Freddy had a rectangular block measuring 10 cm by 3 cm by 6 cm. He painted all the faces of the block.

1. Identify the dimensions of the rectangular block

Length ($l$) = $10\text{ cm}$
Width ($w$) = $3\text{ cm}$
Height ($h$) = $6\text{ cm}$

2. Calculate the area of the three distinct pairs of faces

Area of top and bottom faces = $2 \times (10 \times 3) = 2 \times 30 = 60\text{ cm}^2$
Area of front and back faces = $2 \times (10 \times 6) = 2 \times 60 = 120\text{ cm}^2$
Area of left and right faces = $2 \times (3 \times 6) = 2 \times 18 = 36\text{ cm}^2$

3. Sum the areas to find the total painted area

Total surface area = $60 + 120 + 36 = 216\text{ cm}^2$

1. Determine the side length of the largest possible cube that can be cut from the block.

The dimensions of the block are $10\text{ cm}$, $3\text{ cm}$, and $6\text{ cm}$.
The largest possible cube must have a side length equal to the smallest dimension of the block, which is $3\text{ cm}$.

2. Calculate the number of $3\text{ cm}$ cubes that can fit along each dimension and find the total number.

Number of cubes along the length ($10\text{ cm}$): $10 \div 3 = 3$ with a remainder of $1\text{ cm}$. So, $3$ cubes.
Number of cubes along the width ($3\text{ cm}$): $3 \div 3 = 1$ cube.
Number of cubes along the height ($6\text{ cm}$): $6 \div 3 = 2$ cubes.
Total number of cubes $= 3 \times 1 \times 2 = 6$.