Problems
In the \(6\times7\) large rectangle shown below, how many rectangles are there in total formed by grid lines?
Simplify \(F_0-F_1+F_2-F_3+...-F_{2n-1}+F_{2n}\), where \(n\) is a positive integer.
Is \(\text{Nim}(7,11,15)\) a winning position or a losing position? If it is a winning position, what is the optimal move?
Imagine the Earth is a perfectly round solid ball. Let us drill from the North Pole, London and Beijing simultaneously and meet at the centre of Earth. A ball with three openings is formed. The surface of this ball is shown on the left of the picture below. Describe how to stretch this surface so that it looks like the surface of a donut with two holes as shown on the right.
Show that there are no rational numbers \(a,b\) such that \(a^2 + b^2 = 3\).
There are infinitely many couples at a party. Each pair is separated to form two queues of people, where each person is standing next to their partner. Suppose the queue on the left has the property that every nonempty collection of people has a person (from the collection) standing in front of everyone else from that collection. A jester comes into the room and joins the right queue at the back after the two queues are formed.
Each person in the right queue would like to shake hand with a person in the left queue. However, no two of them would like to shake hand with the same person in the left queue. If \(p\) is standing behind \(q\) in the right queue, \(p\) will only shake hand with someone standing behind \(q\)’s handshake partner. Show that it is impossible to shake hands without leaving out someone from the left queue.
In an office, at various times during the day, the boss gives the secretary a letter to type, each time putting the letter on top of the pile of the secretary’s in-box. When there is time, the secretary takes the top letter off the pile and types it. There are nine letters to be typed during the day, and the boss delivers them in the order \(1,2,3,4,5,6,7,8,9\). While leaving for lunch, the secretary tells a colleague that letter 8 has already been typed, but says nothing else about the morning’s typing. The colleague wonders which of the nine letters remain to be typed after lunch and in what order they will be typed. Base upon the above information, how many such after-lunch orders are possible? (That there are no letters left to be typed is one of the possibilities.)
You meet a group of \(n\) aliens. The first alien asks “is at least one of us a Goop?", the second alien asks “are at least two of us Goops?", the third asks “are at least three of us Goops?" and so on until the final one says “are at least \(n\) of us Goops?".
How many Goops are there?
What’s the sum of the Fibonacci numbers \(F_0+F_1+F_2+...+F_n\)?