What cannot be drawn
Most arithmetically admissible pencil types cannot be drawn. This page lists, for each number of lines, the types that survive the combinatorial test but are still impossible, together with the reason, and the types that are undecided. Types that already fail the combinatorial test (two pencils would share two lines) or one of the classical inequalities are only counted; the full lists are in data/results/classification_n{n}.json.
Three famous impossible pictures
Each of these is a perfectly good linear space, and the last two even exist with complex coordinates, but no drawing with straight real lines exists. Dashed curves mark the lines that would have to bend.
The Fano plane: seven points, seven lines of three; one of them must be a circle
The Fano plane, type {37 } for 7 lines: there is no ordinary point at all, which the Sylvester–Gallai theorem forbids.
The Moebius-Kantor configuration (8_3) on Z/8: lines {i, i+1, i+3}; drawn abstractly, curved where straightness fails
The Möbius–Kantor configuration, type {38 } for 8 lines: it passes every inequality, but admits no orientation.
The Hesse configuration AG(2,3): nine points, twelve lines of three; the four wrapped diagonals cannot be straight
The Hesse configuration: nine points, every pair on a line of three. Dually, nine lines of type {312 }, the dual Hesse arrangement.
4 lines
4 admissible types; 3 realizable; 1 fails the combinatorial test.
5 lines
7 admissible types; 5 realizable; 2 fail the combinatorial test.
6 lines
15 admissible types; 9 realizable; 6 fail the combinatorial test.
7 lines
32 admissible types; 16 realizable; 15 fail the combinatorial test; 1 fails an inequality (Melchior or Csima–Sawyer).
1 type excluded by an inequality
{37 }.
8 lines
66 admissible types; 28 realizable; 37 fail the combinatorial test.
type
points
why it cannot be drawn
{38 }
12
no linear space of this type admits an orientation (exhaustive search)
9 lines
141 admissible types; 53 realizable; 85 fail the combinatorial test; 3 fail an inequality (Melchior or Csima–Sawyer).
3 types excluded by an inequality
{312 }, {311 }, {42 , 37 }.
10 lines
295 admissible types; 98 realizable; 189 fail the combinatorial test; 6 fail an inequality (Melchior or Csima–Sawyer).
type
points
why it cannot be drawn
{313 }
19
no linear space of this type admits an orientation (exhaustive search)
{42 , 39 }
17
no linear space of this type admits an orientation (exhaustive search)
6 types excluded by an inequality
{4, 312 }, {43 , 39 }, {42 , 310 }, {43 , 38 }, {44 , 36 }, {44 , 35 }.
11 lines
619 admissible types; 180 realizable; 417 fail the combinatorial test; 13 fail an inequality (Melchior or Csima–Sawyer).
type
points
why it cannot be drawn
{4, 314 }
22
no linear space of this type admits an orientation (exhaustive search)
{42 , 312 }
21
no linear space of this type admits an orientation (exhaustive search)
{43 , 310 }
20
no linear space of this type admits an orientation (exhaustive search)
{5, 4, 311 }
19
no linear space of this type admits an orientation (exhaustive search)
{52 , 39 }
19
no linear space of this type admits an orientation (exhaustive search)
{5, 43 , 36 }
19
no linear space of this type admits an orientation (exhaustive search)
{46 , 33 }
19
no linear space of this type admits an orientation (exhaustive search)
{52 , 4, 36 }
20
no linear space of this type admits an orientation (exhaustive search)
{5, 44 , 33 }
20
no linear space of this type admits an orientation (exhaustive search)
13 types excluded by an inequality
{317 }, {5, 315 }, {4, 315 }, {42 , 313 }, {5, 314 }, {43 , 311 }, {44 , 39 }, {46 , 36 }, {45 , 37 }, {5, 42 , 39 }, {46 , 35 }, {45 , 36 }, {46 , 34 }.
