Johnson Solid

The Johnson solids are the Convex Polyhedra having regular faces (with the exception of the completely regular Platonic Solids, the ``Semiregular'' Archimedean Solids, and the two infinite families of Prisms and Antiprisms). There are 28 simple (i.e., cannot be dissected into two other regular-faced polyhedra by a plane) regular-faced polyhedra in addition to the Prisms and Antiprisms (Zalgaller 1969), and Johnson (1966) proposed and Zalgaller (1969) proved that there exist exactly 92 Johnson solids in all.

A database of solids and Vertex Nets of these solids is maintained on the Bell Laboratories Netlib server, but a few errors exist in several entries. A concatenated and corrected version of the files is given by Weisstein, together with Mathematica ${}^{\scriptstyle\circledRsymbol}$ (Wolfram Research, Champaign, IL) code to display the solids and nets. The following table summarizes the names of the Johnson solids and gives their images and nets.

1. Square pyramid

$\begin{figure} \BoxedEPSF{J01.epsf scaled 500}\quad\BoxedEPSF{J01_net.epsf scaled 500}\smallskip\end{figure}$

2. Pentagonal pyramid

$\begin{figure} \BoxedEPSF{J02.epsf scaled 500}\quad\BoxedEPSF{J02_net.epsf scaled 500}\smallskip\end{figure}$

3. Triangular cupola

$\begin{figure} \BoxedEPSF{J03.epsf scaled 500}\quad\BoxedEPSF{J03_net.epsf scaled 500}\smallskip\end{figure}$

4. Square cupola

$\begin{figure} \BoxedEPSF{J04.epsf scaled 500}\quad\BoxedEPSF{J04_net.epsf scaled 500}\smallskip\end{figure}$

5. Pentagonal cupola

$\begin{figure} \BoxedEPSF{J05.epsf scaled 500}\quad\BoxedEPSF{J05_net.epsf scaled 500}\smallskip\end{figure}$

6. Pentagonal rotunda

$\begin{figure} \BoxedEPSF{J06.epsf scaled 500}\quad\BoxedEPSF{J06_net.epsf scaled 500}\smallskip\end{figure}$

7. Elongated triangular pyramid

$\begin{figure} \BoxedEPSF{J07.epsf scaled 500}\quad\BoxedEPSF{J07_net.epsf scaled 500}\smallskip\end{figure}$

8. Elongated square pyramid

$\begin{figure} \BoxedEPSF{J08.epsf scaled 500}\quad\BoxedEPSF{J08_net.epsf scaled 500}\smallskip\end{figure}$

9. Elongated pentagonal pyramid

$\begin{figure} \BoxedEPSF{J09.epsf scaled 500}\quad\BoxedEPSF{J09_net.epsf scaled 500}\smallskip\end{figure}$

10. Gyroelongated square pyramid

$\begin{figure} \BoxedEPSF{J10.epsf scaled 500}\quad\BoxedEPSF{J10_net.epsf scaled 500}\smallskip\end{figure}$

11. Gyroelongated pentagonal pyramid

$\begin{figure} \BoxedEPSF{J11.epsf scaled 500}\quad\BoxedEPSF{J11_net.epsf scaled 500}\smallskip\end{figure}$

12. Triangular dipyramid

$\begin{figure} \BoxedEPSF{J12.epsf scaled 500}\quad\BoxedEPSF{J12_net.epsf scaled 500}\smallskip\end{figure}$

13. Pentagonal dipyramid

$\begin{figure} \BoxedEPSF{J13.epsf scaled 500}\quad\BoxedEPSF{J13_net.epsf scaled 500}\smallskip\end{figure}$

14. Elongated triangular dipyramid

$\begin{figure} \BoxedEPSF{J14.epsf scaled 500}\quad\BoxedEPSF{J14_net.epsf scaled 500}\smallskip\end{figure}$

15. Elongated square dipyramid

$\begin{figure} \BoxedEPSF{J15.epsf scaled 500}\quad\BoxedEPSF{J15_net.epsf scaled 500}\smallskip\end{figure}$

16. Elongated pentagonal dipyramid

$\begin{figure} \BoxedEPSF{J16.epsf scaled 500}\quad\BoxedEPSF{J16_net.epsf scaled 500}\smallskip\end{figure}$

