Problems

Theorem: If we mark $n$ points on a circle and connect each point to every other point by a straight line, the lines divide the interior of the circle is into is $2n-1$ regions.
"Proof": First, let’s have a look at the smallest natural numbers.

• When $n=1$ there is one region (the whole disc).

• When $n=2$ there are two regions (two half-discs).

• When $n=3$ there are $4$ regions (three lune-like regions and one triangle in the middle).

• When $n=4$ there are $8$ regions, and if you’re still not convinced then try $n=5$ and you’ll find $16$ regions if you count carefully.

Our proof in general will be by induction on $n$. Assuming the theorem is true for $n$ points, consider a circle with $n+1$ points on it. Connecting $n$ of them together in pairs produces $2n-1$ regions in the disc, and then connecting the remaining point to all the others will divide the previous regions into two parts, thereby giving us $2\times (2n-1)=2n$ regions.

In how many ways can you read the word TRAIN from the picture below, starting from T and going either down or right at each step?

There are $33$ cities in the Republic of Farfarawayland. The delegation of senators wants to pick a new capital city. They want this city to be connected by roads to every other city in the Republic. They know for a fact that given any set of $16$ cities, there will always be some city that is connected by roads to all those selected cities. Show that there exists a suitable candidate for the capital.

There are $25$ bugs sitting on the squares of a $5\times 5$ board, $1$ at each square. When I clap my hands, each bug jumps to a square diagonally from where it was before. Show that after I clap my hands, at least $5$ squares will be empty.

A group of schoolboys are going to walk down a narrow path in a straight line, one behind the other. There are $11$ boys, and among them are Will, Tom, and Alex. If exactly two of them walk directly next to each other, they will start arguing. But if the three of them are all next to each other, in any order, the third one will always break the argument of the other two. We don’t want any arguments to persist. How many ways are there to order all $11$ boys?

For a natural number $n$ consider a regular $2n$-gon, with every vertex coloured either blue or green. It is known that the number of blue vertices equals the number of green vertices. Show that the number of main diagonals (passing through the centre of the $2n$-gon) with both ends blue is the same as the number of main diagonals with both ends green.

Jess and Tess are playing a game colouring points on a blank plane. Jess is moving first, she picks a non-colored point on a plane and colours it red. Then Tess makes a move, she picks $2022$ colourless points on the plane and colours them all green. Jess then moves again, and they take turns. Jess wins if she manages to create a red equilateral triangle on the plane, Tess is trying to prevent that from happening. Will Jess always eventually win?

Can you cover a $10\times 10$ board using only $T$-shaped tetrominos?

Can you cover a $10\times 10$ square with $1\times 4$ rectangles?

Two opposite corners were removed from an $8\times 8$ chessboard. Is it possible to cover this chessboard with $1\times 2$ rectangular blocks?