Question

A unit cube with side lengths of 1 unit is cut into cuboids with heights of 1/2, 1/3, and 1/6 units, respectively. The pieces are then placed adjacent to each other to form a staircase.

The total surface area of the original unit cube was 6 units. What is the total surface area of the the staircase made from slices of the original cube?

Answer 1

I'll assume that the surfaces lying on the ground are not counted. They are all 1x1 anyway, for an additional 3 u^2 area.

The 1/6 block has its top and 3 faces exposed: 1+3*(1/6) = 3/2

The 1/3 block has its top, two sides, and half of another side exposed: 1+2*(1/3)+(1/6) = 11/6

The 1/2 block has its top, two sides, the outside face and 1/3 of its inside face: 1+2*(1/2)+(1/2)+(1/3)(1/2) = 8/3

I was surprised at the answer!

Answer 2

Figured it out. Total surface area is 9. Top and bottom surfaces DO count, which makes 6 total. The sides are 1/6 and 2 x 1/6, 1/6 + 2 x 1/3, and 1/6 + 2 x 1/2=3 more
☺☺☺☺