the column position (0 to 6) of the queen in row i, or state "UNSATISFIABLE" if no solution exists.


Prompt:
Place 8 queens on a 8×8 chessboard such that no two queens attack each other. Queens attack along rows, columns, and diagonals.

Return a list of 8 integers where the i-th integer is the column position (0 to 7) of the queen in row i, or state "UNSATISFIABLE" if no solution exists.


Prompt:
Place 9 queens on a 9×9 chessboard such that no two queens attack each other. Queens attack along rows, columns, and diagonals.

Return a list of 9 integers where the i-th integer is the column position (0 to 8) of the queen in row i, or state "UNSATISFIABLE" if no solution exists.

Partial assignment (fixed values that must be respected):
- q[3]=1
Return a complete solution consistent with these fixed assignments.


Prompt:
Find a 2-coloring of the edges of the complete graph K_12 such that there is no monochromatic red clique of size 4 and no monochromatic blue clique of size 4. Each edge is colored either 0 (red) or 1 (blue).

Return a list of 66 integers (0 or 1) representing the colors of edges listed in lexicographic order of (i,j) for i<j, or state "UNSATISFIABLE" if no such coloring exists.

Partial assignment (fixed values that must be respected):
- c[7]=0, c[9]=0, c[23]=1, c[27]=0, c[28]=0, c[29]=0, c[33]=1, c[34]=0, c[39]=1, c[40]=0, c[53]=0, c[58]=0, c[63]=0
Return a complete solution consistent with these fixed assignments.


Prompt:
Find a 2-coloring of the edges of the complete graph K_13 such that there is no monochromatic red clique of size 4 and no monochromatic blue clique of size 3. Each edge is colored either 0 (red) or 1 (blue).

Return a list of 78 integers (0 or 1) representing the colors of edges listed in lexicographic order of (i,j) for i<j, or state "UNSATISFIABLE" if no such coloring exists.


Prompt:
Find a 2-coloring of the edges of the complete graph K_13 such that there is no monochromatic red clique of size 4 and no monochromatic blue clique of size 4. Each edge is colored either 0 (red) or 1 (blue).

Return a list of 78 integers (0 or 1) representing the colors of edges listed in lexicographic order of (i,j) for i<j, or state "UNSATISFIABLE" if no such coloring exists.

Partial assignment (fixed values that must be respected):
- c[3]=1, c[6]=0, c[13]=0, c[19]=0, c[27]=0, c[34]=1, c[35]=0, c[43]=1, c[56]=1, c[63]=0, c[65]=1, c[74]=0, c[75]=0, c[76]=0, c[77]=0
Return a complete solution consistent with these fixed assignments.


Prompt:
Find a 2-coloring of the edges of the complete graph K_5 such that there is no monochromatic red clique of size 3 and no monochromatic blue clique of size 3. Each edge is colored either 0 (red) or 1 (blue).

Return a list of 10 integers (0 or 1) representing the colors of edges listed in lexicographic order of (i,j) for i<j, or state "UNSATISFIABLE" if no such coloring exists.

Partial assignment (fixed values that must be respected):
- c[3]=1, c[9]=0
Return a complete solution consistent with these fixed assignments.


Prompt:
Find a 2-coloring of the edges of the complete graph K_7 such that there is no monochromatic red clique of size 3 and no monochromatic blue clique of size 3. Each edge is colored either 0 (red) or 1 (blue).

Return a list of 21 integers (0 or 1) representing the colors of edges listed in lexicographic order of (i,j) for i<j, or state "UNSATISFIABLE" if no such coloring exists.


Prompt:
Find a 2-coloring of the edges of the complete graph K_7 such that there is no monochromatic red clique of size 3 and no monochromatic blue clique of size 4. Each edge is colored either 0 (red) or 1 (blue).

Return a list of 21 integers (0 or 1) representing the colors of edges listed in lexicographic order of (i,j) for i<j, or state "UNSATISFIABLE" if no such coloring exists.

Partial assignment (fixed values that must be respected):
- c[7]=1, c[8]=0, c[13]=0, c[14]=1
Return a complete solution consistent with these fixed assignments.


Prompt:
Find a 2-coloring of the edges of the complete graph K_7 such that there is no monochromatic red clique of size 4 and no monochromatic blue clique of size 3. Each edge is colored either 0 (red) or 1 (blue).

Return a list of 21 integers (0 or 1) representing the colors of edges listed in lexicographic order of (i,j) for i<j, or state "UNSATISFIABLE" if no such coloring exists.

Partial assignment (fixed values that must be respected):
- c[3]=0, c[7]=0, c[12]=0, c[18]=0
Return a complete solution consistent with these fixed assignments.


