Problems

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.

Draw the plane tiling with regular hexagons.

Let \(\sigma(n)\) be the sum of the divisors of \(n\). For example, \(\sigma(12)=1+2+3+4+6+12=28\). We use \(\gamma\) to denote the Euler-Mascheroni constant - one way to define this is as \(\gamma:=\lim_{n\to\infty}(\sum_{k=1}^n\frac{1}{n}-\log n)\).

Prove that \(\sigma(n)<e^{\gamma}n\log\log n\) for all integers \(n>5040\).

Let \(n\) be an integer bigger than \(1\), and \(p\) a prime number. Suppose that \(n\) divides \(p-1\) and \(p\) divides \(n^3-1\). Prove that \(4p-3\) is a square number.

Let \(n\) be a composite number. Arrange the factors of \(n\) greater than \(1\) in a circle. When can this be done such that neighbours in the circle are never coprime?

Let \(x\), \(y\), \(z\) and \(w\) be non-negative integers. Find all solutions to \(2^x3^y-5^z7^w=1\).

A robot is programmed to move along the number line starting at \(2\). At each second, the number by which it moves up by must be a factor of the number it’s currently on, but not \(1\). For example, if the robot gets to \(10\), then it can move forward by \(2\), \(5\) or \(10\) steps, going to \(12\), \(15\), or \(20\). What numbers can it land on, and what numbers can’t it land on?

A gang of three jewel thieves has stolen some gold coins and wants to divide them fairly. However, they each have one unusual rule: (i) The first thief wants the number of coins to be divisible by \(3\) so they can split it evenly. (ii) The second thief wants the number of coins to be divisible by \(5\) because she wants to split her share with her four siblings. (iii) The third thief wants the number of coins to be divisible by \(7\) since he wants to split his share amongst seven company stocks.

However, they’re stuck as the number of coins isn’t divisible by any of these numbers. In fact, the number of coins is \(1\) more than a multiple of \(3\), \(3\) more than a multiple of \(5\) and \(5\) more than a multiple of \(7\).
What’s the smallest number of coins they could have? (And if you’re feeling generous, how would you help them out?)