Problems
Let \(n\ge3\) be a positive integer. A regular \(n\)-gon is a polygon with \(n\) sides where every side has the same length, and every angle is the same. For example, a regular \(3\)-gon is an equilateral triangle, and a regular \(4\)-gon is a square.
What symmetries does a regular \(n\)-gon have, and how many?
The set of symmetries of an object (e.g. a square) form an object called a group. We can formally define a group \(G\) as follows.
A is a non-empty set \(G\) with a binary operation \(*\) satisfying the following axioms (you can think of them as rules). A binary operation takes two elements of \(G\) and gives another element of \(G\).
Closure: For all \(g\) and \(h\) in \(G\), \(g*h\) is also in \(G\).
Identity: There is an element \(e\) of \(G\) such that \(e*g=g=g*e\) for all \(g\) in \(G\).
Associativity: For all \(g\), \(h\) and \(k\) in \(G\), \((g*h)*k=g*(h*k)\).
Inverses: For every \(g\) in \(G\), there exists a \(g^{-1}\) in \(G\) such that \(g*g^{-1}=e\).
Prove that the symmetries of the ‘reduce-reuse-recycle’ symbol form a group.
Take a pile of cards, Ace to 7 of Diamonds. Consider the following shuffle: simultaneously move the first card to the third position, the third card to the fifth position, the fifth card to the seventh position, and the seventh card to the first position. Also move the second card to the fourth position, the fourth card to the sixth position and the sixth card to the second position.
How many times do you have to do this one specific shuffle in a row to get back to where you started?
How many permutations are there of 4 cards leaving no card in the same position as before?
In the picture below, we have a regular pentagon. The segments \(AB\) and \(CD\) have equal lengths. What is the angle \(\alpha\)?
We have two squares sharing the same centre, each with side length \(2\). Show that the area of overlap is at least \(3\).
A regular tetrahedron is a three dimensional shape with four faces. Each face of a regular tetrahedron is an equilateral triangle. Describe all rotational symmetries of a regular tetrahedron.
Two lines intersect at a point \(P\) at an angle of \(\alpha\). Show that a rotation in the plane around the point \(P\) through an angle \(2\alpha\) can be achieved by a reflection in one of the two lines followed by a reflection in the other line.
Show that a rotation about an axis on a sphere followed by a rotation about a different axis is again a rotation about some axis.
You have in your possession a rotation of the sphere about an axis \(l\) through an angle \(\alpha\neq k\times360^{\circ}\) for any integer \(k\).
Consider the following funny rules. Suppose you have a rotation \(r_1\) through an angle \(\theta\) around an axis \(m\) and a rotation \(r_2\) through an angle \(\theta'\) around an axis \(m'\). You can add to your possession each of the below:
the rotation \(r_1^{-1}\) through \(-\theta\) around \(m\);
the rotation \(r_2r_1\) obtained by doing \(r_1\) and then \(r_2\);
the rotation \(g^{-1}r_1g\), where \(g\) is any rotation of the sphere.
Can you get all the rotations of the sphere?