12 lines
1286 admissible types; 336 realizable; 902 fail the combinatorial test; 28 fail an inequality (Melchior or Csima–Sawyer).
type
points
why it cannot be drawn
{320 }
26
none of its five linear spaces (Steiner triple systems of order 13 minus a point) admits an orientation (Proposition C3a )
{42 , 316 }
24
no linear space of this type admits an orientation (exhaustive search)
{43 , 314 }
23
no linear space of this type admits an orientation (exhaustive search)
{5, 316 }
25
no linear space of this type admits an orientation (exhaustive search)
{5, 42 , 312 }
23
no linear space of this type admits an orientation (exhaustive search)
{52 , 313 }
22
no linear space of this type admits an orientation (exhaustive search)
{6, 314 }
24
orientable, but every orientable linear space of this type forces an extra collinearity (Gröbner certificate; Theorems C4, C5 )
{45 , 39 }
23
none of its linear spaces admits an orientation: an exhaustive SAT check over the 8 possible configurations of the five 4-pencils (Proposition C3b )
{52 , 312 }
24
no linear space of this type admits an orientation (exhaustive search)
{6, 4, 312 }
23
orientable, but every orientable linear space of this type forces an extra collinearity (Gröbner certificate; Theorems C4, C5 )
{52 , 42 , 38 }
22
no linear space of this type admits an orientation (exhaustive search)
{53 , 39 }
21
no linear space of this type admits an orientation (exhaustive search)
{5, 45 , 35 }
22
no linear space of this type admits an orientation (exhaustive search)
{52 , 43 , 36 }
21
no linear space of this type admits an orientation (exhaustive search)
{53 , 38 }
23
no linear space of this type admits an orientation (exhaustive search)
{48 , 32 }
22
no linear space of this type admits an orientation (exhaustive search)
{52 , 43 , 35 }
23
no linear space of this type admits an orientation (exhaustive search)
{49 }
21
theorem: the unique linear space is the dual of the Hesse configuration, excluded by Sylvester–Gallai (Proposition C3 )
{48 , 3}
24
no linear space of this type admits an orientation (exhaustive search)
{48 }
26
no linear space of this type admits an orientation (exhaustive search)
28 types excluded by an inequality
{43 , 316 }, {42 , 317 }, {43 , 315 }, {44 , 313 }, {5, 4, 315 }, {45 , 311 }, {5, 43 , 312 }, {44 , 312 }, {5, 42 , 313 }, {46 , 39 }, {45 , 310 }, {5, 43 , 311 }, {52 , 4, 312 }, {46 , 38 }, {49 , 34 }, {5, 44 , 39 }, {48 , 35 }, {52 , 4, 311 }, {47 , 36 }, {49 , 33 }, {5, 44 , 38 }, {52 , 42 , 39 }, {48 , 34 }, {47 , 35 }, {5, 45 , 36 }, {49 , 32 }, {48 , 33 }, {49 , 3}.
13 lines
2654 admissible types; 620 realizable; 567 fail the combinatorial test; 1400 fail an inequality (Melchior or Csima–Sawyer); 39 undecided.
For this number of lines the inequalities were checked first, and only the types that pass them went to the combinatorial test, so the count of combinatorial failures covers those types only.
