### Route to Root

A sequence of numbers x1, x2, x3, ... starts with x1 = 2, and, if you know any term xn, you can find the next term xn+1 using the formula: xn+1 = (xn + 3/xn)/2 . Calculate the first six terms of this sequence. What do you notice? Calculate a few more terms and find the squares of the terms. Can you prove that the special property you notice about this sequence will apply to all the later terms of the sequence? Write down a formula to give an approximation to the cube root of a number and test it for the cube root of 3 and the cube root of 8. How many terms of the sequence do you have to take before you get the cube root of 8 correct to as many decimal places as your calculator will give? What happens when you try this method for fourth roots or fifth roots etc.?

### Divided Differences

When in 1821 Charles Babbage invented the `Difference Engine' it was intended to take over the work of making mathematical tables by the techniques described in this article.

### Tournament Scheduling

Scheduling games is a little more challenging than one might desire. Here are some tournament formats that sport schedulers use.

# Sorted

### Why do this problem?

This problem introduces sorting algorithms by encouraging students to explore them using a pack of cards. By performing the algorithms for themselves, we hope students will gain a better understanding of the advantages and limitations of each method.

### Possible approach

Each student will need one suit from a pack of cards.
"Shuffle your cards, and then put them in order from Ace to King. Watch how your partner puts their cards in order. Do you both do it the same way? How efficient is your method?"

Either show students each video and invite them to make sense of the algorithm and have a go themselves,
Or hand out this worksheet for them to make sense of the algorithms on paper.

"For each of the algorithms, perform it a few times to get a feel for it. Then choose two algorithms and compare them. Which is the quickest? Why? Can you put your cards into a worst-case scenario for each of the algorithms, to make it take as long as possible?"

Give students time to explore these, together with the questions from the problem which are on the worksheet:
• On average, which algorithm did you find to be quickest?
• What is the 'worst-case scenario' for each algorithm?
• How long would it take in the worst case?
• Which would you choose if you had to keep the cards in a pile rather than laying them out
• Which would you choose if you only had a limited amount of desk space to arrange the cards on?
Then bring the class together to discuss their findings.

### Possible extension

Invite students to use pseudocode, or a programming language if they know one, to express the algorithms.

### Possible support

Introduce the algorithms one at a time, and then make pairwise comparisons between them.