I4.4 Given that $241 c+214=d^{2}$, find the positive value of $d$.
$$
\begin{array}{l}
d^{2}=241 \times 462+214=111556 \\
d=\sqrt{111556}
\end{array}
$$

Reference：昌爸工作坊圖解直式開平方
Divide 111556 into 3 groups of numbers $11,15,56$.
Find the maximum integer $p$ such that $p^{2} \leq 11 \Rightarrow p=3$
$$
\begin{array}{l}
11-p^{2}=2 \\
3+3=6
\end{array}
$$

Find the maximum integer $q$ such that $(60+q) q \leq 215$
$$
\begin{array}{l}
\Rightarrow q=3 \\
215-63 \times 3=26 \\
60+q+q=66
\end{array}
$$

Find the maximum integer $r$ such that $(660+r) r \leq 2656$
$$
\begin{array}{l}
\Rightarrow r=4 \\
d=334
\end{array}
$$

Method 2: $d^{2}=241 \times 462+214=111556$
$$
\begin{array}{l}
300^{2}=90000<111556<160000=400^{2} \Rightarrow 300<d<400 \\
330^{2}=108900<111556<115600=340^{2} \Rightarrow 330<d<340
\end{array}
$$

The unit digit of $d^{2}$ is $6 \Rightarrow$ the unit digit of $d$ is 4 or 6
$$
335^{2}=112225 \Rightarrow d=334
$$
or 111556 is not divisible by 3 , but 336 is divisible by 3
$$
\therefore d=334
$$

Method 3
Observe the number
patterns:
$$
\begin{array}{l}
34^{2}=1156 \\
334^{2}=111556 \\
3334^{2}=11115556
\end{array}
$$
$$
\therefore d=334
$$

Also, $33^{2}=1089$
$$
\begin{array}{l}
333^{2}=110889 \\
3333^{2}=11108889
\end{array}
$$
$$
\begin{array}{l}
3 \times 4=12 \\
33 \times 34=1122 \\
333 \times 334=111222 \\
3333 \times 3334=11112222
\end{array}
$$