type
points
why it cannot be drawn
{5, 48 , 32 }
25
band theorem: removing the largest pencil leaves a configuration that no arrangement of the remaining lines produces (Theorem B4 )
{62 , 4, 39 }
27
band theorem: removing the largest pencil leaves a configuration that no arrangement of the remaining lines produces (Theorem B4 )
{54 , 4, 36 }
25
band theorem: removing the largest pencil leaves a configuration that no arrangement of the remaining lines produces (Theorem B4 )
{6, 5, 44 , 35 }
25
band theorem: removing the largest pencil leaves a configuration that no arrangement of the remaining lines produces (Theorem B4 )
{62 , 43 , 36 }
23
band theorem: removing the largest pencil leaves a configuration that no arrangement of the remaining lines produces (Theorem B4 )
{54 , 37 }
28
band theorem: removing the largest pencil leaves a configuration that no arrangement of the remaining lines produces (Theorem B4 )
{6, 52 , 4, 37 }
27
band theorem: removing the largest pencil leaves a configuration that no arrangement of the remaining lines produces (Theorem B4 )
{62 , 42 , 37 }
26
band theorem: removing the largest pencil leaves a configuration that no arrangement of the remaining lines produces (Theorem B4 )
{5, 48 , 3}
27
band theorem: removing the largest pencil leaves a configuration that no arrangement of the remaining lines produces (Theorem B4 )
{53 , 44 , 33 }
25
band theorem: removing the largest pencil leaves a configuration that no arrangement of the remaining lines produces (Theorem B4 )
{54 , 42 , 34 }
24
band theorem: removing the largest pencil leaves a configuration that no arrangement of the remaining lines produces (Theorem B4 )
{7, 43 , 37 }
29
band theorem: removing the largest pencil leaves a configuration that no arrangement of the remaining lines produces (Theorem B4 )
{54 , 4, 35 }
27
band theorem: removing the largest pencil leaves a configuration that no arrangement of the remaining lines produces (Theorem B4 )
{6, 5, 44 , 34 }
27
band theorem: removing the largest pencil leaves a configuration that no arrangement of the remaining lines produces (Theorem B4 )
{6, 52 , 42 , 35 }
26
band theorem: removing the largest pencil leaves a configuration that no arrangement of the remaining lines produces (Theorem B4 )
{62 , 43 , 35 }
25
band theorem: removing the largest pencil leaves a configuration that no arrangement of the remaining lines produces (Theorem B4 )
{54 , 36 }
30
band theorem: removing the largest pencil leaves a configuration that no arrangement of the remaining lines produces (Theorem B4 )
{6, 52 , 4, 36 }
29
band theorem: removing the largest pencil leaves a configuration that no arrangement of the remaining lines produces (Theorem B4 )
{62 , 42 , 36 }
28
band theorem: removing the largest pencil leaves a configuration that no arrangement of the remaining lines produces (Theorem B4 )
{5, 48 }
29
band theorem: removing the largest pencil leaves a configuration that no arrangement of the remaining lines produces (Theorem B4 )
{54 , 4, 34 }
29
band theorem: removing the largest pencil leaves a configuration that no arrangement of the remaining lines produces (Theorem B4 )
{54 , 35 }
32
band theorem: removing the largest pencil leaves a configuration that no arrangement of the remaining lines produces (Theorem B4 )
{6, 52 , 4, 35 }
31
band theorem: removing the largest pencil leaves a configuration that no arrangement of the remaining lines produces (Theorem B4 )
{62 , 42 , 35 }
30
band theorem: removing the largest pencil leaves a configuration that no arrangement of the remaining lines produces (Theorem B4 )
{54 , 4, 33 }
31
band theorem: removing the largest pencil leaves a configuration that no arrangement of the remaining lines produces (Theorem B4 )
{54 , 34 }
34
band theorem: removing the largest pencil leaves a configuration that no arrangement of the remaining lines produces (Theorem B4 )
{6, 52 , 4, 34 }
33
band theorem: removing the largest pencil leaves a configuration that no arrangement of the remaining lines produces (Theorem B4 )
{62 , 42 , 34 }
32
band theorem: removing the largest pencil leaves a configuration that no arrangement of the remaining lines produces (Theorem B4 )
Undecided: {324 }, {4, 322 }, {323 }, {42 , 320 }, {4, 321 }, {322 }, {5, 320 }, {43 , 317 }, {5, 4, 318 }, {6, 319 }, {52 , 317 }, {6, 318 }, {45 , 313 }, {5, 43 , 314 }, {46 , 311 }, {52 , 4, 314 }, {6, 42 , 314 }, {53 , 313 }, {6, 5, 314 }, {6, 5, 4, 312 }, {48 , 36 }, {5, 46 , 37 }, {53 , 4, 310 }, {6, 44 , 39 }, {53 , 42 , 38 }, {6, 5, 42 , 39 }, {49 , 33 }, {5, 47 , 34 }, {52 , 45 , 35 }, {53 , 43 , 36 }, {62 , 311 }, {410 , 3}, {6, 52 , 39 }, {52 , 45 , 34 }, {53 , 43 , 35 }, {6, 5, 43 , 36 }, {410 }, {49 , 3}, {6, 46 , 32 }.
