a wooden cube 9*9*9 has three square holes drilled through it each of which forms a 3*3*9 tunnel through the centre of opposite faces.what is the total surface area of exposed wood
My "brain cells that handle 3D" are weak, few, and far apart.
But here's what I'd do:
1: buy a perfect cube of soft cheese (on sale)
2: turn the 9*9*9 and 3*3*9 to 3*3*3 and 1*1*3
3: "line" the surfaces so that each looks like a "X and O" grid
4: going top to bottom and using appropriate kitchen knife,
remove the vertical 1*3 at center; so three 1*1's have been removed
5: perform similar surgery both ways horizontally; here one must remember
that only two 1*1's are removed for each (the center one is gone from step 4),
so four 1*1's have been removed
a = surface area of initial cube: 6 * 9 = 54
b = "removed area"; a 1*1 from each surface: 6 * 1 = 6
c = "added area"; that's 4 surfaces of six 1*1's: 6 * 4 = 24
So exposed area = a - b + c = 72
Now translate that to a 9*9*9.
Well, hope that helps a bit...
1. Draw a sketch.
2. The surface of the original cube: $\displaystyle a_s = 6 \cdot 9^2 = 486$
3. At each face is missing a square: $\displaystyle 6 \cdot 3^2=54$
4. At each face you find 4 squares which form the walls of the holes: $\displaystyle 6 \cdot 4 \cdot 3^2 = 216$
5. Therefore the complete surface area is: $\displaystyle 486-54+216 = 648$