Problems

The case of Brown, Jones and Smith is being considered. One of them committed a crime. During the investigation, each of them made two statements. Brown: “I did not do it. Jones did not do it. " Smith: “I did not do it. Brown did it. “Jones:" Brown did not do it. This was done by Smith. “Then it turned out that one of them had told the truth in both statements, another had lied both times, and the third had told the truth once, and he had lied once. Who committed the crime?

Elephants, rhinoceroses, giraffes. In all zoos where there are elephants and rhinoceroses, there are no giraffes. In all zoos where there are rhinoceroses and there are no giraffes, there are elephants. Finally, in all zoos where there are elephants and giraffes, there are also rhinoceroses. Could there be a zoo in which there are elephants, but there are no giraffes and no rhinoceroses?

Several natives of an island met up (each either a liar or a knight), and everyone said to everyone else: “You are all liars.” How many knights were there among them?

Theorem: All people have the same eye color.

"Proof" by induction: This is clearly true for one person.

Now, assume we have a finite set of people, denote them as \(a_1,\, a_2,\, ...,\,a_n\), and the inductive hypothesis is true for all smaller sets. Then if we leave aside the person \(a_1\), everyone else \(a_2,\, a_3,\,...,\,a_n\) has the same color of eyes and if we leave aside \(a_n\), then all \(a_1,\, a_2,\,a_3,...,\,a_{n-1}\) also have the same color of eyes. Thus any \(n\) people have the same color of eyes.
Find a mistake in this "proof".

King Hattius has three prisoners and gives them the following puzzle. He will put a randomly coloured hat on each of their heads: red, blue or green. He’ll then give them \(10\) seconds for them to each guess their own hat’s colour at the same time.

However! Each prisoner can only see the other two prisoners’ hats, not their own. There are no mirrors in the prison, and they are not allowed to take off their hat, nor talk, mouth, use sign-language, or otherwise communicate with the other two prisoners during those ten seconds.

Hattius tells them that he’ll release them all if at least one correctly guesses their hat’s colour. He gives them an hour to come up with a strategy - what should their strategy be?

Two aliens want to abduct two humans, but aren’t paying attention, so instead run after pigs. They’re all on squares of a \(3\times6\) rectangle, as seen below. On the first move, the aliens move one square horizontally or vertically. Then on the second move, the pigs move horizontally or vertically. The third move is for the aliens, the fourth move is for the pigs, and so on. If an alien lands on a square with a pig on it, then they’ve succeeded. Show that no matter what the pigs do, they’re doomed.

In the diagram below, there are nine discs - each is black on one side, and white on the other side. Two have black face-up right now. Your task is to remove all of the discs by making a series of the following moves. Each move includes choosing a black disc, flipping over its neighbours\(^*\) and removing that black disc. Discs are ‘neighbours’ if they’re adjacent at the beginning - removing a disc creates a gap, so that at later stages, a disc may have two, one or even zero neighbours left. \[\circ\circ\circ\bullet\circ\circ\circ\circ\bullet\] Show that this task is impossible.