Problems

Solve the equation \[\left(x^2-3x+3\right)^2-3\left(x^2-3x+3\right)+3=x\]

Find all integer solutions to \(x^2+y^2-1=4xy\).

Every year the citizens of the planet “Lotsofteeth" enter a contest to see who has the most teeth.
This year the judge notices:

1. Nobody has 0 teeth (everyone has at least 1).

2. There are more people in the contest than the most teeth that any one person has. (For example, if the most teeth anyone has is 27, then there are more than 27 people participating in the contest.)

Must there be two people who have exactly the same number of teeth? Explain why.

Cut a square into three parts and then use these three pieces to form a triangle such that:

1. All its angles are acute (i.e: less than \(90^\circ\)).

2. All its sides are of different lengths.

Seven Smurfs live in seven mushroom houses. There is a tunnel between every pair of houses, so from any house you can walk to any other house. One of the Smurfs, Clumsy, starts walking from his house, but he must not use the same tunnel more than once. He keeps walking until he reaches a house where all the tunnels have already been used. Where will Clumsy’s journey end?

A very important tool in maths is to use symmetries to make problems easier. For today, define a symmetry of a shape as a movement that leaves the shape looking exactly the same as initially. For example, rotating a square by \(90^\circ\) (spinning it by a quarter turn) is a symmetry. Imagine you are playing lights out on a board that has no quiet plans. Explain why if a light pattern has a certain symmetry, then its corresponding plan will also have the same symmetry.

A \(3\times 3\) “Lights Out" board starts with all the lights off. Explain why \(5\) is the smallest number of presses you need to turn the whole board on.

Alice and Jamie each have an identical “Lights Out” board (same size, same rules). Both boards start with all lights off, and on this board size there are no quiet plans. Alice presses a plan \(A\); Jamie presses a different plan \(B\) (not the same set of buttons). Could they end up with exactly the same final pattern of lights?

Friday shows Robinson Crusoe a magic trick:

He asks Robinson to write down any 15 whole numbers of his choice on a piece of paper. Then Friday looks at the list, and is always able to pick two of the numbers so that, when one is subtracted from the other, the result is a multiple of \(13\).

Can you explain why this trick works?

Suppose that \(x_1+y_1\sqrt{d}\) and \(x_2+y_2\sqrt{d}\) give solutions to Pell’s equation \(x^2-dy^2=1\) and \(x_1,x_2,y_1,y_2\geq 0\). Show that the following are equivalent:

1. \(x_1+y_1\sqrt{d} < x_2+y_2\sqrt{d}\),

2. \(x_1<x_2\) and \(y_1<y_2\),

3. \(x_1<x_2\) or \(y_1<y_2\).