Problems
In the government of the planet of liars and truth tellers there are \(101\) ministers. In order to reduce the budget, it was decided to reduce the number of ministers by \(1.\) But each of the ministers said that if they were to be removed from the government, then the majority of the remaining ministers would be liars. How many truth tellers and how many liars are there in the government?
Father Christmas has an infinite number of sweets. A minute before the New Year, Father Christmas gives some children 100 sweets, while the Snow Maiden takes one sweet from them. Within half a minute before the New Year, Father Christmas gives the children 100 more sweets, and the Snow Maiden again takes one sweet. The same is repeated for 15 seconds, for 7.5 seconds, etc. until the new Year. Prove that the Snow Maiden will be able to take away all the sweets from the children by the New Year.
Let \(x\) be a natural number. Among the statements:
\(2x\) is more than 70;
\(x\) is less than 100;
\(3x\) is greater than 25;
\(x\) is not less than 10;
\(x\) is greater than 5;
three are true and two are false. What is \(x\)?
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.
Explain why you can’t rotate the sides on a normal Rubik’s cube to get to the following picture (with no removing stickers, painting, or other cheating allowed).
