Problems

Fred and George each had a square cake. Both of them made two straight cuts across their cake, from edge to edge. How could it happen that Fred ended up with three pieces, while George ended up with four?

Remember that two shapes are congruent if they are the same in shape and size, even if one is flipped or turned around. For example, here are two congruent shapes:

Cut the following shape into four congruent figures:

Welcome back! The topic of this sheet is: dissections and gluings. This means that we will take shapes, break them apart, and put the pieces back together to form interesting objects. Sometimes, we will also “glue" objects together and see how they can be used to construct other shapes. Let’s see a few simple examples:

Welcome back from the summer holidays! We hope you’re rested and ready for an exciting year of problems. Today we’re going to dive right into our first topic with a fun game called “Lights Out."

Imagine a grid of square buttons. Each square has a light that can be ON or OFF. When you press a square, that light flips, and so do the lights directly above, below, left, and right. In today’s pictures, dark blue means OFF and light blue means ON. If you want to try it at home, search “Lights Out (game)” on Wikipedia for an interactive version.

Before we start, let’s write three key words. After we see the examples, their meaning will be much clearer:

1. A light pattern is just which lights are ON or OFF on the board.

2. A plan is the collection of buttons you choose to press to make a pattern

3. A quiet plan is a plan that is not empty (it has at least one button on it) and when pressed, leaves the board looking exactly like it was before. For example: if the board starts all OFF and a quiet plan is pressed, then the board will still be all OFF.

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:

1. If a number is divisible by \(5\), then its last digit must be \(0\) or \(5\).

2. 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!

When Robinson Crusoe was stranded on his island, he found a goat and decided to keep it. To stop the goat from running away, he tied it to a peg in the ground with a rope. The goat wandered around happily, eating all the grass it could reach until the rope pulled tight. For example, Robinson noticed that with just one peg and one rope of length \(5\) meters, the grazed area was exactly a circle of radius \(5\) meters!

In this sheet we will explore what shapes the goat can graze when Robinson uses pegs, ropes, and even small sliding rings in slightly more complicated positions. In mathematics, the shape made by all points that satisfy a condition is called a locus (plural: loci). In our problems, the locus of the goat is the area it can graze. Before we begin, here are the rules of the game:

We will treat the goat as a single point, and the rope as a fixed length that cannot stretch, so the goat can graze any point it can reach before the rope is tight. A peg is simply a point in the ground that does not move. Let’s see some more interesting examples: