萨莉（Sally）正在使用所示的图案用她的正方形瓷砖制作正方形图案。她有黑色和白色的瓷砖。 求她新正方形的尺寸：

1. Relate the number of tiles added to the odd-number sequence.

The tiles added are
$3,5,7,\ldots$
This is an arithmetic sequence with first term $3$ and common difference $2$.

Let $65$ be the $n$th term of this sequence:
$3+2(n-1)=65$

2. Find the resulting square size.

$3+2(n-1)=65$
$2(n-1)=62$
$n=32$

Adding $3$ tiles changes Pattern 1 to Pattern 2, adding $5$ tiles changes Pattern 2 to Pattern 3, and so on.

So the $32$nd addition gives Pattern $33$.

Hence the new square is
$33 \times 33.$

1. Form an equation connecting the number of tiles added to the size of the new square.

For a new square of size $n\times n$, the number of tiles just added is
$2n-1.$
Given that she has just added $4k+3$ tiles,
$2n-1=4k+3.$

2. Solve for the new square size in terms of $k$.

$2n-1=4k+3$
$2n=4k+4$
$n=2k+2$
Hence the size of the new square is
$(2k+2)\times(2k+2).$