Problems
Welcome everybody! In today’s session we will be talking about divisibility tricks. Recall that a number \(a\) is divisible by another number \(b\) if \(a\) divided by \(b\) is a whole number. Often, there are quick ways to check divisibility without doing the full division. For example, a number is divisible by \(5\) if and only if its last digit is \(0\) or \(5\). It is important to remember that this phrase “if and only if" actually means two things:
If a number is divisible by \(5\), then its last digit must be \(0\) or \(5\).
If a number’s last digit is \(0\) or \(5\), then the number is divisible by \(5\).
It is useful to think about this in terms of there being two directions. In today’s sheet we will see many more such tricks, and remember: usually you will need to prove both directions!
Long before meeting Snow White, the seven dwarves lived in seven different mines. There is an underground tunnel connecting any two mines. All tunnels were separate, so you could not start in one tunnel and somehow end up in another. Is it possible to walk through every tunnel exactly once without retracing your path?
There is a queue of \(n\) truth tellers and liars. The first person says, “more than half of us are liars". The second person says, “more than a quarter of us are liars". The third person says, “more than an eighth of us are liars", and so on, until the \(n\)th person says, “more than \(\frac{1}{2^n}\) of us are liars". Describe what the number of truth-tellers and liars could be, as well as their placement in the queue. Note that the solutions are not fixed numbers.
Sam was playing on the \(3\times 3\) lights out board, and starting from all lights off, he pressed every single button on the board. What light pattern did he get in the end?
Find a quiet plan for the \(1\times 5\) “Lights Out" board.
There are \(57\) buttons placed evenly around a circle, each with a light underneath it. Pressing a button changes the state of its own light and also the lights under its two neighbouring buttons. Changing state means that a light which is on becomes off, and a light which is off becomes on. If all the lights start off, what is the smallest number of presses that need to be done so that the entire circle lights up?
Lena lays out a big square blanket that is \(1\) meter on each side. She has a pile of small square tiles, and their total area is \(100\,\text{m}^2\). She wants to place all the tiles on the blanket so that none of them overlap and none hang over the edge. Can she do it?
Without using any wolves, show that Robinson’s goat can only graze shapes that are convex (that means, whenever you pick two points inside the shape, the whole line between them also lies inside). But if Robinson is allowed to use as many wolves as he likes, this restriction disappears. Show that in this case, he can make the goat graze in the shape of any polygon at all.
In a field, there are two pegs, \(A\) and \(B\), placed \(15\) metres apart. Each peg has a small ring on top, and a rope can slide freely through these rings. You have one rope and two goats that want to graze the grass, but they will fight each other if they can both reach the same spot.
The rope may be arranged in any way you choose: it could pass through both rings, only one, or neither, depending on your setup. For what lengths of rope can you arrange things so that the goats cannot fight each other?
Fred starts running from point \(A\) and must reach point \(B\). On the way, he has to touch the fence shown as a straight line in the figure. It doesn’t matter where he touches the fence, as long as he does. What is the shortest path he can take?
