Problems
Certain geometric objects nicely blend when they happen to be together in a problem, one possible example of such a pair of objects is a circle and an inscribed angle.
We will be using the following statements in the examples and problems:
1. The supplementary angles (angles "hugging" a straight line) add up to \(180^{\circ}\).
2. The sum of all internal angles of a triangle is also \(180^{\circ}\).
3. Two triangles are pronounced "congruent" if ALL their corresponding sides and angles are equal.
We recommend solving the problems in this sheet in the order of appearance, as some problems use statements from previous problems as a step in the solution. Specifically, problem 6.2’s statement is required to solve every other problem that comes after it.
Among all natural numbers we can distinguish prime and composite numbers.
A number is composite if it is a product of two smaller, natural numbers. For example, \(6 = 2\times3\). Otherwise, and if the number is not equal to 1, it is called prime. The number 1 is neither prime nor composite.
The Fundamental Theorem of Arithmetic says that any natural number greater than 1 can be uniquely expressed as a product of prime numbers in non-decreasing order. For example: \[630=2\times3\times3\times5\times7=2\times3^2\times5\times7.\]
Modulo operation: Given any two natural numbers \(a\) and \(b\), called the dividend and the divisor respectively, we can divide \(a\) by \(b\) with the remainder. That is to find such non-negative integer numbers \(c\) and \(d\) (\(d<b\)), called the quotient and the remainder respectively, that \(a=c\times b+d\). For example \(41=2\times15+11\) is the division of 41 by 15 with the remainder 11, and \(5=0\times7+5\) is the division of 5 by 7 with the remainder 5.
If \(a\) is divided by \(b\) with zero remainder (without a remainder) we say that "\(a\) is divisible by \(b\)"\(\;\)or "\(b\) divides \(a\)". From the definition of modulo operation for \(a\) the property to be divisible by \(b\) is equivalent to the existence of non-negative integer \(c\) such that \(a=c\times b\). We denote it by \(b|a\) for "\(b\) divides \(a\)". For example \(7 \mid 105\) and \(9|111111111\) because \(105=15\times7\) and \(111111111=12345679\times9\).
We immediately deduce from the Fundamental Theorem of Arithmetic that if a product of two natural numbers is divisible by a prime number, then one of these numbers is divisible by this prime number.
It is often the case in geometric situations that figures look very similar, but not quite equal. Two polygons on a plane are called similar, if and only if ALL their corresponding angles are equal AND the ratio between ALL the corresponding sides is the same.
The relation between the corresponding sides, in our case it is \(\frac{AB}{IH}\) is called the similarity coefficient between the figures. It is common practice to write vertices of similar figures in the order that respects the similarity.
Today we will have a look on some mathematical games. In all the games today, there are two players, who make moves alternately. There is a certain goal in each of these games, common for both players. The player who achieves it, wins, and the game will end at some point, draws are not allowed. It turns out in such games one of the players has a winning strategy - no matter what the other one does, the player following the strategy will always win. Today we can look into finding who has the winning strategy and what it might be.
One important tool we have to investigate these games are winning and losing positions. If you playing the game can win in one go starting from a certain state of the board, number of tokens on the table, set of cards etc, or whatever describes the current state of the game, it is said you are in a winning position. However, if all the moves you can make give the winning position to your opponent, it is said you are in a losing position. You must do a move that will guarantee you opponent to win - or at least give them a certain opportunity if they are smart enough to take it.
We can go further and say that a winning position is such a position that you can always make a move that will change the state of the game to the losing position for your opponent. Whereas a losing position is such a position that you must make a move that handles the winning position to the other player.
In some simple games the strategy can be guessed without going into details about winning or losing positions. But often you can start from the last stages of the game, find out what positions are immediately winning, then find all the losing positions that lead only there. In this way, working your way from the end of the game towards its beginning, one can characterize all the possible positions as winning or losing. The player that starts the game in a winning position has a winning strategy - if they start the game in a losing position, the strategy belongs to their opponent.
Mathematical Induction is a method to prove statements that are usually true for all natural numbers. The method consists of two steps.
The first step, known as the base case, is to prove the given statement for the first natural number.
The second step, known as the inductive step, is to prove that the true statement for the number \(n\) implies that the statement for \(n+1\) is also true.
To understand how the method of induction works we look at dominoes. Have you ever seen a line of dominoes falling? How does it happen?
To prove that a line of dominoes will all fall when we push the first one, we just have to prove that:
The first domino falls down (base case)
The dominoes are close enough that each domino will knock over the next one when it falls (inductive step).
DRAFT
We need to ensure that there isn’t overlap with the first areas problem sheet.
We can introduce the areas of a new shape, e.g. a trapezium more formally. Maybe an ellipse?
Previously, we have explored how to tile the plane using rectangles, but a much more fascinating topic is plane tilings with more intricate shapes such as quadrilaterals, pentagons, and even more unconventional shapes like chickens.
In this exercise sheet, we define a plane tiling as a covering of the entire plane, without any gaps or overlaps, using identical geometric shapes that can be rotated and symmetrical to each other. Usually, it is sufficient to cover a small portion of the plane with a particular pattern that can be extended to cover the entire plane.