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Pebbles printable sheet

Imagine that you're walking along a sandy beach collecting pebbles. You start off with four pebbles and you place them on the sand as the corners of a square. The area inside your pebbles is just 1 square.

Pebble 1

Pebble 1

Pebble 1

Pebble 1

One square with four circles at the corners.

By adding another 2 pebbles you double the area to 2 squares, like this:

Pebble 1

Pebble 1

Pebble 2

Pebble 1

Pebble 1

Pebble 2

Two squares stuck side by side with circles at each corner.

The rule is that you keep the pebbles that you already have, not moving them to any new positions, and add as few new pebbles as necessary to double the previous area.

So, to continue, we add another three pebbles to get an area of 4:

Pebble 1

Pebble 1

Pebble 2

Pebble 1

Pebble 1

Pebble 2

Pebble 3

Pebble 3

Pebble 3

Four squares stuck together in a square arrangement with circles at each corner.

You could have doubled the area by doing this instead:

Pebble 1

Pebble 1

Pebble 2

Pebble 3

Pebble 3

Pebble 1

Pebble 1

Pebble 2

Pebble 3

Pebble 3

Four squares stuck together in an oblong arrangement with circles at each corner.

But this would not be allowed because we are adding as few pebbles as possible each time.

The sixth arrangement of pebbles would look like this:

Pebble 1

Pebble 1

Pebble 2

Pebble 4

Pebble 4

Pebble 6

Pebble 6

Pebble 6

Pebble 6

Pebble 1

Pebble 1

Pebble 2

Pebble 4

Pebble 4

Pebble 6

Pebble 6

Pebble 6

Pebble 6

Pebble 3

Pebble 3

Pebble 3

Pebble 4

Pebble 4

Pebble 6

Pebble 6

Pebble 6

Pebble 6

Pebble 5

Pebble 5

Pebble 5

Pebble 5

Pebble 5

Pebble 6

Pebble 6

Pebble 6

Pebble 6

Pebble 5

Pebble 5

Pebble 5

Pebble 5

Pebble 5

Pebble 6

Pebble 6

Pebble 6

Pebble 6

32 squares stuck together as four rows of eight with circles at each corner.

Now it's time for you to have a go.

What do you notice about the arrangements? What questions could you ask about them?

You might like to explore:

How many extra pebbles are added each time? This starts off as 2, 3, 6...

How many are there around the edges? This starts off as 4, 6, 8...

How big is the area? This starts off as 1, 2, 4...

How many pebbles are there inside each arrangement? This starts off as 0, 0, 1, 3, 9...

 

Don't forget the all-important question to ask - "I wonder what would happen if I ...?''

 

More Pebbles is a three-dimensional version of this task, which you might like to have a go at afterwards.