### 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?

# Shifting Times Tables

##### Stage: 3 Challenge Level:

The numbers in the four times table are
$$4, 8, 12, 16... 36, 40, 44... 100, 104, 108...$$
I could shift the four times table up by 3 and end up with
$$7, 11, 15, 19... 39, 43, 47... 103, 107, 111...$$
What do you notice about the differences between consecutive terms in each sequence?

The interactivity displays five numbers from a shifted times table.
On Levels 1 and 2 it will always be five consecutive terms from the shifted times table.
On Levels 3 and 4 it could be any five terms from the shifted times table.

Use the interactivity to generate some sets of five numbers.
Can you work out the times table and by how much it has been shifted?

Shifting Times Tables

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Always enter the biggest times table it could be.
The shift is always less than the times table.

Table Shifted by

Once you are confident that you can work out the times table and the shift quite easily, here are some questions to consider:

What can you say if the numbers are all odd?
What about if they are all even?
Or a mixture of odd and even?

What can you say if the units digits are all identical?
What if there are only two different units digits?

What can you say if the difference between two numbers is prime?
What can you say if the difference between two numbers is composite (not prime)?

Can you explain how you worked out the table and shift each time, and why your method will always work?