An event has ben generated in a dungeon in which the player finds themselves encountering an incredible powerful and ancient knight that will attack the player if they fail the proceeding challenges.

On the ground of the dungeon floor are numerous circles overlapping each other. The ancient knight challenges the player by telling them to stand in certain positions, which both use the circles and other symbols that appear on the floor.

When playtesting the challenges, you realise that some of the challenges made by the ancient knight seem entirely impossible!

For each question you should:

a$)$ Prove that the challenge is possible by providing all valid solutions to the challenge, or prove that it is impossible. You should provide visuals and supporting logical reasoning in your proof. Remember: your proof should contain all potential valid answers $($not just one$)!$

b$)$ If the challenge is impossible, you should provide a fix that would allow the player to solve the challenge by altering it in as few ways as possible. $($There may be many appropriate solutions$)$

c$)$ The ancient knight poses the following challenges:

d$)2\mathrm{.}1\mathrm{.}1$ Stand in an area that indicates that you are crazy and are a pirate $($reminder: this is regardless of whether you have wings or are an automaton.

e$)($REMEMBER TO PROVIDE ALL SOLUTIONS OR PROVE IT IS INVALID!$)$

f$)2\mathrm{.}1\mathrm{.}2$ Stand in an area that indicates that you are a pirate or an automaton, and in either case you have wings.

g$)2\mathrm{.}1\mathrm{.}3$ Stand in an area that indicates that you are an automaton and a pirate, but do not have wings and are not crazy.

h$)2\mathrm{.}1\mathrm{.}4$ Stand in an area that indicates that you are not crazy and are not an automaton, but you do have wings or you are a pirate but you aren$$t a pirate with wings.

Two new circles appear and the ancient knight states that the following circles represents the treasure you can find in the dungeon. The ancient knight continues by posing the following challenges:

$2\mathrm{.}2\mathrm{.}1$ Speak aloud the number of pieces of treasure in the dungeon that are magical or are weapons.

$2\mathrm{.}2\mathrm{.}2$ Speak aloud how many pieces of treasure in the dungeon are not both magical and weapons.

$2\mathrm{.}2\mathrm{.}3$ Speak aloud how many pieces of treasure in the dungeon are magical and are ancient but are not weapons.First circles:

Second Circles: