1. **Stating the problem:** We have a pattern of square-based shapes made of smaller squares. Step 1 is a 2x2 grid (4 squares), Step 2 is a 3x3 grid (9 squares) with the middle square missing (8 squares), Step 3 is a 4x4 grid (16 squares) with the inner 2x2 square missing (12 squares). We want to find the number of small squares in Step 5 and Step 18, and write an equation relating step number $n$ to the number of squares $y$. 2. **Analyzing the pattern:** - Step 1: grid size $2 \times 2 = 4$ squares, no missing squares. - Step 2: grid size $3 \times 3 = 9$ squares, missing $1 \times 1 = 1$ square, so $9 - 1 = 8$ squares. - Step 3: grid size $4 \times 4 = 16$ squares, missing $2 \times 2 = 4$ squares, so $16 - 4 = 12$ squares. 3. **Generalizing the pattern:** - At step $n$, the grid size is $(n+1) \times (n+1)$, so total squares are $(n+1)^2$. - The missing inner square is of size $(n-1) \times (n-1)$, so missing squares are $(n-1)^2$. 4. **Equation for number of squares:** $$ y = (n+1)^2 - (n-1)^2 $$ 5. **Simplify the equation:** $$ y = \cancel{(n+1)^2} - \cancel{(n-1)^2} = (n+1)^2 - (n-1)^2 $$ Using the difference of squares formula: $$ y = [(n+1) - (n-1)] \times [(n+1) + (n-1)] $$ $$ y = (2) \times (2n) = 4n $$ 6. **Interpretation:** The number of small squares at step $n$ is $y = 4n$. 7. **Calculate for Step 5 and Step 18:** - Step 5: $y = 4 \times 5 = 20$ squares. - Step 18: $y = 4 \times 18 = 72$ squares. 8. **Explanation:** At each step, the pattern adds a layer of squares around the previous shape, increasing the total by $4$ times the step number. **Final answer:** - Step 5 has 20 small squares. - Step 18 has 72 small squares. - The equation relating step number $n$ to number of squares $y$ is: $$ y = 4n $$