Prompt:
Find a 2-coloring of the edges of the complete graph K_8 such that there is no monochromatic red clique of size 3 and no monochromatic blue clique of size 3. Each edge is colored either 0 (red) or 1 (blue).

Return a list of 28 integers (0 or 1) representing the colors of edges listed in lexicographic order of (i,j) for i<j, or state "UNSATISFIABLE" if no such coloring exists.


Prompt:
Find a 2-coloring of the edges of the complete graph K_8 such that there is no monochromatic red clique of size 3 and no monochromatic blue clique of size 4. Each edge is colored either 0 (red) or 1 (blue).

Return a list of 28 integers (0 or 1) representing the colors of edges listed in lexicographic order of (i,j) for i<j, or state "UNSATISFIABLE" if no such coloring exists.


Prompt:
Find a 2-coloring of the edges of the complete graph K_8 such that there is no monochromatic red clique of size 4 and no monochromatic blue clique of size 3. Each edge is colored either 0 (red) or 1 (blue).

Return a list of 28 integers (0 or 1) representing the colors of edges listed in lexicographic order of (i,j) for i<j, or state "UNSATISFIABLE" if no such coloring exists.


Prompt:
Find a 2-coloring of the edges of the complete graph K_8 such that there is no monochromatic red clique of size 4 and no monochromatic blue clique of size 4. Each edge is colored either 0 (red) or 1 (blue).

Return a list of 28 integers (0 or 1) representing the colors of edges listed in lexicographic order of (i,j) for i<j, or state "UNSATISFIABLE" if no such coloring exists.


Prompt:
Find a 2-coloring of the edges of the complete graph K_9 such that there is no monochromatic red clique of size 4 and no monochromatic blue clique of size 3. Each edge is colored either 0 (red) or 1 (blue).

Return a list of 36 integers (0 or 1) representing the colors of edges listed in lexicographic order of (i,j) for i<j, or state "UNSATISFIABLE" if no such coloring exists.


Prompt:
Fill a Sudoku grid with block size 2 (so the full grid is 4x4, containing 4 rows, 4 columns, and 4 non-overlapping 2x2 blocks). Each row, column, and block must contain every integer from 1 to 4 exactly once.

Return a list of 16 integers (the grid in row-major order: cell at row i, column j is at index i*4+j), or state "UNSATISFIABLE" if no solution exists.

Partial assignment (fixed values that must be respected):
- x[7]=2, x[9]=1, x[12]=4
Return a complete solution consistent with these fixed assignments.


Prompt:
Fill a Sudoku grid with block size 3 (so the full grid is 9x9, containing 9 rows, 9 columns, and 9 non-overlapping 3x3 blocks). Each row, column, and block must contain every integer from 1 to 9 exactly once.

Return a list of 81 integers (the grid in row-major order: cell at row i, column j is at index i*9+j), or state "UNSATISFIABLE" if no solution exists.

Partial assignment (fixed values that must be respected):
- x[2]=3, x[11]=6, x[13]=8, x[20]=9, x[22]=2, x[35]=7, x[42]=2, x[46]=9, x[50]=4, x[57]=6, x[58]=4, x[60]=9, x[62]=8, x[69]=5, x[78]=6, x[79]=4
Return a complete solution consistent with these fixed assignments.


Prompt:
Fill a Sudoku grid with block size 4 (so the full grid is 16x16, containing 16 rows, 16 columns, and 16 non-overlapping 4x4 blocks). Each row, column, and block must contain every integer from 1 to 16 exactly once.

Return a list of 256 integers (the grid in row-major order: cell at row i, column j is at index i*16+j), or state "UNSATISFIABLE" if no solution exists.

Partial assignment (fixed values that must be respected):
- x[5]=6, x[13]=14, x[14]=15, x[18]=7, x[22]=3, x[26]=15, x[27]=16, x[33]=10, x[34]=11, x[42]=3, x[53]=10, x[54]=11, x[57]=6, x[58]=7, x[69]=5, x[71]=7, x[80]=6, x[81]=5, x[83]=7, x[84]=2, x[85]=1, x[94]=12, x[103]=15, x[106]=4, x[107]=3, x[109]=5, x[112]=14, x[115]=15, x[120]=6, x[135]=6, x[136]=11, x[140]=15, x[141]=16, x[148]=3, x[150]=1, x[170]=1, x[175]=6, x[185]=8, x[188]=3, x[190]=1, x[196]=8, x[199]=5, x[201]=11, x[203]=9, x[208]=8, x[218]=14, x[240]=16, x[241]=15, x[243]=13, x[244]=12, x[253]=3
Return a complete solution consistent with these fixed assignments.