The 2x4 example, case 2.

Maximize
  Obj: 0
Subject to
  x5 + x8 + x10 = 1
  x1 + x5 + x8 + x9 + x10 = 1
  x5 + x8 + x10 = 1
  x2 + x7 = 1
  x3 + x8 = 1
  x6 + x7 + x9 = 1
  x4 + x6 + x7 + x9 + x10 = 1
  x6 + x7 + x9 = 1
  x1 + x2 + x3 + x4 = 1
  x5 + x6 = 1
  x7 + x8 + x9 + x10 = 1
  x1 + x2 + x3 + x4 + 3 x5 + 3 x6 + 4 x7 + 4 x8 + 4 x9 + 4 x10 = 8
Binary
  x1 x2 x3 x4 x5 x6 x7 x8 x9 x10
End