1400 types excluded by an inequality
{326 }, {4, 324 }, {325 }, {42 , 322 }, {4, 323 }, {43 , 320 }, {42 , 321 }, {5, 322 }, {44 , 318 }, {43 , 319 }, {5, 4, 320 }, {6, 321 }, {45 , 316 }, {5, 321 }, {44 , 317 }, {5, 42 , 318 }, {52 , 319 }, {6, 4, 319 }, {46 , 314 }, {5, 4, 319 }, {6, 320 }, {45 , 315 }, {5, 43 , 316 }, {52 , 4, 317 }, {6, 42 , 317 }, {44 , 316 }, {47 , 312 }, {5, 42 , 317 }, {52 , 318 }, {6, 4, 318 }, {7, 319 }, {46 , 313 }, {5, 44 , 314 }, {52 , 42 , 315 }, {53 , 316 }, {6, 43 , 315 }, {45 , 314 }, {48 , 310 }, {5, 43 , 315 }, {52 , 4, 316 }, {6, 42 , 316 }, {6, 5, 317 }, {7, 4, 317 }, {47 , 311 }, {5, 45 , 312 }, {52 , 43 , 313 }, {53 , 4, 314 }, {6, 4, 317 }, {6, 44 , 313 }, {7, 318 }, {46 , 312 }, {49 , 38 }, {5, 44 , 313 }, {52 , 42 , 314 }, {53 , 315 }, {6, 43 , 314 }, {6, 5, 4, 315 }, {62 , 316 }, {7, 42 , 315 }, {48 , 39 }, {5, 46 , 310 }, {52 , 4, 315 }, {52 , 44 , 311 }, {53 , 42 , 312 }, {6, 42 , 315 }, {6, 45 , 311 }, {6, 5, 316 }, {7, 4, 316 }, {47 , 310 }, {410 , 36 }, {5, 45 , 311 }, {52 , 43 , 312 }, {53 , 4, 313 }, {6, 44 , 312 }, {6, 5, 42 , 313 }, {6, 52 , 314 }, {62 , 4, 314 }, {7, 317 }, {7, 43 , 313 }, {49 , 37 }, {5, 44 , 312 }, {5, 47 , 38 }, {52 , 42 , 313 }, {52 , 45 , 39 }, {53 , 314 }, {53 , 43 , 310 }, {6, 43 , 313 }, {6, 46 , 39 }, {6, 5, 4, 314 }, {62 , 315 }, {7, 42 , 314 }, {7, 5, 315 }, {48 , 38 }, {411 , 34 }, {5, 46 , 39 }, {52 , 44 , 310 }, {53 , 42 , 311 }, {54 , 312 }, {6, 45 , 310 }, {6, 5, 43 , 311 }, {6, 52 , 4, 312 }, {62 , 42 , 312 }, {7, 4, 315 }, {7, 44 , 311 }, {8, 316 }, {47 , 39 }, {410 , 35 }, {5, 45 , 310 }, {5, 48 , 36 }, {52 , 43 , 311 }, {52 , 46 , 37 }, {53 , 4, 312 }, {53 , 44 , 38 }, {6, 44 , 311 }, {6, 47 , 37 }, {6, 5, 42 , 312 }, {6, 52 , 313 }, {62 , 4, 313 }, {7, 43 , 312 }, {7, 5, 4, 313 }, {7, 6, 314 }, {49 , 36 }, {412 , 32 }, {5, 47 , 37 }, {52 , 45 , 38 }, {53 , 43 , 39 }, {54 , 4, 310 }, {6, 46 , 38 }, {6, 5, 4, 313 }, {6, 5, 44 , 39 }, {6, 52 , 42 , 310 }, {6, 53 , 311 }, {62 , 314 }, {62 , 43 , 310 }, {7, 42 , 313 }, {7, 45 , 39 }, {7, 5, 314 }, {8, 4, 314 }, {48 , 37 }, {411 , 33 }, {5, 46 , 38 }, {5, 49 , 34 }, {52 , 44 , 39 }, {52 , 47 , 35 }, {53 , 42 , 310 }, {53 , 45 , 36 }, {54 , 311 }, {6, 45 , 39 }, {6, 48 , 35 }, {6, 5, 43 , 310 }, {6, 52 , 4, 311 }, {62 , 42 , 311 }, {62 , 5, 312 }, {7, 44 , 310 }, {7, 5, 42 , 311 }, {7, 52 , 312 }, {7, 6, 4, 312 }, {8, 315 }, {410 , 34 }, {413 }, {5, 48 , 35 }, {52 , 46 , 36 }, {53 , 4, 311 }, {53 , 44 , 37 }, {54 , 42 , 38 }, {55 , 39 }, {6, 44 , 310 }, {6, 47 , 36 }, {6, 5, 42 , 311 }, {6, 5, 45 , 37 }, {6, 52 , 312 }, {6, 52 , 43 , 38 }, {6, 53 , 4, 39 }, {62 , 4, 312 }, {62 , 44 , 38 }, {7, 43 , 311 }, {7, 46 , 37 }, {7, 5, 4, 312 }, {7, 6, 313 }, {8, 42 , 312 }, {8, 5, 313 }, {49 , 35 }, {412 , 3}, {5, 47 , 36 }, {5, 410 , 32 }, {52 , 45 , 37 }, {52 , 48 , 33 }, {53 , 43 , 38 }, {53 , 46 , 34 }, {54 , 4, 39 }, {6, 46 , 37 }, {6, 49 , 33 }, {6, 5, 44 , 38 }, {6, 52 , 42 , 39 }, {6, 53 , 310 }, {62 , 43 , 39 }, {62 , 5, 4, 310 }, {63 , 311 }, {7, 45 , 38 }, {7, 5, 313 }, {7, 5, 43 , 39 }, {7, 52 , 4, 310 }, {7, 6, 42 , 310 }, {8, 4, 313 }, {9, 314 }, {411 , 32 }, {5, 49 , 33 }, {52 , 44 , 38 }, {52 , 47 , 34 }, {53 , 42 , 39 }, {53 , 45 , 35 }, {54 , 310 }, {54 , 43 , 36 }, {55 , 4, 37 }, {6, 45 , 38 }, {6, 48 , 34 }, {6, 5, 43 , 39 }, {6, 5, 46 , 35 }, {6, 52 , 4, 310 }, {6, 52 , 44 , 36 }, {6, 53 , 42 , 37 }, {62 , 42 , 310 }, {62 , 45 , 36 }, {62 , 5, 311 }, {7, 44 , 39 }, {7, 47 , 35 }, {7, 5, 42 , 310 }, {7, 52 , 311 }, {7, 6, 4, 311 }, {72 , 312 }, {8, 43 , 310 }, {8, 5, 4, 311 }, {410 , 33 }, {5, 48 , 34 }, {5, 411 }, {52 , 46 , 35 }, {52 , 49 , 3}, {53 , 44 , 36 }, {53 , 47 , 32 }, {54 , 42 , 37 }, {55 , 38 }, {6, 47 , 35 }, {6, 410 , 3}, {6, 5, 45 , 36 }, {6, 52 , 43 , 37 }, {6, 53 , 4, 38 }, {62 , 4, 311 }, {62 , 44 , 37 }, {62 , 5, 42 , 38 }, {62 , 52 , 39 }, {63 , 4, 39 }, {7, 43 , 310 }, {7, 46 , 36 }, {7, 5, 4, 311 }, {7, 5, 44 , 37 }, {7, 52 , 42 , 38 }, {7, 53 , 39 }, {7, 6, 312 }, {7, 6, 43 , 38 }, {8, 42 , 311 }, {8, 