1. (i) (Grade 10) Let the sets be
$$
A=\left\{a^{2}+8 \mid a \in \mathbf{N}\right\}, B=\left\{b^{2}+29 \mid b \in \mathbf{N}\right\} \text {. }
$$

If $A \cap B=P$, then $P$ contains $(\quad)$ elements.
(A) 0
(B) 1
(C) 2
(D) at least 3
(ii) (Grade 11) Three non-coincident planes can divide space into $n$ parts. Then all possible values of $n$ are ( ).
(A) $4,6,8$
(B) $4,6,7$
(C) $4,5,7,8$
(D) $4,6,7,8$
-1. (i) C.
From $a^{2}+8=b^{2}+29$, we get $a^{2}-b^{2}=21$, which is $(a-b)(a+b)=21$.
By $\left\{\begin{array}{l}a-b=1, \\ a+b=21,\end{array}\right.$ we solve to get $\left\{\begin{array}{l}a=11, \\ b=10 .\end{array}\right.$
By $\left\{\begin{array}{l}a-b=3, \\ a+b=7,\end{array}\right.$ we solve to get $\left\{\begin{array}{l}a=5, \\ b=2 .\end{array}\right.$
Therefore, $P=\{129,33\}$, with a total of 2 elements.
(ii) D.
Three parallel planes divide the space into 4 parts; when the three planes pass through the same line, they divide the space into 6 parts; the three side faces of a triangular prism divide the space into 7 parts; the three coordinate planes in space divide the space into 8 parts (8 octants).