Suppose we have a function HALTS which when applied to any arbitrary function f and its arguments will say TRUE if function f terminates for those arguments and FALSE otherwise. Example, Given the following function definition.
FACTORIAL (N) = IF(N=0) THEN 1 ELSE N*FACTORIAL (N-1)
Then HALTS(FACTORIAL 4) = TRUE and HATS(FACTORIAL - 5) = FALSE

Let us define the function FUNNY(f) = IF HALTS(ff) THEN not(ff) ELSE TRUE
(a) Show that FUNNY terminates for all functions f.
(b) Use (a) to prove (by contradiction) that it is not possible to have a function like HALTS which for arbitrary functions and inputs says whether it will terminate on that input or not.

In the land of Twitter, there are two kinds of people: knights (also called outragers), who always tell the truth, and knaves (also called trolls), who always lie. It so happened that a person with handle @anand tweeted something offensive. It was not known whether @anand was a knight or a knave. A crack team, headed by Inspector Chitra, rounded up three suspects and interrogated them.
The first interrogation went as follows.
Chitra : What do you know about @anand?
Suspect 1 : @anand once claimed that I was a knave.
Chitra : Are you by any chance @anand?
Suspect 1 : Yes.
The second interrogation:
Chitra : Have you ever claimed you were @anand?
Suspect 2 : No.
Chitra : Did you ever claim you are not @anand?
Suspect 2 : Yes.
The third suspect arrived with a defense lawyer (also on Twitter):
Lawyer : My client is indeed a knave, but he is not @anand.
Suspect 3 : My lawyer always tells the truth.
Which of the above suspects are innocent, and which are guilty?
Explain your reasoning.

Consider a degree-5 polynomial function f : (−∞,∞) → (−∞,∞). If f exhibits at least four local maxima, which of the following is necessarily true? (Note : A local maximum is a point where the function value is the maximum in a sufficiently small neighbourhood.)