### Roots and Coefficients

If xyz = 1 and x+y+z =1/x + 1/y + 1/z show that at least one of these numbers must be 1. Now for the complexity! When are the other numbers real and when are they complex?

### Target Six

Show that x = 1 is a solution of the equation x^(3/2) - 8x^(-3/2) = 7 and find all other solutions.

### 8 Methods for Three by One

This problem in geometry has been solved in no less than EIGHT ways by a pair of students. How would you solve it? How many of their solutions can you follow? How are they the same or different? Which do you like best?

# Complex Countdown

### Why play this game?

Confidence with basic manipulation of complex numbers is very important in school courses in complex numbers and critical in university mathematics, physical sciences and engineering courses. This game is a fun (to maths folk, at the very least) way to practice these skills in a way which will increase fluency and understanding far more than simply working through a bank of exercises: striving to hit a target requires the players to consider all aspects of the complex numbers on offer; as there is no 'recipe' players must actively engage with the mathematical properties of each complex number leading to a far richer learning experience.

### Possible approach

Complex countdown is initially best played at a slow and thoughtful pace, where students collaborate to try to hit the targets. It is possible that certain games might seem difficult, but (in our experience) it is possible to hit most targets exactly. In the hint to the problem we give a few worked solutions so that you can get an idea for the possibilities. Once students are more fluent you might consider timed competitions.

Extension ideas: The scoring system provides an interesting mathematical question: how can we determine whether two complex numbers are 'close'. Do not be tempted to provide up-front the default answer that any mathematician would come up with (look at the modulus of the difference) - this would be a very good opportunity for students to come to this realisation themselves (perhaps as a homework) that 'closeness' can be summarised by distance in the complex plane. Some different scoring systems might be offered, such as "Look at the difference between the modulus of the target and the modulus of the answer", "Look at the differences between the real and imaginary parts separately and add them" or "Hitting either the real or imaginary part exactly always beats a score which hits neither" or simply voting on the "nicest" solution.

As with any of our countdown games, a nice extension is for students to create their own set of cards and targets. Some motivation might be to deduce a set of impossible targets for a given set of cards. Can the impossibility be proved or strongly justified?

Impossible solutions: In the event that a game seems to provide a set of cards and a target which appears to be impossible you can easily pose the very difficult challenge 'Can you prove that there is no solution to this particular game?'. Take care to record the values of the card before moving on!