Gerry had a rectangular block of wood 15 cm by 8 cm by 20 cm. He painted all the faces of the block.

1. Identify the dimensions of the three distinct pairs of rectangular faces.

The dimensions of the cuboid are $15\text{ cm}$, $8\text{ cm}$, and $20\text{ cm}$.
The three pairs of faces have dimensions:

1. $15\text{ cm} \times 8\text{ cm}$
2. $15\text{ cm} \times 20\text{ cm}$
3. $8\text{ cm} \times 20\text{ cm}$

2. Calculate the area of each pair of faces.

Area of top and bottom faces $= 2 \times (15 \times 8) = 2 \times 120 = 240\text{ cm}^2$
Area of front and back faces $= 2 \times (15 \times 20) = 2 \times 300 = 600\text{ cm}^2$
Area of left and right faces $= 2 \times (8 \times 20) = 2 \times 160 = 320\text{ cm}^2$

3. Sum the areas to find the total painted area.

Total painted area $= 240 + 600 + 320 = 1160\text{ cm}^2$

1. Identify the location of cubes with exactly 2 painted faces

Cubes with exactly 2 painted faces are located along the edges of the rectangular block, excluding the 8 corner cubes (which have 3 painted faces). A cuboid has 12 edges: 4 of length $15\text{ cm}$, 4 of length $8\text{ cm}$, and 4 of length $20\text{ cm}$.

2. Calculate the number of such cubes on each edge and sum them up

For each edge, we subtract the 2 corner cubes.
Number of cubes on the $15\text{ cm}$ edges: $4 \times (15 - 2) = 4 \times 13 = 52$
Number of cubes on the $8\text{ cm}$ edges: $4 \times (8 - 2) = 4 \times 6 = 24$
Number of cubes on the $20\text{ cm}$ edges: $4 \times (20 - 2) = 4 \times 18 = 72$
Total number of cubes = $52 + 24 + 72 = 148$