### Pebbles

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

### Happy Numbers

Take any whole number between 1 and 999, add the squares of the digits to get a new number. Make some conjectures about what happens in general.

### Intersecting Circles

Three circles have a maximum of six intersections with each other. What is the maximum number of intersections that a hundred circles could have?

# Odds, Evens and More Evens

### Why do this problem?

This problem offers a really straightforward starting point for discussion of sequences that can lead on to generalisations, and perhaps for some students thinking about orders of infinity!

### Possible approach

Begin by displaying the sequences below, or give students the top section of this worksheet, and ask:
"What do you notice?"

$A_0 = 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...$

$A_1 = 2, 6, 10, 14, 18, 22, 26, 30, 34, 38, 42...$

$A_2 = 4, 12, 20, 28, 36, 44, 52, 60...$

$A_3 = 8, 24, 40, 56, 72, 88, 104...$

$A_4 = 16, 48, 80, 112, 144...$

$A_5 = 32, 96, 160...$

$A_6 = 64...$

$A_7 = ...$

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Allow students some time to consider on their own or in pairs, noting down their thoughts before sharing them with the class.

Then pose the following question and allow students to continue working on their own or in pairs:
"Which sequences will contain the number 1000?"

After a few minutes, hand out this scaffolding sheet

"When you've finished or can't make any further progress, look at the worksheet showing three approaches used by students working on this task."

"What might each student do next? Can you take each of their starting ideas and develop it into a solution?"

Here are some prompts that could be offered to students working on each approach if they get stuck:

For Alison's approach:

"What happens to the numbers as you go down the rows?"
"So what happens as you go up the rows?"

For Bernard's approach:

"Which numbers end in a 0 in row $A_2$?"
"Which numbers end in a 0 in row $A_3$?"
"Which of these sequences will hit 1000?"

For Charlie's approach:

"Can you find a similar method to Charlie's to describe the other rows?"
"Which descriptions include 1000?"

Select a few students to report back on how each approach was developed, and invite students to share their own alternative approaches.

In a follow-up lesson, return to the very first question "What do you notice?".
Invite students to phrase their noticings as questions and conjectures.

Here are some key questions that students might suggest, or which could be offered if students' ideas are not forthcoming (these appear at the bottom of the worksheet):

• How many of the numbers from 1 to 63 appear in the first sequence? The second sequence? ...

• Do all positive whole numbers appear in a sequence?
• Do any numbers appear more than once?
• Which sequence will be the longest?

• Given any number, how can you work out in which sequence it belongs?
• How can you describe the $n^{th}$ term in the sequence $A_0$? $A_1$? $A_2$? ... $A_m$?

### Possible extension

Ask students to provide convincing proofs of their answers to two of the questions above:

• Do all positive whole numbers appear in a sequence?
• Do any numbers appear more than once?

### Possible support

Shifting Times Tables provides some preliminary work on sequences that may prepare students for tackling this task.