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Interesting colourings of a cube

Source: Chinese MO 1999

March 19, 2012

geometry3D geometrycombinatorics proposedcombinatorics

Problem Statement

4\times4\times4

cube is composed of

64

unit cubes. The faces of

16

unit cubes are to be coloured red. A colouring is called interesting if there is exactly

1

red unit cube in every

1\times1\times 4

rectangular box composed of

4

unit cubes. Determine the number of interesting colourings.