1. **Problem statement:** We have a shape made by joining $n$ cubes in a row, each cube is $1$ cm on each side. 2. **Find an expression for the surface area of the shape made from $n$ cubes:** - Each cube has $6$ faces, each face is $1 \times 1 = 1$ cm². - For $n$ cubes in a row, the total surface area is not simply $6n$ because the cubes share faces where they join. - When two cubes join, they share one face, so that face is not visible. - There are $n-1$ shared faces between $n$ cubes. 3. **Formula for surface area:** $$\text{Surface Area} = 6n - 2(n-1)$$ - We subtract $2(n-1)$ because each shared face removes $2$ visible faces (one from each cube). 4. **Simplify the expression:** $$6n - 2(n-1) = 6n - 2n + 2 = 4n + 2$$ 5. **Answer for part (a):** The surface area of the shape made from $n$ cubes is: $$\boxed{4n + 2}$$ 6. **Find $n$ when surface area is $254$ cm²:** Set the formula equal to $254$: $$4n + 2 = 254$$ 7. **Solve for $n$:** $$4n = 254 - 2$$ $$4n = 252$$ $$n = \frac{252}{4}$$ 8. **Show cancellation:** $$n = \frac{\cancel{252}}{\cancel{4}} = 63$$ 9. **Answer for part (b):** The number of cubes needed is: $$\boxed{63}$$