5, 312 }, {9, 4, 312 }, {412 }, {5, 47 , 35 }, {5, 410 , 3}, {52 , 45 , 36 }, {52 , 48 , 32 }, {53 , 43 , 37 }, {53 , 46 , 33 }, {54 , 4, 38 }, {54 , 44 , 34 }, {55 , 42 , 35 }, {56 , 36 }, {6, 46 , 36 }, {6, 49 , 32 }, {6, 5, 44 , 37 }, {6, 5, 47 , 33 }, {6, 52 , 42 , 38 }, {6, 52 , 45 , 34 }, {6, 53 , 39 }, {6, 53 , 43 , 35 }, {62 , 43 , 38 }, {62 , 46 , 34 }, {62 , 5, 4, 39 }, {63 , 310 }, {7, 45 , 37 }, {7, 48 , 33 }, {7, 5, 43 , 38 }, {7, 52 , 4, 39 }, {7, 6, 42 , 39 }, {7, 6, 5, 310 }, {72 , 4, 310 }, {8, 4, 312 }, {8, 44 , 38 }, {8, 5, 42 , 39 }, {8, 52 , 310 }, {9, 313 }, {411 , 3}, {5, 49 , 32 }, {52 , 47 , 33 }, {53 , 45 , 34 }, {53 , 48 }, {54 , 43 , 35 }, {55 , 4, 36 }, {6, 48 , 33 }, {6, 5, 46 , 34 }, {6, 52 , 4, 39 }, {6, 52 , 44 , 35 }, {6, 53 , 42 , 36 }, {6, 54 , 37 }, {62 , 42 , 39 }, {62 , 45 , 35 }, {62 , 5, 310 }, {62 , 5, 43 , 36 }, {62 , 52 , 4, 37 }, {63 , 42 , 37 }, {7, 44 , 38 }, {7, 47 , 34 }, {7, 5, 42 , 39 }, {7, 5, 45 , 35 }, {7, 52 , 310 }, {7, 52 , 43 , 36 }, {7, 53 , 4, 37 }, {7, 6, 4, 310 }, {7, 6, 44 , 36 }, {72 , 311 }, {8, 43 , 39 }, {8, 5, 4, 310 }, {8, 6, 311 }, {9, 42 , 310 }, {410 , 32 }, {5, 48 , 33 }, {52 , 46 , 34 }, {52 , 49 }, {53 , 44 , 35 }, {53 , 47 , 3}, {54 , 42 , 36 }, {54 , 45 , 32 }, {55 , 37 }, {55 , 43 , 33 }, {56 , 4, 34 }, {6, 47 , 34 }, {6, 410 }, {6, 5, 45 , 35 }, {6, 5, 48 , 3}, {6, 52 , 43 , 36 }, {6, 52 , 46 , 32 }, {6, 53 , 4, 37 }, {6, 53 , 44 , 33 }, {62 , 44 , 36 }, {62 , 47 , 32 }, {62 , 5, 42 , 37 }, {62 , 52 , 38 }, {63 , 4, 38 }, {7, 46 , 35 }, {7, 49 , 3}, {7, 5, 44 , 36 }, {7, 52 , 42 , 37 }, {7, 53 , 38 }, {7, 6, 311 }, {7, 6, 43 , 37 }, {7, 6, 5, 4, 38 }, {7, 62 , 39 }, {72 , 42 , 38 }, {8, 42 , 310 }, {8, 45 , 36 }, {8, 5, 311 }, {8, 5, 43 , 37 }, {8, 52 , 4, 38 }, {9, 4, 311 }, {5, 410 }, {52 , 48 , 3}, {53 , 46 , 32 }, {54 , 4, 37 }, {54 , 44 , 33 }, {55 , 42 , 34 }, {56 , 35 }, {6, 49 , 3}, {6, 5, 44 , 36 }, {6, 5, 47 , 32 }, {6, 52 , 42 , 37 }, {6, 52 , 45 , 33 }, {6, 53 , 38 }, {6, 53 , 43 , 34 }, {6, 54 , 