Problems
Michael made a cube with edge \(1\) out of eight bars as in the picture. All \(8\) bars have the same volume. The dimensions of the grey bars are the same as each other. Similarly, the dimensions of the white bars are the same as each other. Find the lengths of the edges of the white bars.
Peter went to the Museum of Modern Art and saw a square painting in a frame of an unusual shape. The frame consisted of \(21\) congruent triangles. Peter was interested in what the angles of these triangles were equal to. Help him find these angles.
Red, blue and green chameleons live on an island. One day \(35\) chameleons stood in a circle. A minute later, they all changed colour at the same time, each changing into the colour of one of their neighbours. A minute later, everyone again changed their colours at the same time into the colour of one of their neighbours. Is it ever possible that each chameleon was each of the colours red, blue and green at some point? For example, it’s allowed for a chameleon to start off blue, turn green after one minute, then turn red after the second minute. It’s not allowed for a chameleon to start off blue, turn green after one minute, but then turn back to blue after the second minute.
Let \(p\) and \(q\) be two prime numbers such that \(q = p + 2\). Prove that \(p^q + q^p\) is divisible by \(p + q\).
Let us define XOR (or addition mod 2). XOR is defined for 0 and 1 only. Here is a table recording the values of XOR:
| XOR | 0 | 1 |
|---|---|---|
| 0 | 0 | 1 |
| 1 | 1 | 0 |
Now we define the important concept of nim-sum. Given two natural numbers \(x\) and \(y\), we first convert them into binary representations and then compute XOR on individual digits. The resulting number, denoted \(x \oplus y\), is the nim-sum of \(x\) and \(y\). Here is an example.
| 1 | 0 | 1 | 1 | 0 | |
| XOR | 0 | 0 | 1 | 0 | 1 |
| 1 | 0 | 0 | 1 | 1 |
This is simply saying \(22 \oplus 5 = 19\). Note that \(22=(10110)_2\) and \(5=(00101)_2\).
Verify \((x \oplus y) \oplus z = x \oplus (y \oplus z)\), so we can speak of \(x \oplus y \oplus z\) with no ambiguity.
Show that \(x \oplus y = 0\) if and only if \(x = y\). Remember that \(x \oplus y\) denotes the nim-sum of \(x\) and \(y\).
Show that \(\text{Nim}(x,y,z)\) is a losing position if and only if \(x \oplus y \oplus z = 0\). Remember that \(x \oplus y\) denotes the nim-sum of \(x\) and \(y\).
Is \(\text{Nim}(7,11,15)\) a winning position or a losing position? If it is a winning position, what is the optimal move?
Show that \(\text{Nim}(x_1,\dots,x_k)\) is an losing position if and only if \(x_1 \oplus \dots \oplus x_k = 0\). \(x \oplus y\) denotes the nim-sum of \(x\) and \(y\).
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.