17. Gyroelongated square dipyramid

$\begin{figure} \BoxedEPSF{J17.epsf scaled 500}\quad\BoxedEPSF{J17_net.epsf scaled 500}\smallskip\end{figure}$

18. Elongated triangular cupola

$\begin{figure} \BoxedEPSF{J18.epsf scaled 500}\quad\BoxedEPSF{J18_net.epsf scaled 500}\smallskip\end{figure}$

19. Elongated square cupola

$\begin{figure} \BoxedEPSF{J19.epsf scaled 500}\quad\BoxedEPSF{J19_net.epsf scaled 500}\smallskip\end{figure}$

20. Elongated pentagonal cupola

$\begin{figure} \BoxedEPSF{J20.epsf scaled 500}\quad\BoxedEPSF{J20_net.epsf scaled 500}\smallskip\end{figure}$

21. Elongated pentagonal rotunda

$\begin{figure} \BoxedEPSF{J21.epsf scaled 500}\quad\BoxedEPSF{J21_net.epsf scaled 500}\smallskip\end{figure}$

22. Gyroelongated triangular cupola

$\begin{figure} \BoxedEPSF{J22.epsf scaled 500}\quad\BoxedEPSF{J22_net.epsf scaled 500}\smallskip\end{figure}$

23. Gyroelongated square cupola

$\begin{figure} \BoxedEPSF{J23.epsf scaled 500}\quad\BoxedEPSF{J23_net.epsf scaled 500}\smallskip\end{figure}$

24. Gyroelongated pentagonal cupola

$\begin{figure} \BoxedEPSF{J24.epsf scaled 500}\quad\BoxedEPSF{J24_net.epsf scaled 500}\smallskip\end{figure}$

25. Gyroelongated pentagonal rotunda

$\begin{figure} \BoxedEPSF{J25.epsf scaled 500}\quad\BoxedEPSF{J25_net.epsf scaled 500}\smallskip\end{figure}$

26. Gyrobifastigium

$\begin{figure} \BoxedEPSF{J26.epsf scaled 500}\quad\BoxedEPSF{J26_net.epsf scaled 500}\smallskip\end{figure}$

27. Triangular orthobicupola

$\begin{figure} \BoxedEPSF{J27.epsf scaled 500}\quad\BoxedEPSF{J27_net.epsf scaled 500}\smallskip\end{figure}$

28. Square orthobicupola

$\begin{figure} \BoxedEPSF{J28.epsf scaled 500}\quad\BoxedEPSF{J28_net.epsf scaled 500}\smallskip\end{figure}$

29. Square gyrobicupola

$\begin{figure} \BoxedEPSF{J29.epsf scaled 500}\quad\BoxedEPSF{J29_net.epsf scaled 500}\smallskip\end{figure}$

30. Pentagonal orthobicupola

$\begin{figure} \BoxedEPSF{J30.epsf scaled 500}\quad\BoxedEPSF{J30_net.epsf scaled 500}\smallskip\end{figure}$

31. Pentagonal gyrobicupola

$\begin{figure} \BoxedEPSF{J31.epsf scaled 500}\quad\BoxedEPSF{J31_net.epsf scaled 500}\smallskip\end{figure}$

32. Pentagonal orthocupolarontunda

$\begin{figure} \BoxedEPSF{J32.epsf scaled 500}\quad\BoxedEPSF{J32_net.epsf scaled 500}\smallskip\end{figure}$

33. Pentagonal gyrocupolarotunda

$\begin{figure} \BoxedEPSF{J33.epsf scaled 500}\quad\BoxedEPSF{J33_net.epsf scaled 500}\smallskip\end{figure}$

34. Pentagonal orthobirotunda

$\begin{figure} \BoxedEPSF{J34.epsf scaled 500}\quad\BoxedEPSF{J34_net.epsf scaled 500}\smallskip\end{figure}$

35. Elongated triangular orthobicupola

$\begin{figure} \BoxedEPSF{J35.epsf scaled 500}\quad\BoxedEPSF{J35_net.epsf scaled 500}\smallskip\end{figure}$

36. Elongated triangular gyrobicupola

$\begin{figure} \BoxedEPSF{J36.epsf scaled 500}\quad\BoxedEPSF{J36_net.epsf scaled 500}\smallskip\end{figure}$

37. Elongated square gyrobicupola

$\begin{figure} \BoxedEPSF{J37.epsf scaled 500}\quad\BoxedEPSF{J37_net.epsf scaled 500}\smallskip\end{figure}$

