In the figure below, Maria arranged 24 sticks to form a $3 \times 3$ square.

![](https://cdn.mathpix.com/cropped/2024_05_01_80f7f1185898b39e7876g-08.jpg?height=403&width=394&top_left_y=1626&top_left_x=731)

a) How many sticks would she need to form a $4 \times 4$ square?

b) What is the side length of the largest square she could form with 100 sticks? If there are any sticks left over, determine how many.

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Solution

a) The figure below shows that she will need to use 40 sticks

![](https://cdn.mathpix.com/cropped/2024_05_01_80f7f1185898b39e7876g-09.jpg?height=565&width=576&top_left_y=497&top_left_x=840)

b) If we want to form a square with a side composed of $n$ sticks, we will need $n+1$ rows of $n$ sticks and $n+1$ columns also of $n$ sticks. This totals to $n(n+1)+n(n+1)=2n(n+1)$ sticks. Therefore, if

$$
\begin{aligned}
& n=5 \Rightarrow 2n(n+1)=60 \\
& n=6 \Rightarrow 2n(n+1)=84 \\
& n=7 \Rightarrow 2n(n+1)=112
\end{aligned}
$$

Thus, with 100 sticks, we can form a square with a side of 6 and there will be $100-84=16$ sticks left over.