some problems about pigeonhole principle

Question

The Problems

In question A, I know I can set the hole as the picture below. Then I can prove it by pigeonhole principle.

question A

In question B, is it true that I just say that I can place the 16 pieces of kings as the picture below and it is proved?

question B

As for questions C and D, I have no idea how to do next. How should I set the holes? Any suggestion or hint is appreciated. Thanks a lot!

Answer 1

(Fill in the gaps. If you're stuck, show your work and explain where you're stuck.)

D - Do this first (yourself).
It provides a bit of a hint for how to proceed with C, in terms of determining the structure we want to find/avoid. (This is quite common for such setups, where studying the counter example provides some hints.)

C - We want the kings "far apart".
We're told to apply PP. Let's deconstruct the solution (backward thinking from the problem statement).
In particular, $ 5 = \lceil \frac{17}{4} \rceil, 4 = \lceil \frac{16}{4} \rceil$, which suggests that there are 4 holes.
Look at your 16 counterexample. We have 4 holes, each having 4 pigeons.
What could these holes be, to contain kings that definitely do not attack each other?

Hint: 2 kings cannot attack each other if ...

the squares that they are on have the same parity.

(Note: It's possible that a king cannot attack another king that's in a different hole. We don't care about that.)