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## 'Complex Countdown' printed from http://nrich.maths.org/

### 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!