38. Elongated pentagonal orthobicupola

$\begin{figure} \BoxedEPSF{J38.epsf scaled 500}\quad\BoxedEPSF{J38_net.epsf scaled 500}\smallskip\end{figure}$

39. Elongated pentagonal gyrobicupola

$\begin{figure} \BoxedEPSF{J39.epsf scaled 500}\quad\BoxedEPSF{J39_net.epsf scaled 500}\smallskip\end{figure}$

40. Elongated pentagonal orthocupolarotunda

$\begin{figure} \BoxedEPSF{J40.epsf scaled 500}\quad\BoxedEPSF{J40_net.epsf scaled 500}\smallskip\end{figure}$

41. Elongated pentagonal gyrocupolarotunda

$\begin{figure} \BoxedEPSF{J41.epsf scaled 500}\quad\BoxedEPSF{J41_net.epsf scaled 500}\smallskip\end{figure}$

42. Elongated pentagonal orthobirotunda

$\begin{figure} \BoxedEPSF{J42.epsf scaled 500}\quad\BoxedEPSF{J42_net.epsf scaled 500}\smallskip\end{figure}$

43. Elongated pentagonal gyrobirotunda

$\begin{figure} \BoxedEPSF{J43.epsf scaled 500}\quad\BoxedEPSF{J43_net.epsf scaled 500}\smallskip\end{figure}$

44. Gyroelongated triangular bicupola

$\begin{figure} \BoxedEPSF{J44.epsf scaled 500}\quad\BoxedEPSF{J44_net.epsf scaled 500}\smallskip\end{figure}$

45. Gyroelongated square bicupola

$\begin{figure} \BoxedEPSF{J45.epsf scaled 500}\quad\BoxedEPSF{J45_net.epsf scaled 500}\smallskip\end{figure}$

46. Gyroelongated pentagonal bicupola

$\begin{figure} \BoxedEPSF{J46.epsf scaled 500}\quad\BoxedEPSF{J46_net.epsf scaled 500}\smallskip\end{figure}$

47. Gyroelongated pentagonal cupolarotunda

$\begin{figure} \BoxedEPSF{J47.epsf scaled 500}\quad\BoxedEPSF{J47_net.epsf scaled 500}\smallskip\end{figure}$

48. Gyroelongated pentagonal birotunda

$\begin{figure} \BoxedEPSF{J48.epsf scaled 500}\quad\BoxedEPSF{J48_net.epsf scaled 500}\smallskip\end{figure}$

49. Augmented triangular prism

$\begin{figure} \BoxedEPSF{J49.epsf scaled 500}\quad\BoxedEPSF{J49_net.epsf scaled 500}\smallskip\end{figure}$

50. Biaugmented triangular prism

$\begin{figure} \BoxedEPSF{J50.epsf scaled 500}\quad\BoxedEPSF{J50_net.epsf scaled 500}\smallskip\end{figure}$

51. Triaugmented triangular prism

$\begin{figure} \BoxedEPSF{J51.epsf scaled 500}\quad\BoxedEPSF{J51_net.epsf scaled 500}\smallskip\end{figure}$

52. Augmented pentagonal prism

$\begin{figure} \BoxedEPSF{J52.epsf scaled 500}\quad\BoxedEPSF{J52_net.epsf scaled 500}\smallskip\end{figure}$

53. Biaugmented pentagonal prism

$\begin{figure} \BoxedEPSF{J53.epsf scaled 500}\quad\BoxedEPSF{J53_net.epsf scaled 500}\smallskip\end{figure}$

54. Augmented hexagonal prism

$\begin{figure} \BoxedEPSF{J54.epsf scaled 500}\quad\BoxedEPSF{J54_net.epsf scaled 500}\smallskip\end{figure}$

55. Parabiaugmented hexagonal prism

$\begin{figure} \BoxedEPSF{J55.epsf scaled 500}\quad\BoxedEPSF{J55_net.epsf scaled 500}\smallskip\end{figure}$

56. Metabiaugmented hexagonal prism

$\begin{figure} \BoxedEPSF{J56.epsf scaled 500}\quad\BoxedEPSF{J56_net.epsf scaled 500}\smallskip\end{figure}$

