### 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

#### If you are a teacher, click here for a version of the problem suitable for classroom use, together with supporting materials. Otherwise, read on...

Here are the first few sequences from a family of related sequences:

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

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

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

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

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

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

$A_6 = 64...$

$A_7 = ...$

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Which sequences will contain the number 1000?

Once you've had a chance to think about it, click below to see how three different students began working on the task.

Alison started by thinking:

"I have noticed that each number is double the number in the row above. I wonder if I can work out what would go in the rows above 1000?"

Bernard started by thinking:

"I have noticed that in $A_1$, the numbers which end in a 0 are 10, 30, 50... If I carry on going up in 20s I won't hit 1000, so I know 1000 isn't in $A_1$."

Charlie started by thinking:

"I have noticed that each number in $A_1$ is 2 more than a multiple of 4. I know 1000 is $250 \times 4$ so it can't be in $A_1$."

Can you take each of their starting ideas and develop them into a solution?

Here are some further questions you might like to consider:

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 $A_0$? $A_1$? $A_2$? ... $A_m$?