Triangles and cubes ...?

Question

a) 20 1cm edge cubes have white faces: 44 1 cm edge have blue faces. The 64 cubes are glued together to form a cube with edge 4 cm. What is the minimum surface area that could be white?

b) In the equilateral triangle ABC with triangle PQR situated inside, AB = 5cm, AP = 4cm, BQ = 4cm and CR = 3cm.
If S is the aread of triangle ABC, what is the area of triangle PQR in terms of S.

Answer 1

a) 16cm squared - 8 hidden in middle, 24 can show 1 face - only need 16
b) unanswerable - anywhere on arc

Answer 2

a) Put the white cubes inside: 8 go hidden. Remain 12 whites. These will be on the faces/edge. Since there are 6 faces on the big cube, you could arrange the white cubes only 2 per face visible (at the center of the cube's face). this will give you a minimum surface of TWO white cubes ON EACH FACE.
However, if you allow for white areas to be different, you could say "ONE" (just one of the white facing outside, the others on the other faces).
b). Not possible to answer without the real configuration:
Trace your triangle ABC. From A, go towards B and, at 4cm, place the point P. Do the same for the other two points. The inside triangle is pretty big.
Other choice: from A, go towards segment BC up to 4cm and place your point P. Repeat for the others: you have a small triangle inside of the big one.
The geometry of the triangle PQR is not defined, hence, no answerable.

Answer 3

8 of the cubes could be in the centre so that no faces show and the remaining 12 can be places so that one face shows. This gives a total of 12 cm^2.

I don't think there's enough information in the second question to fix the area of PQR. Draw ABC and then draw an arc radius 4cm centred on A, P could lie anywhere on this arc, similarly for Q and R. You could draw any number of triangles by changing the positions of P,Q and R and they wouldn't all have the same areas.

Answer 4

answer to a is definitely 16 cm^2

i would need to draw b, and dont want to.

Answer 5

spaghetti hoops !!