57. Triaugmented hexagonal prism

$\begin{figure} \BoxedEPSF{J57.epsf scaled 500}\quad\BoxedEPSF{J57_net.epsf scaled 500}\smallskip\end{figure}$

58. Augmented dodecahedron

$\begin{figure} \BoxedEPSF{J58.epsf scaled 500}\quad\BoxedEPSF{J58_net.epsf scaled 500}\smallskip\end{figure}$

59. Parabiaugmented dodecahedron

$\begin{figure} \BoxedEPSF{J59.epsf scaled 500}\quad\BoxedEPSF{J59_net.epsf scaled 500}\smallskip\end{figure}$

60. Metabiaugmented dodecahedron

$\begin{figure} \BoxedEPSF{J60.epsf scaled 500}\quad\BoxedEPSF{J60_net.epsf scaled 500}\smallskip\end{figure}$

61. Triaugmented dodecahedron

$\begin{figure} \BoxedEPSF{J61.epsf scaled 500}\quad\BoxedEPSF{J61_net.epsf scaled 500}\smallskip\end{figure}$

62. Metabidiminished icosahedron

$\begin{figure} \BoxedEPSF{J62.epsf scaled 500}\quad\BoxedEPSF{J62_net.epsf scaled 500}\smallskip\end{figure}$

63. Tridiminished icosahedron

$\begin{figure} \BoxedEPSF{J63.epsf scaled 500}\quad\BoxedEPSF{J63_net.epsf scaled 500}\smallskip\end{figure}$

64. Augmented tridiminished icosahedron

$\begin{figure} \BoxedEPSF{J64.epsf scaled 500}\quad\BoxedEPSF{J64_net.epsf scaled 500}\smallskip\end{figure}$

65. Augmented truncated tetrahedron

$\begin{figure} \BoxedEPSF{J65.epsf scaled 500}\quad\BoxedEPSF{J65_net.epsf scaled 500}\smallskip\end{figure}$

66. Augmented truncated cube

$\begin{figure} \BoxedEPSF{J66.epsf scaled 500}\quad\BoxedEPSF{J66_net.epsf scaled 500}\smallskip\end{figure}$

67. Biaugmented truncated cube

$\begin{figure} \BoxedEPSF{J67.epsf scaled 500}\quad\BoxedEPSF{J67_net.epsf scaled 500}\smallskip\end{figure}$

68. Augmented truncated dodecahedron

$\begin{figure} \BoxedEPSF{J68.epsf scaled 500}\quad\BoxedEPSF{J68_net.epsf scaled 500}\smallskip\end{figure}$

69. Parabiaugmented truncated dodecahedron

$\begin{figure} \BoxedEPSF{J69.epsf scaled 500}\quad\BoxedEPSF{J69_net.epsf scaled 500}\smallskip\end{figure}$

70. Metabiaugmented truncated dodecahedron

$\begin{figure} \BoxedEPSF{J70.epsf scaled 500}\quad\BoxedEPSF{J70_net.epsf scaled 500}\smallskip\end{figure}$

71. Triaugmented truncated dodecahedron

$\begin{figure} \BoxedEPSF{J71.epsf scaled 500}\quad\BoxedEPSF{J71_net.epsf scaled 500}\smallskip\end{figure}$

72. Gyrate rhombicosidodecahedron

$\begin{figure} \BoxedEPSF{J72.epsf scaled 500}\quad\BoxedEPSF{J72_net.epsf scaled 500}\smallskip\end{figure}$

73. Parabigyrate rhombicosidodecahedron

$\begin{figure} \BoxedEPSF{J73.epsf scaled 500}\quad\BoxedEPSF{J73_net.epsf scaled 500}\smallskip\end{figure}$

74. Metabigyrate rhombicosidodecahedron

$\begin{figure} \BoxedEPSF{J74.epsf scaled 500}\quad\BoxedEPSF{J74_net.epsf scaled 500}\smallskip\end{figure}$

75. Trigyrate rhombicosidodecahedron

$\begin{figure} \BoxedEPSF{J75.epsf scaled 500}\quad\BoxedEPSF{J75_net.epsf scaled 500}\smallskip\end{figure}$

76. Diminished rhombicosidodecahedron

$\begin{figure} \BoxedEPSF{J76.epsf scaled 500}\quad\BoxedEPSF{J76_net.epsf scaled 500}\smallskip\end{figure}$

