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Painted Cube 5x5x5: 3D Geometry and Counting Exploration

Painted Cube 5x5x5: 3D Geometry and Counting Exploration
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2016-03-06 22:47
PolygonSequenceSolid GeometryGeometric Construction
This GeoGebra example demonstrates a 5x5x5 cube with its exterior surface painted, leaving inner unit cubes unpainted, to visually explore the classic painted cube counting problem. Core mathematical concepts include solid geometry (cube structure), surface area vs. volume relationships, and combinatorial classification of unit cubes by the number of painted faces (3, 2, 1, or 0). The construction uses GeoGebra commands: Point to define vertices, Cube to build the main solid, PointIn to mark interior points, and Segment to draw edges and auxiliary lines. Objects include faces (faceABCD, faceADHE, etc.), edges (edgeAD, edgeAB, etc.), and points (I, J, K, L, M, N, O, P, Q) for interactive exploration. Suitable for middle and high school students, this example supports spatial reasoning, combinatorial counting practice, and introductory solid geometry lessons, and can be used for inquiry-based learning or math competition preparation.

GeoGebra Commands & Parameters

  1. Point(xAxis) → A
  2. Point(xAxis) → B
  3. Point(Circle[B, Distance[A, B], Segment[A, B]]) → C
  4. Cube(A, B, C) → a, D, faceCDHG, faceEFGH, edgeAD, edgeAB, edgeBC, edgeCD, edgeAE, edgeDH, edgeEH, edgeEF, E, edgeBF, edgeFG, edgeCG, edgeGH, F, G, H, faceABCD, faceADHE, faceABFE, faceBCGF
  5. Point(edgeBC) → I
  6. PointIn(faceBCGF) → J
  7. Segment(I, J) → b
  8. Point(edgeBC) → K
  9. Point(edgeFG) → L
  10. Segment(K, L) → c
  11. Point(edgeBC) → M
  12. Point(edgeFG) → N
  13. Segment(M, N) → d
  14. Point(edgeBC) → O
  15. PointIn(faceBCGF) → P
  16. Segment(O, P) → e
  17. Point(edgeCD) → Q
  18. Point(edgeGH) → R
  19. Segment(Q, R) → f
  20. Point(edgeCD) → S
  21. Point(edgeGH) → T
  22. Segment(S, T) → g
  23. Point(edgeCD) → U
  24. Point(edgeGH) → V
  25. Segment(U, V) → h
  26. Point(edgeCD) → W
  27. PointIn(faceCDHG) → Z
  28. Segment(W, Z) → i
  29. Point(yAxis) → A_1
  30. Point(edgeEH) → B_1
  31. Segment(A_1, B_1) → j
  32. Point(yAxis) → C_1
  33. Point(edgeEH) → D_1
  34. Segment(C_1, D_1) → k
  35. Point(yAxis) → E_1
  36. PointIn(faceADHE) → F_1
  37. Segment(E_1, F_1) → l
  38. Point(yAxis) → G_1
  39. PointIn(faceADHE) → H_1
  40. Segment(G_1, H_1) → m
  41. Point(xAxis) → I_1
  42. Point(edgeEF) → J_1
  43. Segment(I_1, J_1) → n
  44. Point(xAxis) → K_1
  45. Point(edgeEF) → L_1
  46. Segment(K_1, L_1) → p
  47. Point(xAxis) → M_1
  48. Point(edgeEF) → N_1
  49. Segment(M_1, N_1) → q
  50. Point(xAxis) → O_1
  51. Point(edgeEF) → P_1
  52. Segment(O_1, P_1) → r
  53. Segment(P, B_1) → s
  54. Segment(D_1, N) → t
  55. Segment(L, F_1) → a_1
  56. Segment(J, H_1) → b_1
  57. Segment(Z, P_1) → c_1
  58. Segment(N_1, V) → d_1
  59. Segment(T, L_1) → e_1
  60. Segment(R, J_1) → f_1
  61. Segment(Q, I_1) → g_1
  62. Segment(S, K_1) → h_1
  63. Segment(U, M_1) → i_1
  64. Segment(W, O_1) → j_1
  65. Segment(A_1, O) → k_1
  66. Segment(M, C_1) → l_1
  67. Segment(E_1, K) → m_1
  68. Segment(G_1, I) → n_1
  69. Point(zAxis) → Q_1
  70. Point(edgeBF) → R_1
  71. Segment(Q_1, R_1) → p_1
  72. Point(zAxis) → S_1
  73. Point(edgeBF) → T_1
  74. Segment(S_1, T_1) → q_1
  75. Point(zAxis) → U_1
  76. Point(edgeBF) → V_1
  77. Segment(U_1, V_1) → r_1
  78. Point(zAxis) → W_1
  79. PointIn(faceABFE) → Z_1
  80. Segment(W_1, Z_1) → s_1
  81. Point(edgeDH) → A_2
  82. Segment(Q_1, A_2) → t_1
  83. Point(edgeDH) → B_2
  84. Segment(S_1, B_2) → a_2
  85. Point(edgeDH) → C_2
  86. Segment(U_1, C_2) → b_2
  87. Point(edgeDH) → D_2
  88. Segment(W_1, D_2) → c_2
  89. Point(edgeCG) → E_2
  90. Segment(D_2, E_2) → d_2
  91. Point(edgeCG) → F_2
  92. Segment(C_2, F_2) → e_2
  93. Point(edgeCG) → G_2
  94. Segment(B_2, G_2) → f_2
  95. Point(edgeCG) → H_2
  96. Segment(A_2, H_2) → g_2
  97. Segment(Z_1, E_2) → h_2
  98. Segment(V_1, F_2) → i_2
  99. Segment(T_1, G_2) → j_2
  100. Segment(R_1, H_2) → k_2