Problems

Josie and Kevin are each thinking of a two digit positive integer. Josie’s number is twice as big as Kevin’s. One digit of Kevin’s number is equal to the sum of digits of Josie’s number. The other digit of Kevin’s number is equal to the difference between the digits of Josie’s number. What is the sum of Kevin and Josie’s numbers?

A rectangular sheet of paper is folded so that one corner lies on top of the corner diagonally opposite. The resulting shape is a pentagon whose area is \(20\%\) one-sheet-thick, and \(80\%\) two-sheets-thick. Determine the ratio of the two sides of the original sheet of paper.

A shop sells golf balls, golf clubs and golf hats. Golf balls can be purchased at a rate of \(25\) pennies for two balls. Golf hats cost \(\mathsterling1\) each. Golf clubs cost \(\mathsterling10\) each. At this shop, Ross purchased \(100\) items for a total cost of exactly \(\mathsterling100\) (Ross purchased at least one of each type of item). How many golf hats did Ross purchase?

Two players are playing a game with a heap of \(100\) rocks, and they take turns removing rocks from the heap. The rules are the following: the first player takes one rock, the second can take either one or two rocks, then the first player can take one, two or three rocks, then the second can take \(1\), \(2\), \(3\) or \(4\) rocks from the pile and so on. That is, on each turn, the players have one more option for the number of rocks that they can take. The one who takes the last rock wins. Who has the winning strategy?

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 an alien, who you learn is thinking of a positive integer \(n\). They ask the following three questions.

“Am I the kind who could ask whether \(n\) is divisible by no primes other than \(2\) or \(3\)?"

“Am I the kind who could ask whether the sum of the divisors of \(n\) (including \(1\) and \(n\) themselves) is at least twice \(n\)?"

“Is \(n\) divisible by 3?"

Is this alien a Crick or a Goop?