77. Paragyrate diminished rhombicosidodecahedron

$\begin{figure} \BoxedEPSF{J77.epsf scaled 500}\quad\BoxedEPSF{J77_net.epsf scaled 500}\smallskip\end{figure}$

78. Metagyrate diminished rhombicosidodecahedron

$\begin{figure} \BoxedEPSF{J78.epsf scaled 500}\quad\BoxedEPSF{J78_net.epsf scaled 500}\smallskip\end{figure}$

79. Bigyrate diminished rhombicosidodecahedron

$\begin{figure} \BoxedEPSF{J79.epsf scaled 500}\quad\BoxedEPSF{J79_net.epsf scaled 500}\smallskip\end{figure}$

80. Parabidiminished rhombicosidodecahedron

$\begin{figure} \BoxedEPSF{J80.epsf scaled 500}\quad\BoxedEPSF{J80_net.epsf scaled 500}\smallskip\end{figure}$

81. Metabidiminished rhombicosidodecahedron

$\begin{figure} \BoxedEPSF{J81.epsf scaled 500}\quad\BoxedEPSF{J81_net.epsf scaled 500}\smallskip\end{figure}$

82. Gyrate bidiminished rhombicosidodecahedron

$\begin{figure} \BoxedEPSF{J82.epsf scaled 500}\quad\BoxedEPSF{J82_net.epsf scaled 500}\smallskip\end{figure}$

83. Tridiminished rhombicosidodecahedron

$\begin{figure} \BoxedEPSF{J83.epsf scaled 500}\quad\BoxedEPSF{J83_net.epsf scaled 500}\smallskip\end{figure}$

84. Snub disphenoid

$\begin{figure} \BoxedEPSF{J84.epsf scaled 500}\quad\BoxedEPSF{J84_net.epsf scaled 500}\smallskip\end{figure}$

85. Snub square antiprism

$\begin{figure} \BoxedEPSF{J85.epsf scaled 500}\quad\BoxedEPSF{J85_net.epsf scaled 500}\smallskip\end{figure}$

86. Sphenocorona

$\begin{figure} \BoxedEPSF{J86.epsf scaled 500}\quad\BoxedEPSF{J86_net.epsf scaled 500}\smallskip\end{figure}$

87. Augmented sphenocorona

$\begin{figure} \BoxedEPSF{J87.epsf scaled 500}\quad\BoxedEPSF{J87_net.epsf scaled 500}\smallskip\end{figure}$

88. Sphenomegacorona

$\begin{figure} \BoxedEPSF{J88.epsf scaled 500}\quad\BoxedEPSF{J88_net.epsf scaled 500}\smallskip\end{figure}$

89. Hebesphenomegacorona

$\begin{figure} \BoxedEPSF{J89.epsf scaled 500}\quad\BoxedEPSF{J89_net.epsf scaled 500}\smallskip\end{figure}$

90. Disphenocingulum

$\begin{figure} \BoxedEPSF{J90.epsf scaled 500}\quad\BoxedEPSF{J90_net.epsf scaled 500}\smallskip\end{figure}$

91. Bilunabirotunda

$\begin{figure} \BoxedEPSF{J91.epsf scaled 500}\quad\BoxedEPSF{J91_net.epsf scaled 500}\smallskip\end{figure}$

92. Triangular hebesphenorotunda

$\begin{figure} \BoxedEPSF{J92.epsf scaled 500}\quad\BoxedEPSF{J92_net.epsf scaled 500}\smallskip\end{figure}$

The number of constituent -gons ( $\{n\}$ ) for each Johnson solid are given in the following table.

