You’re probably familiar with 20 cent and 50 cent coins. But different countries have different values for their coins. So what’s the best way to number coins?

#### You will need

- Australian silver coins:
- 5 x 50 cents
- 5 x 20 cents
- 5 x 10 cents
- 5 x 5 cents

- Pen and paper
- Scissors
- Stopwatch

#### Setting up

- You’re going to test the Australian coin system as it was first designed. That means you need one cent and two cent coins. Cut out five small circles of paper and write ‘1’ on them. Then cut out five more circles and write ‘2’ on them. Keep the silver coins and these cut out coins together as the Australian coins.
- To compete with the Australian system, mathematicians have developed a different way of making coins. So cut out 30 circles of paper, and then make five of each of these value coins: 1, 2, 5, 13, 29 and 64. Keep these coins together as the mathematician’s coins.
- To test these two systems, you need some prices. Choose five numbers at random, between 1 and 100. To make sure they are random, you could use the internet – try typing ‘random number between 1 and 100’ into your favourite search engine!

#### The challenge

- Start with the Australian coins. Start the timer, and try to make each of the five prices with your coins. When you finish each price, write down the number of coins you used, and then put the coins back. When you’re finished, stop the timer and write down the time.
- Do a second trial with the mathematician’s coins. Once again, time your attempt, and record the number of coins you used for each price.
- Which trial used the fewest coins? Which was fastest?

#### What’s happening?

It turns out there’s no single best way to number coins. It depends on what you want the coins to do.

Would you like to make any number from 1 to 100 with the fewest number of coins? The obvious way is to make every possible coin – one cent, two cent, three cent, and so on. This is a bad idea, as it would be difficult to fit 100 drawers in a cash register! Mathematician Jeffrey Shallit suggests minting 1, 4, 6, 21, 30 and 37 cent coins. On average, it takes less than three coins to make any price under a dollar. However, it can be tricky to find the best way to make change.

The Australian system is not quite as efficient as Jeffrey’s, but it has another nice feature. If you’re not certain how to make a price, always choose the biggest coin that doesn’t go over. For example, if you need to make 81 cents, start with a 50 cent coin. Then a 20 cent coin brings you up to 70 cents, and a 10 cent brings you to 80 cents. Finally, add a 1 cent coin. This system is called a greedy algorithm, and it’s a quick and easy way to make up money totals. And with the Australian system of coins, the greedy algorithm always finds the fewest number of coins to make a price!

The special prices in this activity, 1, 2, 5, 13, 29 and 64 were also developed by Jeffrey, and they also work well with greedy change making. However, it can be tricky adding the numbers up in your head. So which set of coins do you prefer – the Australian way, or Jeffrey’s?

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