4, 35 }, {62 , 43 , 37 }, {62 , 46 , 33 }, {62 , 5, 4, 38 }, {62 , 5, 44 , 34 }, {62 , 52 , 42 , 35 }, {62 , 53 , 36 }, {63 , 39 }, {63 , 43 , 35 }, {7, 45 , 36 }, {7, 48 , 32 }, {7, 5, 43 , 37 }, {7, 5, 46 , 33 }, {7, 52 , 4, 38 }, {7, 52 , 44 , 34 }, {7, 53 , 42 , 35 }, {7, 6, 42 , 38 }, {7, 6, 45 , 34 }, {7, 6, 5, 39 }, {72 , 4, 39 }, {8, 44 , 37 }, {8, 5, 42 , 38 }, {8, 52 , 39 }, {8, 6, 4, 39 }, {9, 312 }, {9, 43 , 38 }, {411 }, {5, 49 , 3}, {52 , 47 , 32 }, {53 , 45 , 33 }, {54 , 43 , 34 }, {54 , 46 }, {55 , 4, 35 }, {55 , 44 , 3}, {56 , 42 , 32 }, {6, 48 , 32 }, {6, 5, 46 , 33 }, {6, 52 , 44 , 34 }, {6, 52 , 47 }, {6, 53 , 42 , 35 }, {6, 53 , 45 , 3}, {6, 54 , 36 }, {62 , 45 , 34 }, {62 , 48 }, {62 , 5, 43 , 35 }, {62 , 52 , 4, 36 }, {63 , 42 , 36 }, {63 , 5, 37 }, {7, 47 , 33 }, {7, 5, 45 , 34 }, {7, 52 , 39 }, {7, 52 , 43 , 35 }, {7, 53 , 4, 36 }, {7, 6, 4, 39 }, {7, 6, 44 , 35 }, {7, 6, 5, 42 , 36 }, {7, 6, 52 , 37 }, {7, 62 , 4, 37 }, {72 , 310 }, {72 , 43 , 36 }, {8, 43 , 38 }, {8, 46 , 34 }, {8, 5, 4, 39 }, {8, 5, 44 , 35 }, {8, 52 , 42 , 36 }, {8, 6, 310 }, {9, 42 , 39 }, {9, 5, 310 }, {53 , 44 , 34 }, {53 , 47 }, {54 , 42 , 35 }, {54 , 45 , 3}, {55 , 36 }, {55 , 43 , 32 }, {56 , 4, 33 }, {6, 47 , 33 }, {6, 5, 45 , 34 }, {6, 5, 48 }, {6, 52 , 43 , 35 }, {6, 52 , 46 , 3}, {6, 53 , 4, 36 }, {6, 53 , 44 , 32 }, {6, 54 , 42 , 33 }, {6, 55 , 34 }, {62 , 44 , 35 }, {62 , 47 , 3}, {62 , 5, 42 , 36 }, {62 , 5, 45 , 32 }, {62 , 52 , 37 }, {62 , 52 , 43 , 33 }, {62 , 53 , 4, 34 }, {63 , 4, 37 }, {63 , 44 , 33 }, {7, 46 , 34 }, {7, 49 }, {7, 5, 44 , 35 }, {7, 5, 47 , 3}, {7, 52 , 42 , 36 }, {7, 52 , 45 , 32 }, {7, 53 , 37 }, {7, 53 , 43 , 33 }, {7, 6, 43 , 36 }, {7, 6, 46 , 32 }, {7, 6, 5, 4, 37 }, {7, 62 , 38 }, {72 , 42 , 37 }, {72 , 5, 38 }, {8, 45 , 35 }, {8, 5, 43 , 36 }, {8, 52 , 4, 37 }, {8, 6, 42 , 37 }, {8, 6, 5, 38 }, {9, 4, 310 }, {9, 44 , 36 }, {10, 311 }, {52 , 48 }, {53 , 46 , 3}, {54 , 44 , 32 }, {55 , 42 , 33 }, {56 , 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