$\{3\}$ $\{4\}$ $\{5\}$ $\{6\}$ $\{8\}$ $\{10\}$

1 4 1 47 35 5 7

2 5 1 48 40 12

3 4 3 1 49 6 2

4 4 5 1 50 10 1

5 5 5 1 1 51 14

6 10 6 1 52 4 4 2

7 4 3 53 8 3 2

8 4 5 54 4 5 2

9 5 5 1 55 8 4 2

10 12 1 56 8 4 2

11 15 1 57 12 3 2

12 6 58 5 11

13 10 59 10 10

14 6 3 60 10 10

15 8 4 61 15 9

16 10 5 62 10 2

17 16 63 5 3

18 4 9 1 64 7 3

19 4 13 1 65 8 3 3

20 5 15 1 1 66 12 5 5

21 10 10 6 1 67 16 10 4

22 16 3 1 68 25 5 1 11

23 20 5 1 69 30 10 2 10

24 25 5 1 1 70 30 10 2 10

25 30 6 1 71 35 15 3 9

26 4 4 72 20 30 12

27 8 6 73 20 30 12

28 8 10 74 20 30 12

29 8 10 75 20 30 12

30 10 10 2 76 15 25 11 1

31 10 10 2 77 15 25 11 1

32 15 5 7 78 15 25 11 1

33 15 5 7 79 15 25 11 1

34 20 12 80 10 20 10 2

35 8 12 81 10 20 10 2

36 8 12 82 10 20 10 2

37 8 18 83 5 15 9 3

38 10 20 2 84 12

39 10 20 2 85 24 2

40 15 15 7 86 12 2

41 15 15 7 87 16 1

42 20 10 12 88 16 2

43 20 10 12 89 18 3

44 20 6 90 20 4

45 24 10 91 8 2 4

46 30 10 2 92 13 3 3 1

References

Bell Laboratories. http://netlib.bell-labs.com/netlib/polyhedra/.

Bulatov, V. ``Johnson Solids.'' http://www.physics.orst.edu/~bulatov/polyhedra/johnson/.

Cromwell, P. R. Polyhedra. New York: Cambridge University Press, pp. 86-92, 1997.

Hart, G. W. ``NetLib Polyhedra DataBase.'' http://www.li.net/~george/virtual-polyhedra/netlib-info.html.

Holden, A. Shapes, Space, and Symmetry. New York: Dover, 1991.

Hume, A. Exact Descriptions of Regular and Semi-Regular Polyhedra and Their Duals. Computer Science Technical Report #130. Murray Hill, NJ: AT&T Bell Laboratories, 1986.

Johnson, N. W. ``Convex Polyhedra with Regular Faces.'' Canad. J. Math. 18, 169-200, 1966.

Pugh, A. ``Further Convex Polyhedra with Regular Faces.'' Ch. 3 in Polyhedra: A Visual Approach. Berkeley, CA: University of California Press, pp. 28-35, 1976.

Weisstein, E. W. ``Johnson Solids.'' Mathematica notebook JohnsonSolids.m.

Weisstein, E. W. ``Johnson Solid Netlib Database.'' Mathematica notebook JohnsonSolids.dat.

Zalgaller, V. Convex Polyhedra with Regular Faces. New York: Consultants Bureau, 1969.

	$\{3\}$	$\{4\}$	$\{5\}$	$\{6\}$	$\{8\}$	$\{10\}$
1	4	1					47	35	5	7
2	5		1				48	40		12
3	4	3		1			49	6	2
4	4	5			1		50	10	1
5	5	5	1			1	51	14
6	10		6			1	52	4	4	2
7	4	3					53	8	3	2
8	4	5					54	4	5		2
9	5	5	1				55	8	4		2
10	12	1					56	8	4		2
11	15		1				57	12	3		2
12	6						58	5		11
13	10						59	10		10
14	6	3					60	10		10
15	8	4					61	15		9
16	10	5					62	10		2
17	16						63	5		3
18	4	9		1			64	7		3
19	4	13			1		65	8	3		3
20	5	15	1			1	66	12	5			5
21	10	10	6			1	67	16	10			4
22	16	3		1			68	25	5	1			11
23	20	5			1		69	30	10	2			10
24	25	5	1			1	70	30	10	2			10
25	30		6			1	71	35	15	3			9
26	4	4					72	20	30	12
27	8	6					73	20	30	12
28	8	10					74	20	30	12
29	8	10					75	20	30	12
30	10	10	2				76	15	25	11			1
31	10	10	2				77	15	25	11			1
32	15	5	7				78	15	25	11			1
33	15	5	7				79	15	25	11			1
34	20		12				80	10	20	10			2
35	8	12					81	10	20	10			2
36	8	12					82	10	20	10			2
37	8	18					83	5	15	9			3
38	10	20	2				84	12
39	10	20	2				85	24	2
40	15	15	7				86	12	2
41	15	15	7				87	16	1
42	20	10	12				88	16	2
43	20	10	12				89	18	3
44	20	6					90	20	4
45	24	10					91	8	2	4
46	30	10	2				92	13	3	3	1