729 small cubelets are painted pink on each face and then arranged together so as to form 27 identical medium-sized cubes. Each of these 27 medium-sized cubes is painted black on all the outside faces. The 27 medium-sized cubes are now arranged to form one large cube and the faces of this large cube are painted pink again. What is the number of small cubelets that have at least one face painted? What is the number of small cubelets that have at least one face painted black? What is the number of small cubelets that have at least one face painted pink? What is the number of small cubelets which have an equal number of faces painted pink and black?
Answers
Answer:
The painting of the 27 middle-sized cubes produces 27⋅8 small cubes with a black corner, 27⋅12 small cubes with a black edge and 27⋅6 small cubes with a black face (and leaves 27 small cubes in the centres all pink).
The painting of the big cube destroys 8 corners. Along each of the 12 edges of the big cube, there are 3 small cubes that had only that edge black, so 12⋅3=36 edges are destroyed. And on each of the 6 faces of the big cube, there are 9 small cubes that had only that face black, so 6⋅9=54 faces are destroyed.
That leaves (27−1)⋅8=208 (former) corners, (27−3)⋅12=288 (former) edges and (27−9)⋅6=108 faces, for a total of 208+288+108=604 small cubes that still have some black.
Step-by-step explanation:
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The painting of the 27 middle-sized cubes produces 27⋅8 small cubes with a black corner, 27⋅12 small cubes with a black edge and 27⋅6 small cubes with a black face (and leaves 27 small cubes in the centres all pink).
The painting of the big cube destroys 8 corners. Along each of the 12 edges of the big cube, there are 3 small cubes that had only that edge black, so 12⋅3=36 edges are destroyed. And on each of the 6 faces of the big cube, there are 9 small cubes that had only that face black, so 6⋅9=54 faces are destroyed.
That leaves (27−1)⋅8=208 (former) corners, (27−3)⋅12=288 (former) edges and (27−9)⋅6=108 faces, for a total of 208+288+108=604 small cubes that still have some black.