To further one's understanding of mathematics, doing actual problems is a great aid, and can also help encourage students in mathematics. It is quite satisfying to do a problem and to get it right! It may also be beneficial to give students an idea of some other forms of questions than are usually expected, to get experience. But the advantages are not only for the students - for teachers, looking at these sorts of questions may help gear their teaching to aid the transition between high school teaching and university-level teaching. Note that the problem we present to you below is not really based on knowing a lot of obscure facts, but depends on your understanding of the mathematical processes behind the scenes, and perhaps remembering how to solve a simple quadratic equation.

These problems are from Parabola, a mathematics publication from UNSW and the School of Mathematics and Statistics, for high schools.

Q. 1053, Vol 35 No. 2 (answer in No. 3)

A continued fraction is an expression of the form \[ [ n_0; n_1, n_2, n_3,\ldots ] = n_0 + \frac{1}{n_1 + \frac{1}{n_2 + \frac{1}{n_3 + \frac{1}{\cdots}}}}.\]

A useful feature of continued fractions is that they can be used to provide a set of integers to approximate an irrational number. As an example consider \[ \sqrt{14} = 3.7416573867739413856\]

What is the continued fraction for the Golden Section, below? \[ \phi = \frac{1+\sqrt{5}}{2}.\]

Answer: The Golden Section \( \phi \) can be written as the continued fraction \[ [1; 1,1,1,\ldots ] = 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{\cdots}}}}.\]

It is easy to verify this result, but not so easy to derive it. To verify the result, note that if \[ [1; 1,1,1,\ldots ] = 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{\cdots}}}}\] then \[ \phi = 1 + \frac{1}{\phi}\] which is equivalent to the quadratic \[ \phi^2 - \phi -1 = 0.\] The formula for the roots of a quadratic then gives the result \[ \phi = \frac{1 +\sqrt{5}}{2}.\]

Also try the following questions for yourself, or give them to your students, or check out the latest edition of Parabola.

Q. 1054 Vol 35 No. 2 (answer in No. 3)

What is the value of the infinite square root \[ \sqrt{1+ \sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}} \]

Q1060 Vol 35 No. 3 (answer in Vol 36 No. 1)

The colour of each side of a wooden cube is chosen randomly, and independently of all other sides, from one of the three colours red, green, and blue. What is the probability that the cube has at least one pair of opposite faces which have the same colour?